GEOMETRY of ENTANGLEMENT and DECOHERENCE in quantum …homepage.univie.ac.at/Reinhold.Bertlmann/pdfs/dipl_diss/Durst... · GEOMETRY of ENTANGLEMENT and DECOHERENCE in quantum systems

GEOMETRY

of

ENTANGLEMENT

and

DECOHERENCE

in quantum systems

Dissertation

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

an der

UNIVERSITAT WIEN

eingereicht von

Magistra Katharina Durstberger

betreut von

Ao. Univ. Prof. Dr. Reinhold A. Bertlmann

Wien, im Oktober 2005

Meiner Familie gewidmet.

Ich mochte mich bei allen bedanken,die zum Entstehen und Gelingen dieser Arbeit beigetragen haben.

III

IV

Contents

Abstract 1

1 Introduction 3

I Basic Concepts 5

2 Entanglement 72.1 QM with density matrices . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 QM for pure states . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 QM for mixed states . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Example: spin-1

2 particle – qubit . . . . . . . . . . . . . . . . 92.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Definition of entanglement . . . . . . . . . . . . . . . . . . . . 102.2.2 Properties of the subsystems . . . . . . . . . . . . . . . . . . 112.2.3 Examples for C2 ⊗ C2 . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Separability criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Schmidt rank criterion . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Peres-Horodecki criterion – positive partial transposition . . . 142.3.3 Reduction criterion . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Examples for C2 ⊗ C2 . . . . . . . . . . . . . . . . . . . . . . 152.3.5 Positive and complete positive maps . . . . . . . . . . . . . . 162.3.6 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . 172.3.7 Connection entanglement witnesses – positive maps . . . . . 18

2.4 Entanglement measures . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . 192.4.2 Von Neumann entropy . . . . . . . . . . . . . . . . . . . . . . 202.4.3 Concurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.4 Entanglement of formation . . . . . . . . . . . . . . . . . . . 222.4.5 Entanglement of distillation . . . . . . . . . . . . . . . . . . . 232.4.6 Relative entropy of entanglement . . . . . . . . . . . . . . . . 232.4.7 Properties and relations . . . . . . . . . . . . . . . . . . . . . 24

2.5 Bell inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.1 Bell-CHSH inequality . . . . . . . . . . . . . . . . . . . . . . 25

V

CONTENTS Contents

3 Decoherence 273.1 Nomenclature and classifications . . . . . . . . . . . . . . . . . . . . 273.2 Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Quantum master equation . . . . . . . . . . . . . . . . . . . . 293.2.2 Microscopic derivation of the master equation . . . . . . . . . 31

3.3 Representations of the dissipator . . . . . . . . . . . . . . . . . . . . 333.4 Quantum operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Operator-sum representation . . . . . . . . . . . . . . . . . . 343.4.2 Correspondence Lindblad operators ↔ Kraus operators . . . 353.4.3 Depolarizing channel . . . . . . . . . . . . . . . . . . . . . . . 363.4.4 Phase damping channel . . . . . . . . . . . . . . . . . . . . . 373.4.5 Amplitude damping channel . . . . . . . . . . . . . . . . . . . 37

4 Geometric Phase 394.1 Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Spin-1

2 particle – qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Experimental approach . . . . . . . . . . . . . . . . . . . . . 424.3.2 Example: qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Neutron-Interferometry 455.1 Properties of neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.1 Particle properties . . . . . . . . . . . . . . . . . . . . . . . . 455.1.2 Wave properties . . . . . . . . . . . . . . . . . . . . . . . . . 465.1.3 Production, moderation, detection . . . . . . . . . . . . . . . 46

5.2 Neutron optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2.1 Various types of interferometers . . . . . . . . . . . . . . . . . 475.2.2 Perfect crystal interferometry . . . . . . . . . . . . . . . . . . 48

5.3 Entanglement in neutron-interferometry . . . . . . . . . . . . . . . . 495.3.1 Definition of entanglement . . . . . . . . . . . . . . . . . . . . 505.3.2 Contextuality . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Neutral K–mesons 536.1 QM description of K–mesons . . . . . . . . . . . . . . . . . . . . . . 53

6.1.1 Quantum operations . . . . . . . . . . . . . . . . . . . . . . . 546.1.2 Strangeness oscillation . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Quasi-spin formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2.1 Single kaons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2.2 Entangled kaons . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3 Unitary time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 57

VI

Contents CONTENTS

7 Spin-Geometry 597.1 Hilbert-Schmidt space . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2 Hilbert-Schmidt space for two qubits . . . . . . . . . . . . . . . . . . 617.3 Spin geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.3.1 Equivalence classes of states . . . . . . . . . . . . . . . . . . . 627.3.2 Singular value decomposition . . . . . . . . . . . . . . . . . . 627.3.3 Geometric picture . . . . . . . . . . . . . . . . . . . . . . . . 63

8 Quaternions and Hopf-fibration 658.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658.1.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.2 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688.3 Quaternions and rotations . . . . . . . . . . . . . . . . . . . . . . . . 69

8.3.1 Rotations in 2 dimensions . . . . . . . . . . . . . . . . . . . . 708.3.2 Rotations in 3 dimensions . . . . . . . . . . . . . . . . . . . . 70

8.4 Quaternionic conjugation map . . . . . . . . . . . . . . . . . . . . . . 728.4.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.4.2 Hopf map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.4.3 Hopf fibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.4.4 Pre-image of the Hopf fibration . . . . . . . . . . . . . . . . . 74

8.5 Stereographic projection . . . . . . . . . . . . . . . . . . . . . . . . . 758.5.1 Simple case S2 −→ R2 . . . . . . . . . . . . . . . . . . . . . . 758.5.2 Generalization Sn −→ Rn . . . . . . . . . . . . . . . . . . . . 77

II Applications 79

9 Berry Phase and Entanglement 819.1 Theoretical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

9.1.1 Berry phase for qubits . . . . . . . . . . . . . . . . . . . . . . 819.1.2 Spin-echo method . . . . . . . . . . . . . . . . . . . . . . . . 829.1.3 Berry phase and entangled qubits . . . . . . . . . . . . . . . . 829.1.4 Joint measurements . . . . . . . . . . . . . . . . . . . . . . . 839.1.5 Bell-CHSH inequality . . . . . . . . . . . . . . . . . . . . . . 849.1.6 Analysis of the S-function . . . . . . . . . . . . . . . . . . . . 84

9.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

10 Decoherence for entangled Kaons 8910.1 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10.1.1 Decoherence model for single kaons . . . . . . . . . . . . . . . 8910.1.2 Decoherence model for entangled kaons . . . . . . . . . . . . 90

10.2 Connection to experiment . . . . . . . . . . . . . . . . . . . . . . . . 9110.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 9110.2.2 Mathematical description of measurements . . . . . . . . . . 9210.2.3 Bounds from experimental data . . . . . . . . . . . . . . . . . 93

VII

CONTENTS Contents

10.3 Connection to phenomenological model . . . . . . . . . . . . . . . . . 9410.3.1 Phenomenological model . . . . . . . . . . . . . . . . . . . . . 9410.3.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

10.4 Connection to quantum information . . . . . . . . . . . . . . . . . . 9510.4.1 Quasi-spin picture . . . . . . . . . . . . . . . . . . . . . . . . 9510.4.2 Mixing and entanglement . . . . . . . . . . . . . . . . . . . . 9610.4.3 Measures of entanglement . . . . . . . . . . . . . . . . . . . . 9710.4.4 Discussion of the results . . . . . . . . . . . . . . . . . . . . . 99

11 Decoherence modes in a two qubit system 10111.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

11.1.1 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.1.2 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10211.1.3 Different bases . . . . . . . . . . . . . . . . . . . . . . . . . . 102

11.2 Mode A: E ⊗ E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10311.3 Mode B: GR⊗ E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11.3.1 Mode B1: R⊗ E . . . . . . . . . . . . . . . . . . . . . . . . . 10511.3.2 Mode B2: IR⊗ E . . . . . . . . . . . . . . . . . . . . . . . . 106

11.4 Mode C: GR⊗GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10711.4.1 Mode C1: R⊗R . . . . . . . . . . . . . . . . . . . . . . . . . 10711.4.2 Mode C2: IR⊗ IR . . . . . . . . . . . . . . . . . . . . . . . . 109

11.5 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11011.5.1 Mode A: E ⊗ E . . . . . . . . . . . . . . . . . . . . . . . . . . 11011.5.2 Mode B: GR⊗ E . . . . . . . . . . . . . . . . . . . . . . . . . 11111.5.3 Mode C: GR⊗GR . . . . . . . . . . . . . . . . . . . . . . . . 112

12 Decoherence in neutron interferometry 11312.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

12.1.1 State preparation and detection . . . . . . . . . . . . . . . . . 11312.1.2 Decoherence via random magnetic fields . . . . . . . . . . . . 11412.1.3 Mode A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.1.4 Mode B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11612.1.5 Mode C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

12.2 Kraus operator decomposition . . . . . . . . . . . . . . . . . . . . . . 11812.2.1 Mode A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11912.2.2 Mode B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

12.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

13 Geometry of decoherence modes 12113.1 Mode A: E ⊗ E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12113.2 Mode B: GR⊗ E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12213.3 Mode C: GR⊗GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12313.4 Kraus operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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Contents CONTENTS

14 Hopf geometry for pure qubit states 12714.1 Hilbert space for one qubit . . . . . . . . . . . . . . . . . . . . . . . 127

14.1.1 State space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.1.2 First Hopf map . . . . . . . . . . . . . . . . . . . . . . . . . . 128

14.2 Visualizing the fibration . . . . . . . . . . . . . . . . . . . . . . . . . 12914.2.1 U(1) fibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 12914.2.2 Stereographic S3 picture . . . . . . . . . . . . . . . . . . . . . 129

14.3 Hilbert space for two qubits . . . . . . . . . . . . . . . . . . . . . . . 13314.3.1 State space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13314.3.2 Quaternionic formulation . . . . . . . . . . . . . . . . . . . . 13414.3.3 Second Hopf map . . . . . . . . . . . . . . . . . . . . . . . . . 13414.3.4 Relation to state representation . . . . . . . . . . . . . . . . . 135

14.4 Hilbert space for three qubits . . . . . . . . . . . . . . . . . . . . . . 13814.4.1 Third Hopf map . . . . . . . . . . . . . . . . . . . . . . . . . 13814.4.2 Families of Hopf maps . . . . . . . . . . . . . . . . . . . . . . 139

Bibliography 140

List of publications 151

Curriculum Vitae 181

IX

X

Abstract

The whole work is settled within the framework of quantum theory in particularwe consider the theory of entanglement and the theory of decoherence. The theoryof entanglement and its protection from environmental decoherence is importantbecause entanglement acts as the resource for quantum information theory. Most ofthe time we restrict our considerations to the two qubit system.From the basic topics we can go either into some mathematical concepts such as spingeometry and Hopf fibration or we turn to the experimental ground, represented byneutron interferometry and K–meson particle physics.The more mathematical concepts aim at visualizing the Hilbert space and the dy-namics generated by decoherence thus providing new insides from other perspectivesinto the field of quantum theory. The abstract mathematical concept of the Hopffibration can be related to the entanglement contained in a system.The experimental part is the anchorage of physical conepts in the real world andtherefore essential. We restrict to two different experimental fields where most effectsof quantum theory can be implemented. In particular the neutrons allow for theconcept of contextuality whereas the K–mesons are really heavy particles which candecay where quantum phenomena can be observed.The above mentioned concepts are introduced and discussed in some detail in thefirst part of the thesis. The second part is devoted to the connections between thesebasic concepts.Thus a rather complex and nested picture settled within quantum theory is drawnon the following pages.Most of the works presented in this thesis have been worked out during the Ph.D.time.

Kurzfassung

Die Dissertation beschaftigt sich mit der Quantentheorie und im Besonderen mitder Theorie der Verschrankung und mit der Theorie der Dekoharenz. Die Theorieder Verschrankung und der Schutz dieser Verschrankung vor Umgebungseinflussenist von großter Wichtigkeit, da Verschrankung als Resource in der Quanteninforma-tionstheorie dient. Die meisten Untersuchungen in der vorliegenden Arbeit werdenim Zwei-Qubit-System durchgefuhrt.Von diesen beiden großen Gebieten kann man nun entweder zu etwas mathematis-chen Konzepten weitergehen, wie z.B. Spin-Geometrie oder Hopf-Faserungen, oderman wendet sich eher den experimentellen Grundlagen zu, die durch Neutronen-Interferometrie und K–Mesonen Physik vertreten sind.Auf der mathematischen Seite versucht man den Hilbertraum und die Dynamik,die durch Dekoharenz verursacht wird, darzustellen, wobei sich neuartige Zugangeund Blickwinkel auf die Quantentheorie ergeben. Es stellt sich heraus, dass dasabstrakte Konzept der Hopf Faserungen eng mit der Verschrankung, die in einemSystem enthalten ist, verknupft ist.Der experimentelle Teil bildet den Anknupfungspunkt der physikalischen Konzepte

1

an die reale Welt und ist damit uberaus wichtig. Es werden zwei unterschiedlicheexperimentelle Arbeitsgebiete betrachtet, in denen die meisten Effekte der Quanten-theorie uberpruft werden konnen. Bei Neutronen finden sich Effekte wie die Kontex-tualitat und bei K–Mesonen, schweren Elementarteilchen mit Zerfallseigenschaften,konnen ebenfalls Quantenphanomene beobachtet werden.Der erste Teil der Arbeit beinhaltet eine Einfuhrung und Diskussion der oben ange-sprochenen Konzepte. Der zweite Teil ist den Verbindungen und Anwendungendieser grundlegenden Bereiche gewidmet.Somit wird in der vorliegenden Arbeit ein complexes und verschachteltes Bild rundum die Quantentheorie gezeichnet.Ein Großteil dieser Arbeiten entstand wahrend der Dissertationszeit.

2

Chapter 1

Introduction

The following graphic may serve as a road map through this thesis.

Spin GeometryChapt.7

Hopf fibrationChapt.8

Theory ofEntanglement

Chapt.2

QUANTUMTHEORY

Theory ofDecoherenceChapts.3, 11

Geometric phaseChapt.4

NeutronInterferometry

Chapt.5

K-MesonSystemChapt.6

Chapt.13Chapt.14

Chapt.9Chapt.12 Chapt.10

Chapt.10

The whole work is settled within the framework of quantum theory in particular weconsider the theory of entanglement and the theory of decoherence. These two big

3

CHAPTER 1. Introduction

blocks form the basis and can be regarded in a way as counterparts of each other ifwe interpret decoherence as the “destruction” of entanglement.Schrodinger already noted that entanglement is at the heart of quantum theory but ittook a long time since the real importance of this concept as a resource for quantuminformation was figured out. One step in this direction was made by Bell who endedthe debate about hidden variables and the completeness of quantum mechanics byexperimental evidence. The theory of entanglement has developed and for higherdimensional system is still revealing new phenomena but in this thesis we restrictmore or less to two qubit systems.Due to the applications of entanglement in quantum information as a resource it isof great interest to keep entanglement as long as possible in your system. The pro-tection of entanglement against environmental effects is only feasible if the so calleddecoherence mechanisms are understood. Thus we are investigating decoherencemodels of open quantum systems which allow for a detailed study of the occurringphenomena.From the basic topics we can go either into some mathematical concepts such as spingeometry and Hopf fibration or we turn to the experimental ground, represented byneutron interferometry and K–meson particle physics.The more mathematical concepts aim at visualizing the Hilbert space and thusproviding new insides from other perspectives into the field of quantum theory. Oneinteresting concept is the spin geometry picture which allows to visualize the statespace for two qubits in a rather appealing and instructive way and to show howdifferent decoherence modes act on the system. Another concepts deals with themore abstract Hopf fibrations which are related to the entanglement contained in asystem.The experimental part is the anchorage of physical conepts in the real world andtherefore essential. We restrict to two different experimental fields where most effectsof quantum theory can be implemented. In particular the neutrons allow for theconcept of contextuality whereas the K–mesons are really heavy particles wherequantum phenomena can be tested.The above mentioned concepts are introduced and discussed in some detail in thefirst part of the thesis. The second part is devoted to applications or more specif-ically to the connections between these basic concepts which are indicated by theconnecting lines in the graphic. Most of these connections have been worked outduring this thesis by the author and her coworkers and are published or will bepublished in international journals (see the list of publications at the end of thethesis).Thus a rather nested picture settled within quantum theory is drawn on the followingpages.

4

Part I

Basic Concepts

5

Chapter 2

Entanglement

Entanglement is one of the properties of quantum mechanics (QM) which causedEinstein and others to dislike the theory. In 1935 Schrodinger published a seminalpaper about the present situation of quantum mechanics [126].Therein he coins the term ”entanglement“ (german: “Verschrankung”) to describethe peculiar connection between quantum systems.

When two systems, of which we know the states by their respectiverepresentatives, enter into temporary physical interaction due to knownforces between them, and when after a time of mutual influence thesystems separate again, then they can no longer be described in thesame way as before, viz. by endowing each of them with a representativeof its own. I would not call that one but rather the characteristic traitof quantum mechanics, the one that enforces its entire departure fromclassical lines of thought. By the interaction the two representatives (thequantum states) have become entangled.

Another way of expressing the peculiar situation is: the best possibleknowledge of a whole does not necessarily include the best possibleknowledge of all its parts, even though they may be entirely separateand therefore virtually capable of being “best possibly known”, i.e., ofpossessing, each of them, a representative of its own.1

In the same year Einstein, Podolsky, and Rosen (EPR) [56] formulated a para-dox demonstrating that due to entanglement QM gets an incomplete and non-localtheory. The study of entanglement was ignored for thirty years until Bell [15] re-considered and extended the EPR argument in 1964. He showed that the statisticalcorrelations between the measurement outcomes of suitably chosen different quanti-ties on the two systems are inconsistent with an inequality derived from Einstein’sseparability and locality assumptions. The so-called Bell inequalities (see Sect.2.5)are experimentally testable and show clear significance that QM is complete andnon-local.

1Quoted from Ref.[127].

7

CHAPTER 2. Entanglement 2.1. QM with density matrices

Bell’s investigation generated an ongoing debate on the foundations of quantum me-chanics. But it was not until the 1980s that physicists, computer scientists, and cryp-tographers began to regard the non-local correlations of entangled quantum statesas a new kind of resource that could be exploited, rather than an embarrassmentto be explained away. The fields of quantum information, quantum communicationand quantum computation were born (for an overview see e.g. Refs.[4, 36, 39, 110]).

In the following sections we give a short introduction to the field of quantum in-formation. In Sect.2.2 we define entanglement in a mathematically sound way andexplore some separability criteria (Sect.2.3) which tell you immediately if a state isentangled or not. We discuss some ways to quantify entanglement (Sect.2.4) and inSect.2.5 investigate the basics of Bell inequalities.

2.1 QM with density matrices

In the following section we will just briefly recall the basic conception of pure andmixed states and the basic properties of density operators. For a more detaileddescription see for instance the books of Ballentine [13], Nielsen and Chuang [110]and Peres [116].

2.1.1 QM for pure states

In QM states are described by state vectors |ψ〉 which are elements of the Hilbertspace H. State vectors are normalized 〈ψ|ψ〉 = 1.The time evolution is given by the time dependent Schrodinger equation withthe Hamiltonian H(t)

∂

∂t|ψ(t)〉 = −iH(t)|ψ(t)〉 , (2.1.1)

where we have set � = 1.The expectation value of an observable A ∈ B(H) with respect to the state |ψ〉 iscalculated by

〈A〉 = 〈A〉ψ = 〈ψ|A|ψ〉 , (2.1.2)

where B(H) denotes the Algebra of bounded operators on H which forms a Hilbertspace and is called Hilbert-Schmidt space (see Sect.7.1).The above description in terms of state vectors is only valid for pure states.

2.1.2 QM for mixed states

Mixed states arise if a system has the possibility to be in one state out a whole bunchof states |ψi〉 each with respective probabilities pi. It is not possible to describe thisensemble {pi, |ψi〉} of pure states – which is also called a mixed state – by a uniquestate vector. We have to use the concept of density operators.Mixed states are described by the so called density operator ρ ∈ B(H). For a purestate the density operator is given by the projection operator ρ = |ψ〉〈ψ|. General

8

2.1. QM with density matrices CHAPTER 2. Entanglement

states are convex combinations of pure state projectors and can be written as

ρ =∑

i

pi|ψi〉〈ψi| =∑

i

pi ρi with∑

i

pi = 1 and pi ≥ 0 . (2.1.3)

Note, that the same ρ can be described by several ensembles {pi, ρi} which are notdistinguishable. Physics depends only on the density operator ρ.Density operators2 satisfy the following properties:

• ρ† = ρ hermiticity

• ρ ≥ 0 positivity

• Trρ = 1 normalization

• positive and real eigenvalues

The time evolution of mixed states is given by the Liouville-von Neumann equa-tion

∂

∂tρ(t) = −i[H(t), ρ(t)] , (2.1.4)

where [·, ·] denotes the commutator.The expectation value of an observable A in a state ρ is defined by

〈A〉 = 〈A〉ρ = Tr ρA . (2.1.5)

We can distinguish between pure and mixed states via the trace-criterion:

Trρ2 = 1 the state is pure,

Trρ2 < 1 the state is mixed.(2.1.6)

This condition defines a measure of purity or mixedness of a state δ = Trρ2.For pure states δ = 1 and for maximally mixed states δ = 1

d where d denotes thedimension of the system.

2.1.3 Example: spin-12

particle – qubit

A spin-12 particle represents the simplest example of a quantum system with a two-

dimensional complex Hilbert space H = C2. We also speak of a qubit.A pure state of a qubit is given by |ψ〉 = α|0〉+β|1〉, where the complex amplitudessatisfy |α|2 + |β|2 = 1. The states |0〉 and |1〉 form an orthonormal basis of C2

which is called the computational or standard basis. Due to the normalizationcondition the state vector can be rewritten in another parametrization

|ψ〉 = α|0〉+ β|1〉 = cosθ

2|0〉+ sin

θ

2eiφ|1〉 , (2.1.7)

where the parameters θ and φ define a point on the unit sphere in 3 dimensionalspace – the Bloch sphere.

2The word density matrix is used equivalently.

9

CHAPTER 2. Entanglement 2.2. Entanglement

A general (pure and mixed) qubit density matrix can be written in terms of Paulimatrices 3

ρ =12(� + r · σ) =

12

(1 + r3 r1 − ir2

r1 + ir2 1− r3

), (2.1.8)

where the vector r = (r1, r2, r3)T , ri ∈ R is called the Bloch vector. The coordi-nates of the Bloch vector are obtained as the expectation values of the correspondingPauli matrices r = 〈σ〉ρ = Tr(ρσ). Due to the positivity of the density matrix we getthe condition |r|2 ≤ 1 for the Bloch vector. It turns out that pure states, satisfying|r|2 = 1, are sitting on the surface (Bloch sphere) whereas mixed states, satisfying|r|2 < 1, are located inside the sphere (Bloch ball).For pure states we can establish the relation between the Bloch vector r and theparameters of Eq.(2.1.7)

r1 = 〈σ1〉ψ = 2e(αβ) = αβ + αβ = sin θ cos φ

r2 = 〈σ2〉ψ = 2�m(αβ) = i(αβ − αβ) = sin θ sinφ

r3 = 〈σ3〉ψ = |α|2 − |β|2 = cos θ ,

(2.1.9)

which gives in matrix form

ρ =(|α|2 αβαβ |β|2

)=(

cos2 θ2

12 sin θ e−iφ

12 sin θ eiφ sin2 θ

2

). (2.1.10)

2.2 Entanglement

2.2.1 Definition of entanglement

It turns out that systems consisting of two or more parties reveal a new featurecalled entanglement. Due to Schrodinger [126] entanglement is the essence of QM.In the following treatment we will restrict ourselves to bipartite systems.The Hilbertspace H can be written as the tensor product of the Hilbertspaces of thetwo subsystems H = H(1) ⊗ H(2). As it turns out the density operator can not bewritten in the same product form for all states. We can formalize this observationby the following definition.

Definition 1 (Separability). A state ρ is called separable if it can be written as

a convex combination of product states

ρ =∑

j

pj ρ(1)j ⊗ ρ

(2)j , (2.2.1)

with positive weights pj ≥ 0 which sum up to one∑

j pj = 1. The convex set of

separable states is denoted by S.

3The Pauli matrices are given by σ1 =

(0 1

1 0

)σ2 =

(0 −i

i 0

)σ3 =

(1 0

0 −1

).

10

2.2. Entanglement CHAPTER 2. Entanglement

Definition 2 (Entanglement). A state ρ is called entangled if it is not separable.

The set of entangled states is denoted by SC = B(H) \ S.

This definition of entanglement is due to Werner [149] who called states satisfyingEq.(2.2.1) classically correlated states. A separable state can be produced by localpreparation of the two parties, so to say in a “classical” way, whereas entanglementis a global property of the whole system.Note that the property of being entangled is not changed by local unitary operations4

of the following form

ρ→ ρ′ = (U1 ⊗ U2) ρ (U †1 ⊗ U †

2) , (2.2.2)

where Ui ∈ B(Hi). Therefore we call ρ and ρ′ equivalent.

2.2.2 Properties of the subsystems

Suppose we have a general bipartite state in the common state ρ ∈ B(H). We areonly interested in the states of a subsystems ρ1 and ρ2, respectively. Then we haveto calculate the reduced density matrix by tracing over the other subsystem whichwe write as

ρ1 = Tr2ρ ρ2 = Tr1ρ . (2.2.3)

We assume in each subspace an orthonormal basis {|ei〉} and {|fi〉}, respectively.Then the partial trace is defined

ρ1 = Tr2ρ =∑

i

〈fi|ρ|fi〉 ρ2 = Tr1ρ =∑

i

〈ei|ρ|ei〉 . (2.2.4)

and represents a generalization of the trace operation. Whereas the trace is a scalarvalued function on operators, the partial trace is an operator-valued function. Wecan express this in a more mathematical way by the unique mapping

Tr1 : B(H) = B(H1)⊗ B(H2)→ B(H2)ρ �→ ρ2 ,

(2.2.5)

such that the following properties are satisfied

Tr1(�H) = dimH1 �H2

Tr1(σ(τ ⊗ �H2)) = Tr1((τ ⊗ �H2)σ) σ ∈ B(H) , τ ∈ B(H1) .(2.2.6)

Even though an original bipartite state is pure, that means we know everythingabout it, after tracing away one subsystem we end up with a mixed state, where ourknowledge is limited. It is also possible to construct a pure state out of a mixed ina higher dimensional state space. This procedure is called purification, see for in-stance Ref.[110]. The property that the reduced density matrix of a pure maximallyentangled state gives a mixture is one of the characteristics of entanglement.

4A unitary operation is given by an operator U satisfying U†U = UU† = �.

11

CHAPTER 2. Entanglement 2.2. Entanglement

2.2.3 Examples for C2 ⊗ C2

Let us consider two spin-12 particles where the Hilbert space is given by C2⊗C2. We

write down the examples in the so-called computational basis {|00〉, |01〉, |10〉, |11〉},which corresponds for instance to eigenstates of the σz ⊗ σz operator and forms thebasis for the matrix representation.Pure states (δ = 1) are for example

|ψ〉 = |00〉 = |0〉(1) ⊗ |0〉(2) , separable state,

|Φ±〉 = 1√2

(|00〉 ± |11〉

), |Ψ±〉 = 1√

2

(|01〉 ± |10〉

), entangled states.

The states |Φ±〉 and |Ψ±〉 are called the Bell-states5. They are maximally entangledand form a basis of C2⊗C2. The density matrices for these pure states are given by

ρ = |ψ〉〈ψ| =

⎛⎜⎜⎝1 0 0 00 0 0 00 0 0 00 0 0 0

⎞⎟⎟⎠ ,

ω± = |Φ±〉〈Φ±| = 12

⎛⎜⎜⎝1 0 0 ±10 0 0 00 0 0 0±1 0 0 1

⎞⎟⎟⎠ , ρ± = |Ψ±〉〈Ψ±| = 12

⎛⎜⎜⎝0 0 0 00 1 ±1 00 ±1 1 00 0 0 0

⎞⎟⎟⎠ .

Important examples for mixed states are on the one hand the identity ρ = 14�4 =

14�2 ⊗ �2 which represents a maximally mixed (δ = 1

4) but separable state and onthe other hand the so called Werner-state (δ = 1+3γ2

4 ) [149]

ρW =1− γ

4�2 ⊗ �2 + γ|Φ+〉〈Φ+| = 1

4

⎛⎜⎜⎝1 + γ 0 0 2γ

0 1− γ 0 00 0 1− γ 02γ 0 0 1 + γ

⎞⎟⎟⎠ . (2.2.7)

In general mixtures of the identity and of one of the Bell-states are called Werner-states. We can scale continuously between separability and entanglement with theparameter γ. The state is separable for γ ≤ 1

3 otherwise it is entangled. We see thatas long as a state is “close enough” to the identity it is separable.Bell-diagonal states, which are diagonal states in the Bell basis

ρBD =4∑

i=1

νi|Ψi〉〈Ψi| =12

⎛⎜⎜⎝ν1 + ν2 0 0 ν1 − ν2

0 ν3 + ν4 ν3 − ν4 00 ν3 − ν4 ν3 + ν4 0

ν1 − ν2 0 0 ν1 + ν2

⎞⎟⎟⎠ , (2.2.8)

with∑

i νi = 1 are also mixed states (δ = ν21 + ν2

2 + ν23 + ν2

4). They are entangled aslong as νi > 1

2 for the largest weight.

5Sometimes we make the identification |Φ±〉 ≡ |Ψ1,2〉 and |Ψ±〉 ≡ |Ψ3,4〉

12

2.3. Separability criteria CHAPTER 2. Entanglement

Bell-diagonal states are generalizations of Werner-states. If we set ν1 = 1+3γ4 and

ν2 = ν3 = ν4 = 1−γ4 in Eq.(2.2.8) we reach exactly the Werner-state given in

Eq.(2.2.7).

2.3 Separability criteria

The definition of separability (2.2.1) is mathematically exact but in practice it isdifficult to work with. It is not easy to say whether a state is separable or notbecause there are infinitely many possibilities to decompose a density matrix andyou have to check all of them. If there is one decomposition of the form (2.2.1)then the state is called separable regardless of the fact that there may be otherdecompositions which are not of that form.Therefore people tried to look for more operational criteria to test separability (orentanglement) that tell us immediately if a state is entangled or not. As it turnsout we can find such criteria, which are differently powerful.First we will discuss the case of pure states, where only one criterion is applicable: theSchmidt rank criterion. In the case of mixed states we know several criteria: Peres-Horodecki criterion (positive partial transposition), reduction criterion, completepositive maps, entanglement witness.

2.3.1 Schmidt rank criterion

We work in the bipartite Hilbertspace H = H(1) ⊗ H(2) and fix some arbitraryorthonormal bases {φk}n1 ⊂ H(1) and {ψl}m1 ⊂ H(2). Then we can expand anarbitrary pure state |Ψ〉 in the corresponding product basis {|φk〉⊗|ψl〉}n·m1 ⊂ H(1)⊗H(2) of the composite Hilbertspace in the following way

|Ψ〉 =∑k,l

akl |φk〉 ⊗ |ψl〉 , (2.3.1)

where the coefficients are given by akl = 〈φk| ⊗ 〈ψl|Ψ〉. This general decomposition(in this case via a double sum) works also for more than two tensor factors.In the case of a bipartite tensor product we can expand |Ψ〉 in a more economic wayby a single sum, which is called the Schmidt decomposition. We use (possiblyincomplete) orthonormal systems {|φk〉}n≤n

1 ⊂ H(1) and {|ψl〉}m≤m1 ⊂ H(2) instead

of the previously used product basis. Then we get the following decomposition

|Ψ〉 =r∑

i=1

√λi |φi〉 ⊗ |ψi〉 . (2.3.2)

The coefficients are called Schmidt coefficients and are positive√

λi ≥ 0 and sumup to unity

∑i λi = 1. r ≤ min(n, m) is called the Schmidt rank of the decompo-

sition.It turns out that the coefficients λi are the (identical) eigenvalues of the reduceddensity matrices of the system which are diagonal matrices in the Schmidt basis

ρ1 =∑

i

λi|φi〉〈φi| , ρ2 =∑

i

λi|ψi〉〈ψi| . (2.3.3)

13

CHAPTER 2. Entanglement 2.3. Separability criteria

The Schmidt decomposition is calculated via the eigensystem of the reduced densitymatrices of the system. The Schmidt rank and the Schmidt coefficients are preservedunder unitary transformations of the system. The expansion is uniquely determined(except when the reduced density matrices are degenerated) without any furtherconditions but only for two subspaces.Eq.(2.3.1) and Eq.(2.3.2) are related by unitary transformations |φi〉 =

∑k Uik|φk〉

and |ψi〉 =∑

l Vil|ψl〉 [118]. Up to now people have found extensions of the Schmidtdecomposition for more than two subspaces, see e.g. [117].The Schmidt decomposition provides us with a very powerful criterion for pure statesto decide wether a state is separable or entangled.

Theorem 1 (Schmidt rank criterion). A pure state is separable iff the Schmidt

rank r of the Schmidt decomposition is equal to 1.

That means if the Schmidt rank r = 1 we have a separable state and the reduceddensity matrix is pure. If the Schmidt rank r > 1 then the state is entangled andthe reduced density matrix is mixed.

2.3.2 Peres-Horodecki criterion – positive partial transposition

In 1996 the first operational criterion for mixed states was found by Peres and theHorodeckis [118, 80].The transpose of a state is defined with respect to a certain basis. So we need thematrix elements of the state ρ in the product basis

ρkα,lβ = 〈k| ⊗ 〈α| ρ |l〉 ⊗ |β〉 , (2.3.4)

where the latin indices refer to the first subsystem and the greek indices refer to thesecond subsystem. Partial transposition with respect to the first or second subsystemis now defined in the following way

(ρT1)kα,lβ = ρlα,kβ , (ρT2)kα,lβ = ρkβ,lα . (2.3.5)

The actual form of the partially transposed states depends on the choice of basisbut its eigenvalues do not. They are invariant under basis transformations. Thisleads us to the following theorem.

Theorem 2 (ppt-criterion). The partial transpose of a separable state with respect

to any subsystem is positive.

That means the partially transposed state has only positive or zero eigenvalues. Itwas shown for a bipartite system that the converse is only true for low dimensions(2× 2 and 2× 3). In this case the positivity of the partial transpose is a necessaryand sufficient criterion for separability. In higher dimensions it is only necessary.Therefore we are motivated to use the following nomenclature.

Definition 3. A state ρ is called a ppt-state (positive partial transpose) if it fulfills

ρT1,2 ≥ 0 and npt-state (negative partial transpose) otherwise.

14


In lower dimensions all ppt-states are separable, but in higher dimensions thereare also ppt-entangled states, which are also called bound entangled states [82]because their entanglement does not seem to be “useful”, that means it can not bedistilled into a maximally entangled state (see Sect.2.4.5).In Fig.2.1 the situation is illustrated for 2 × 2 and 2 × 3 systems and for higherdimensional systems.

npt, entangled

ppt, separable

(a)

npt, entangled

ppt, entangled

ppt, separable

(b)

Figure 2.1: A schematical picture of ppt- and npt-states for (a) 2 × 2 and 2 × 3 systems

and (b) higher dimensional systems.

2.3.3 Reduction criterion

The reduction criterion was found by the Horodeckis [79] in 1999. It is anotheroperational criterion.

Theorem 3 (Reduction criterion). A separable state ρ must satisfy the following

inequalities

ρ1 ⊗ �− ρ ≥ 0 and �⊗ ρ2 − ρ ≥ 0 , (2.3.6)

where ρ1,2 = Tr2,1ρ denotes the reduced density matrix of the system.

This criterion is necessary and sufficient only for 2×2 and 2×3 dimensions where thecriterion is equivalent to the ppt-criterion. In higher dimensions it is only a neces-sary condition and the partial transposition criterion is stronger than the reductioncriterion.

2.3.4 Examples for C2 ⊗ C2

We consider the examples of a Werner state and a Bell-diagonal state

ρW =14

⎛⎜⎜⎜⎜⎝1 + γ 0 0 2γ

0 1− γ 0 0

0 0 1− γ 0

2γ 0 0 1 + γ

⎞⎟⎟⎟⎟⎠ , ρBD =12

⎛⎜⎜⎜⎜⎝ν1 + ν2 0 0 ν1 − ν2

0 ν3 + ν4 ν3 − ν4 0

0 ν3 − ν4 ν3 + ν4 0

ν1 − ν2 0 0 ν1 + ν2

⎞⎟⎟⎟⎟⎠ ,

15


and test via the ppt-criterion and the reduction criterion whether they are entangledor not (compare Section 2.2.3).The partial transpose of the state ρW has the eigenvalues 1−3γ

4 , 1+γ4 , 1+γ

4 , 1+γ4 . The

first one is positive for γ ≤ 13 where the state is separable, and negative for γ > 1

3where the states is entangled.For the state ρBD the partial transpose has the eigenvalues 1−2νi. They are positivefor νi ≤ 1

2 where the state is separable and negative for νi > 12 where the state is

entangled. But due to the condition∑

i νi = 1 only one eigenvalue can turn negative.The same statements hold for the reduction criterion.

2.3.5 Positive and complete positive maps

Now we will consider a rather powerful separability criterion which was presentedby the Horodeckis [80] in 1996.

Definition 4 (Positivity – complete positivity). Consider a linear map between

two Hilbert-Schmidt spaces L : B(H1)→ B(H2).

• The map L is positive if L(A) ≥ 0 for all operators A ≥ 0 and A ∈ B(H1).

(Positive operators are mapped to positive operators.)

• The map L is complete positive (cp) if the induced map

L ⊗ �n : B(H1)⊗M→ B(H2)⊗M ≥ 0

is positive for all n ∈ N and B(H1) � A ≥ 0. Here �n denotes the identity on

the space of matrices M with dimension n.

(The map stays positive for all possible extensions.)

• The map L is unital if L(�) = � holds.

• The map L is trace preserving if TrL(A) = TrA for any A ≥ 0 holds.

Complete positivity is a much stronger condition for a map than positivity alone.The most general physical process a system can undergo is described by cp maps[79, 93]. That means cp maps are the physical relevant maps. It turns out thatfor the detection of separability and entanglement positive maps which are not cpmaps are useful. Figure 2.2 illustrates this connection between positive maps andcp maps.Consider a positive but not necessarily cp map L. For a product state ρ1 ⊗ ρ2 itis easy to see that the extension (� ⊗ L)(ρ1 ⊗ ρ2) = ρ1 ⊗ L(ρ2) ≥ 0. Therefore(� ⊗ L)ρ ≥ 0 for positive L is a necessary condition for ρ to be separable. In 1996the Horodeckis [80] showed that the condition is also sufficient.

Theorem 4 (Positivity criterion). A state ρ is separable if for any positive map

L ≥ 0 the induced map (�⊗ L)ρ ≥ 0 is positive.

16


positive maps

entanglement detection

cp-maps

physical

Figure 2.2: Illustration of the relation between positive maps and cp maps.

We see that only those maps L are important which are positive but not cp becausefor cp maps the above statement is trivially fulfilled.For the positivity criterion one has to check all possible positive but not cp mapsL. In low dimensions (e.g. L : M2 → M2 and L : M3 → M2) these maps L canbe written in the so-called decomposable form [45, 153] L = L1

cp + L2cp ◦ T , where

Licp are some cp maps and T denotes transposition. Therefore it is sufficient in low

dimensions to check separability via the transposition map.In higher dimensions not all positive maps are decomposable which makes the situ-ation much more complicated and the problem is still unsolved.

2.3.6 Entanglement witnesses

Entanglement witnesses are observables (operators) which allow us to detect entan-glement. We have the following theorem of the Horodeckis [80].

Theorem 5 (Entanglement witness). A density matrix ρ is entangled if and

only if there exists a hermitian operator W ∈ B(H(1) ⊗H(2)) with the properties

• Tr(Wρ) < 0

• Tr(Wρsep) ≥ 0 for all separable states ρsep

• TrW = 1 (optional)

The operator W is called an entanglement witness.

This is a necessary and sufficient condition for separability. A state ρ is entangledif and only if there exists an entanglement witness that detects it. The theoremfollows from the Hahn-Banach theorem in convex analysis which states that betweenthe compact and convex set of separable states and a point (the entangled densitymatrix ρ) there exists a separating hyperplane. The hyperplane is characterized bythe vector W that is orthogonal to it and can be determined by the set of densitymatrices τ satisfying Tr(Wτ) = 0.Entanglement witnesses do not really solve the problem of separability because wehave to construct all possible entanglement witnesses and check wether ρ is entangled

17


or not. Another problem is to find the “best” entanglement witness that means thehyperplane which is tangential to the set of separable states. Therefore we have tocompare certain entanglement witness [98]. First we define DW = {ρ ≥ 0|Tr(Wρ) <0}, the set of all by W “detected” states.

Definition 5. The entanglement witness W1 is finer than W2 if DW2 ⊆ DW1.

An entanglement witness is optimal if there exists no other entanglement witness

which is finer.

This means a finer entanglement witness can detect more states and an optimal onedetects all states that are possible.Two examples for entanglement witnesses are on the one hand the famous Bell-entanglement witness studied by Terhal [133, 134]

WBell = 2�− B , (2.3.7)

with the Bell operator B = a · σ ⊗ (b + b′) · σ − a′ · σ ⊗ (b− b′) · σ and on the otherhand the witness introduced by Bertlmann, Narnhofer and Thirring [35]

WBNT = σ ⊗ σ . (2.3.8)

This kind of entanglement witness is used in Ref.[30] to investigate the separabilityproblem for isotropic states for qubits and qutrits.The Schmidt number which tells you how many degrees of freedom of a bipartitesystem are entangled can also be considered as an entanglement witness [124, 135].

2.3.7 Connection entanglement witnesses – positive maps

There exists a direct relation between entanglement witnesses and positive linearmaps because there exits an isomorphism between positive maps and operators whichare positive on product states [88]. Suppose we have a maximally entangled state inH(1) ⊗H(2), for example the |Ψ+〉 state, and a linear map L : B(H(1)) → B(H(2)).We can construct an Operator W in the following way

W = (�⊗ L)(|Ψ+〉〈Ψ+|) . (2.3.9)

This operator serves as an entanglement witness if and only if the linear map L isa positive but not cp map. On the other hand each entanglement witness defines apositive but not cp map L : B(H(1))→ B(H(2)) via

L(ρ1) = Tr1(WABρT1 ) , (2.3.10)

where ρ1 ∈ H(1) and WAB ∈ B(H(1) ⊗H(2)) and T denotes transposition.

18

2.4. Entanglement measures CHAPTER 2. Entanglement

2.4 Entanglement measures

It is useful to quantify the amount of entanglement contained in a certain state.Therefore we have to define an appropriate entanglement measure. As it turnsout this task is not that easy and up to now there exist many different kinds ofentanglement measures serving for different purposes. The question if there exists aunique measure has not been solved yet.For pure states the problem is already solved because the von Neumann entropycan be used as an entanglement measure (see Sect.2.4.2). But for mixed states thesituation is much more complicated. There is no “unique” measure to characterizeentangled states but all entanglement measures should coincide on pure bipartitestates and be equal to the von Neumann entropy of the reduced density matrix.This is stated by the so called uniqueness theorem [52]. In this section we wantto discuss some entanglement measures for mixed states, such as the concurrence(Sect.2.4.3), the entanglement of formation (Sect.2.4.4), the entanglement of distil-lation (Sect.2.4.5) and the relative entropy of entanglement (Sect.2.4.6).In the following section we investigate the properties a “good” entanglement measureshould satisfy.

2.4.1 General properties

There are several properties which an entanglement measure should fulfill (see e.g.[83, 91]). Some of them are quite essential, others can be handled rather freely.Until now it is not clear which candidate (see following section) is the right or thebest one which depends severely on the situation.We assume a bipartite system with Hilbert space H = H(1) ⊗H(2) and dimH(1) =dimH(2) = d.

Axiom 1. An entanglement measure is a function E which assigns to each state ρ

of a finite dimensional bipartite system a positive real number E(ρ) ∈ R+.

Axiom 2 (Normalization). The entanglement measure E vanishes for separable

states E(ρsep) = 0 for ρsep ∈ S and takes its maximum on maximally entangled

states E(ρME) = log2(d).

In section 2.2.3 we have introduced the maximally entangled Bell-states in 2 × 2dimensions. The state |Φ〉 = 1√

d

∑di=1|i, i〉 represents a maximally entangled pure

state for a d × d dimensional system. The reduced density matrices for maximallyentangled states are maximally mixed Tr1ρME = Tr2ρME = 1

d� (proportional to theidentity).

Axiom 3 (LOCC monotonicity). E can not increase under LOCC6. That means

E(T (ρ)) ≤ E(ρ) for all states ρ and all LOCC channels T .6LOCC=local operations and classical communication, such as teleportation schemes and distil-

lation protocols.

19

CHAPTER 2. Entanglement 2.4. Entanglement measures

Local unitary operations are a special case of LOCC. Therefore the next axiom is aweakened version of the previous one.

Axiom 4 (Local unitary invariance). E is invariant under local unitary opera-

tions U, V which means E(U ⊗ V ρ U † ⊗ V †) = E(ρ) for all states ρ.

Entanglement can not be created by mixing two states.

Axiom 5 (Convexity). E is a convex function that means for two states ρ and σ

the equation E(λρ + (1− λ)σ ≤ λE(ρ) + (1− λ)E(σ) holds for 0 ≤ λ ≤ 1.

The next axiom considers that if we perturb the state a little bit the change of theentanglement measure should be small.

Axiom 6 (Continuity). In the limit of vanishing distance between two states ρ

and σ ‖ρ − σ‖ → 0 the difference between their entanglement should tend to zero

E(ρ)− E(σ)→ 0.

Axiom 7 (Additivity). For two states ρ and σ we have E(ρ⊗ σ) = E(ρ) + E(σ).

In fact this axiom is a little bit too strong and excludes reasonable candidates forentanglement measures. Therefore we state the following.

Axiom 8 (Subadditivity). For two states ρ and σ we have E(ρ⊗σ) ≤ E(ρ)+E(σ).

Nevertheless the additivity axiom should hold in the case σ = ρ.

Axiom 9 (Weak additivity). For a state ρ we have E(ρ⊗N ) = NE(ρ).

2.4.2 Von Neumann entropy

The Shannon entropy7 H(X) = −∑

X pX log pX , where the random variable Xoccurs with probability px, quantifies the amount of information when we learn thevalue of the random variable or the amount of uncertainty in the value of X beforewe learn its value. In the case of maximal uncertainty the Shannon entropy is 1 bit,for the case of maximal knowledge or zero uncertainty about the alternatives theShannon entropy is zero.Since information is always embodied in the state of a physical system, we canalso think of the Shannon entropy as quantifying the physical resources required tostore classical information. In the case of quantum information the unit of quantuminformation is the “qubit”, representing the amount of information that can be storedin the state of a qubit (e.g. the polarization state of a photon). An arbitrarily largeamount of classical information can be encoded in a qubit but due to measurementwe only have access to one bit.

7Note that we use the symbol log to indicate log2 unless it is stated other.

20


The von Neumann entropy [113] of a state ρ, defined by

SvN (ρ) = −Tr(ρ log ρ) , (2.4.1)

can be seen as a straight forward “quantization” of the classical Shannon entropy.An entanglement measure for pure states is the von Neumann entropy of thereduced density matrix of the bipartite system

EvN (ρ) = SredvN = −Tr(ρ2 log ρ2) = −Tr(ρ1 log ρ1) , (2.4.2)

where ρ2,1 = Tr1,2ρ are the reduced density matrices. The van Neumann entropysatisfies the axioms 1, 2, 3, 5, 9 and is the only candidate for an entanglementmeasure which is physically reasonable.

2.4.3 Concurrence

This measure was introduced by Bennett, DiVincenzo, Smolin and Wootters [19] in1996 and generalized by Wootters and Hill [77, 151, 152]. It is only valid for qubitsbut rather easy to compute.Suppose we consider a pure state |ψ〉 which can be written as a linear combination|ψ〉 =

∑i αi|ei〉 in the “magic basis”8

|e1〉 =12(|00〉+ |11〉) |e3〉 =

i

2(|01〉+ |10〉)

|e2〉 =i

2(|00〉 − |11〉) |e4〉 =

12(|01〉 − |10〉) .

(2.4.3)

The concurrence C is defined as [19]

C(ψ) =∣∣∑

i

α2i

∣∣ , (2.4.4)

and ranges from zero (separable) to one (maximally entangled). To get an expressionapplicable for mixed states [77, 151] we re-express the concurrence for pure states

C(ψ) = |〈ψ|ψ〉| with |ψ〉 = (σ2 ⊗ σ2) |ψ∗〉 , (2.4.5)

where |ψ∗〉 denotes the complex conjugate in the computational basis {|0〉, |1〉}. |ψ〉represents a kind of spin-flipped state which is compared with the original statewhich gives a measure of entanglement [77].For matrices the spin-flip operation is given by

ρ = (σ2 ⊗ σ2)ρ∗(σ2 ⊗ σ2) , (2.4.6)

where ρ∗ is the complex conjugate in the standard basis. We define a quantityR2(ρ) = ρ · ρ where TrR ranges from 0 to 1 and is a measure of the degree ofequality of the two matrices. The concurrence is then given by

C(ρ) = max{0, λ1 − λ2 − λ3 − λ4} , (2.4.7)

where the λi are the square roots of the eigenvalues of the matrix R2 = ρρ indescending order. A general discussion of concurrence by Wootters can be found inRef.[152].

8The magic basis is a kind of Bell basis. It is created by |e1〉 = 12

∑1j=0|jj〉, |ek〉 = i(�⊗σk−1)|e1〉,

where σk denotes the Pauli matrices.

21

CHAPTER 2. Entanglement 2.4. Entanglement measures

2.4.4 Entanglement of formation

The entanglement of formation (EoF) EF (ρ) is the minimal convex extension ofthe von Neumann entropy of the reduced density matrix to mixed states [19].The name EoF is justified in the sense that the two parties must already share anamount of pure singlet Bell states in order to create the state ρ without transferringquantum states between the parties (in the asymptotic limit). EF stands for theminimum cost.Every state can be decomposed as a convex combination of pure state projectorsρ =

∑i pi|ψi〉〈ψi|.

Definition 6. The EoF is defined as the averaged von Neumann entropy of the

reduced density matrices of the pure states |ψi〉 realizing ρ minimized over all possible

decompositions

EF (ρ) = infρ=

∑i pi|ψi〉〈ψi|

∑i

pi EvN (|ψi〉〈ψi|) . (2.4.8)

The EoF reduces to the von Neumann entropy of the reduced matrices for purestates and satisfies the axioms 1, 2, 3, 4, 5, 6 and 8.In general it is difficult to find the minimal decomposition of ρ and therefore it isdifficult to compute the EoF. But there exists a connection between EoF and theconcurrence for two qubit systems, shown in Ref.[19]. There exists a lower boundfor the EoF for general mixed states of a two qubit system. We introduce a quantity,called the fully entangled fraction F

F (ρ) = maxψ〈ψ|ρ|ψ〉 , (2.4.9)

where the maximum is taken over all maximally entangled states |ψ〉. For Belldiagonal states the fully entangled fraction is simply given by the largest eigenvalue ofρ. For all states of a two qubit system the following lower bound for the entanglementof formation holds

EF (ρ) ≥ h[F (ρ)] . (2.4.10)

For pure states and Bell diagonal states the EoF is equal to this bound. The functionh is defined by

h(F ) =

{H[12 +

√F (1− F )] for F ≥ 1

2

0 for F < 12 ,

(2.4.11)

with the binary entropy function

H[x] = −x log x− (1− x) log(1− x) . (2.4.12)

The above definition can also be reformulated in terms of the well known concur-rence, defined in Sect.2.4.3. The entanglement of formation is then given by thesimple formula [77, 151]

EF (ρ) = H[12(1 +

√1− C2(ρ))] , (2.4.13)

where the concurrence C is defined by equation (2.4.7).

22


2.4.5 Entanglement of distillation

The following definitions trace back to the Horodeckis in 1998 [82].

Definition 7. A density operator ρ is called distillable (free entangled) if one

can produce a maximally entangled state out of (several copies of) it.

Definition 8. An entangled state which is not distillable is called bound entangled.

In 2 × 2 dimensional systems all entangled states are distillable and therefore nobound entanglement exists, but in higher dimensions bound entanglement can occur.Examples for bound entangled states are all ppt-states which are not separable. Upto now it is not known, if bound entangled npt-states do exist.There are several distillation protocols, e.g. the first proposed by Bennett et al. [18]in 1996, for an overview see Ref.[84].Distillation works in the following way. Alice sends one subsystem of an entangledstate ρ to Bob. She repeats this n times, such that they share n copies of thestate ρ. Due to the noisy channel the original amount of entanglement is decreased.Then both make LOCC operations according to a distillation protocol and thereforeincrease the amount of entanglement or correct the occurring errors of the channel.The entanglement of distillation ED defines the amount of entanglement of a stateas the proportion of the number of singlet states nmax.ent.,out that can be distilledout of a number of input states nin using a distillation protocol. It should beindependent of the used protocol and satisfies the axioms 1, 2, 3, 9. There is noclosed analytical expression for this measure but in the limit of infinitely many inputsthe entanglement of distillation reads

ED(ρ) = supDist.prot

limnin→∞

nmax.ent.,out

nin, (2.4.14)

where the supremum is taken over all possible distillation protocols.

2.4.6 Relative entropy of entanglement

It is possible to define the amount of entanglement via distance measures D suchthat we measure the distance between the entangled state ρ and the closest separablestate σ

E(ρ) = minσ∈S

D(ρ ‖ σ) , (2.4.15)

where S denotes the set of separable states. We can interpret the amount of entan-glement given by this equation as finding a certain state ρ0 ∈ S which is closest toρ under the measure D. The state ρ0 approximates the classical correlations of thestate ρ as close as possible thus E(ρ) measures the remaining quantum correlations.One way of defining a distance measure is via the relative entropy

ER(ρ) = minσ∈S

Tr(ρ(log ρ− log σ)) = minσ∈S

Tr(ρ logρ

σ) , (2.4.16)

introduced by Vedral, Plenio, Rippin and Knight [144] (see also [142, 143]).

23

CHAPTER 2. Entanglement 2.5. Bell inequalities

The relative entropy is not a distance in a strict mathematical sense since it fails tobe symmetric. It satisfies the axioms 1, 2, 3, 5, 6, 8.Another way of defining a distance measure is via the Bures metric [90, 110]

DB(ρ ‖ σ) =√

2− 2F (ρ, σ) , (2.4.17)

where the fidelity F is given by

F (ρ, σ) = Tr√√

ρ σ√

ρ , (2.4.18)

or the Hilbert-Schmidt distance (see Sect.7.1 and Ref.[150])

DHS(ρ ‖ σ) = ‖σ − ρ‖2 , (2.4.19)

with the Hilbert-Schmidt norm

‖A‖2 = Tr(A†A) . (2.4.20)

2.4.7 Properties and relations

The following table summarizes the important properties of the above introducedentanglement measures.

measure norm. LOCC mono. loc. U -inv. convexity continuity additivityEvN � � � � �(weak)EF � � � � � �(sub)ED � � �(weak)ER � � � � �(weak)

It can be shown [83] that the following relation holds

ED(ρ) ≤ E(ρ) ≤ EF (ρ) , (2.4.21)

where E(ρ) denotes any entanglement measure.

2.5 Bell inequalities

In 1935 Schrodinger [126] denoted entanglement as a fundamental concept of Quan-tum mechanics. Einstein, Podolsky and Rosen [56] pointed out that superpositionscan lead to paradoxical quantum nonlocality, known as the EPR-paradox. Theyconcluded form nonlocality that quantum mechanics cannot be complete and mustbe modified by some “hidden variables”. The following philosophical debate was puton an objective level by John Bell in 1964 [15]. He derived an inequality based onhidden variable theory for joined measurement probabilities in entangled systems.The Bell inequality has to be satisfied by (all) local realistic theories but can be vio-lated by QM. The version published by Clauser, Horne, Shimony and Holt (CHSH)[46] allowed an experimental decision between hidden variable theories and quantummechanics.

24

2.5. Bell inequalities CHAPTER 2. Entanglement

A violation of a Bell inequality demonstrates the presence of entanglement and thusaccording to Bell’s Theorem the occurrence of nonlocal features in the quantumsystems. BI serve as criteria for separability and entanglement and represent somekind of entanglement witness. Generalized BI in the sense of optimal entanglementwitnesses are considered in Refs.[30, 35] (see Sect.2.3.6).So far all experiments (e.g. with photons [8, 148], kaons [34, 40], atoms [59]) confirmthat they are not compatible with any local hidden variable theory. The violationof Bell’s inequality confirms that quantum mechanic is a nonlocal theory.But nonlocality does not conflict with Einstein’s relativity, because it cannot beused for superluminal communication. Nevertheless, Bell’s work provides the basisfor new physics: quantum information and quantum communication [4, 36, 39, 110].In the following section we introduce the most prominent example of a Bell inequality– the Bell-CHSH inequality.

2.5.1 Bell-CHSH inequality

The Bell-CHSH inequality [46] is a special kind of a Bell inequality suited for exper-imental testing because it assumes no perfect correlations.Suppose we have a system of two spin-1

2 particles and measure the observablesA(α) and B(β) corresponding to a spin measurement along the direction α on oneside and along β on the other side, see Fig.2.3. The crucial assumption is Bell’slocality hypothesis which states that the outcomes of the individual measurementsare independent of each other. The two spins are well separated such that one canassure that no communication between them can occur (Einstein locality).

α βα1

β1

β2

z

Figure 2.3: The setup for a Bell experiment. The measurement directions are specified by

polar angles (indicated by the index 1) and azimuthal angles (indicated by 2).

A local realistic theory based on hidden variables tells you that the following inequal-ity for the expectation values of joint measurements must hold for each measurementconfiguration

S(α, α′, β, β′) = |E(α, β)− E(α, β′)|+ |E(α′, β) + E(α′, β′)| ≤ 2 , (2.5.1)

and is called Bell-CHSH inequality.

25

CHAPTER 2. Entanglement 2.5. Bell inequalities

The quantum mechanical expectation value

EQM (α, β) = 〈AQM (α)⊗BQM (β)〉 , (2.5.2)

for the Bell singlet state |Ψ−〉 = 1√2

(|⇑〉 ⊗ |⇓〉 − |⇓〉 ⊗ |⇑〉

)written in the eigenbasis

of the σ3-operator gives

EQM (α, β) = 〈Ψ−|σ · α ⊗ σ · β|Ψ−〉 = − cos(α1 − β1) . (2.5.3)

In the above calculation we have assumed that the measurement directions α, βspecified by polar angles α1, β1 and azimuthal angles α2, β2 have an azimuthaldifference of zero α2 − β2 = 0 which corresponds to parallel measurement planes,see Fig.2.3.The Bell-CHSH inequality (2.5.1) can be violated by the quantum mechanical expec-tation value for certain angles. The maximal violation by the amount of SQM

max = 2√

2occurs at the so-called Bell angles given by9

SQM (0,π

4,3π

4,π

2) = 2

√2 � 2 . (2.5.4)

The experimental value achieved in a photon experiment under strict Einstein lo-cality [148] is given by Sexp = 2.73 ± 0.02 which is quite close to the theoreticalpredictions.

9The azimuthal angles of all measurement configurations are assumed to be equal and without

loss of generality set to zero.

26

Chapter 3

Decoherence

Quantum mechanical systems must be regarded as open systems due to the factthat any realistic system is subjected to a coupling to an uncontrollable environmentwhich influences it in a non-negligible way. The theory of open quantum systems thusplays a major role in many applications of quantum physics since perfect isolation ofa quantum system is not possible and a complete description of the the environmentaldegrees of freedom is not feasible. Therefore we have to seek for a simpler andeffective description of the dynamics of the system – the master equation.Another reason for invoking open quantum systems is of more fundamental origin:the measurement process can be interpreted as a kind of open system dynamics.This leads to a destruction of superpositions between the states of the system whichexplains in a way the origin of the classical world out of quantum theory (see forinstance Refs.[66, 158, 159]).This chapter provides a short introduction to the theme of open quantum systemsand their description in terms of quantum master equations based on dynamicalmaps satisfying semigroup laws (Sect.3.2.1). In Sect.3.2.2 we derive the masterequation via several approximations from first principle and discuss different rep-resentations of the dissipator in two dimensions (Sect.3.3). Furthermore (Sect.3.4)we give a short review of the Kraus operator representation of quantum operationsfor various quantum channels which is is another way of describing open quantumsystems. This approach is used in quantum information theory (for an introductionsee [110, 120]).

3.1 Nomenclature and classifications

Up to now we have studied systems that can be described by the Liouville-vonNeumann equation (for pure and mixed states)

∂

∂tρ(t) = −i[H(t), ρ(t)] . (3.1.1)

This defines a unitary time evolution

ρ(t) = U(t, t0)ρ(t0)U †(t, t0) , (3.1.2)

27

CHAPTER 3. Decoherence 3.2. Open quantum systems

where the unitary operator satisfies

∂

∂tU(t, t0) = −iH(t)U(t, t0) , (3.1.3)

with the initial condition U(t0, t0) = �.We call a system closed if its dynamics are governed by Eq.(3.1.1). A system iscalled closed and isolated if the Hamiltonian is time independent. In the casewhere the system is driven by external forces (e.g. external electromagnetic field) itis possible to formulate the dynamics in terms of a Liouville-von Neumann equationbut with time dependent Hamiltonian (see for example Sect.4.1).An open quantum system is a system S which is coupled to another quantum systemcalled the environment E. System and environment together S + E form a closedsystems whose dynamics is again determined by unitary evolution. The dynamicsof the system S can not be modelled by Eq.(3.1.1). We need the quantum masterequation (see Sect.3.2) which in general describes a non-unitary evolution.Special kinds of environments are reservoirs and heat baths. A reservoir is is a sys-tem with infinite number of degrees of freedom, where the frequencies of the reservoirmodes form a continuum. A heat bath is a reservoir in a thermal equilibrium state.

3.2 Open quantum systems

We consider the open quantum system S coupled to an environment E. The Hilbertspace of the total system S + E is given by the tensor product H = HS ⊗HE . Thetotal Hamiltonian can be written in the form

H(t) = HS ⊗ � + �⊗HE + HI(t) (3.2.1)

where HS and HE are the free Hamiltonians of the system and the environment,respectively, and HI(t) denotes the interaction Hamiltonian. The state of the systemρS

1 is given by the reduced density matrix ρS = TrEρ.The evolution of the total system S + E is assumed to be unitary

U(t, t0) = T e−i∫ t0

t H(t′)dt′ , (3.2.2)

where T denotes timeordering.The time-evolved state of the system S is obtained by taking the partial trace overthe environmental degrees of freedom

ρS(t) = TrE(U(t, t0)ρ(t0)U †(t, t0)) . (3.2.3)

The dynamic of the system S is given by the equation of motion

∂

∂tTrEρ(t) =

∂

∂tρS(t) = −iTrE [H(t), ρ(t)] . (3.2.4)

1States of the subsystems S and E are labelled accordingly, states of the total system do not

have any index.

28

3.2. Open quantum systems CHAPTER 3. Decoherence

In practise these equations are not applicable because they consist of the state ofthe total system ρ. In general this state is unknown due to the fact that the modesof the environment nor are exactly known neither are controllable. Therefore we tryto develop an approximation for Eq.(3.2.4) that consists only of the reduced stateρS which is of primary interest.

3.2.1 Quantum master equation

Under the assumptions of

• no memory effects and

• weak coupling to the environment

we can derive a so-called quantum master equation describing an open quantumsystem and depending only on ρS (see Sect.3.2.2).We assume the initial state of the total system to be uncorrelated

ρ(0) = ρS(0)⊗ ρE , (3.2.5)

where the state of the environment can be written in spectral decomposition withthe orthonormal basis {|φi〉}

ρE =∑

i

λi|φi〉〈φi| . (3.2.6)

Quantum dynamical maps

We can define the following quantum dynamical map

V (t) : B(HS)→ B(HS)ρS(0) �→ ρS(t) = V (t)ρS(0) .

(3.2.7)

The map V (t) is convex-linear, complete positive and trace preserving.The most general form of a dynamical map is in terms of Kraus operators [93]defined by

V (t)ρS(0) =∑ij

Eij(t)ρS(0)E†ij(t) . (3.2.8)

The Kraus operators satisfy∑

ij E†ij(t)Eij(t) = � and are related to the unitary

evolution operator byEij(t) =

√λj〈φi|U(t, 0)|φj〉 . (3.2.9)

The one parameter family of dynamical maps {V (t), t ≥ 0} determines the timeevolution of an open system. If memory effects of the environment can be neglectedthe dynamical map satisfies the semigroup property

V (t1)V (t2) = V (t1 + t2) for t1, t2 ≥ 0 , (3.2.10)

that means the time evolution depends only on the state the time step before (seeRefs.[5, 61]).

29


Quantum master equation

Under certain conditions it is possible to represent the time evolution of the reduceddensity matrix ρS as a linear map

∂

∂tρS(t) = L(ρS(t)) , (3.2.11)

where the linear map L is related to the dynamical map V (t) by V (t) = eL(ρS(t))t.Eq.(3.2.11) is called Markovian2 quantum master equation.The most general form of the generator L of the semigroup derived in 1976 byLindblad [99] and independently by Gorini, Kossakowski and Sudarshan [68] is givenby the so-called Lindblad form

∂

∂tρS = −i[HU , ρS ]− 1

2

n2−1∑k=1

γk

(A†

kAkρS + ρSA†kAk − 2AkρSA†

k

), (3.2.12)

where Ak represent the so-called Lindblad operators and n is the dimension of thesystem. The unitary part of the evolution is generated by the Hamiltonian HU whichin general can not be identified with the Hamiltonian of the system HS . Sometimesthe nonunitary part is called the dissipator

D(ρS) =12

n2−1∑k=1

γk

(A†

kAkρS + ρSA†kAk − 2AkρSA†

k

), (3.2.13)

and the Lindblad master equation is of the form

∂

∂tρS = −i[HU , ρS ]−D(ρS) . (3.2.14)

The Lindblad operators and the Hamiltonian part are not uniquely determined. TheLindblad equation is invariant under the following transformations:

• unitary transformations

Ai −→ A′i =

∑j

UijAj , (3.2.15)

where Uij is a unitary matrix,

• inhomogeneous transformations

Ai −→ A′i = Ai + ai ,

HU −→ H ′U = HU +

12i

∑j

γk(a∗jAj − ajA†j) + b , (3.2.16)

where ai ∈ C and b ∈ R.2A Markovian process is a stochastic process with a short memory that forgets rapidly its past

history. Mathematically this is expressed by the semigroup property.

30

3.2. Open quantum systems CHAPTER 3. Decoherence

3.2.2 Microscopic derivation of the master equation

In this section we derive the master equation from the underlying Hamiltonian dy-namics of the total system by using various approximations (see also Ref.[43]).The Hamiltonian of the total system is given by

H = HS ⊗ � + �⊗HE + HI . (3.2.17)

The following manipulations are performed in the interaction picture where theLiouville-von Neumann equation is given by

d

dtρ(t) = −i[HI(t), ρ(t)] , (3.2.18)

which can be formally solved by

ρ(t) = ρ(0)− i

∫ t

0ds[HI(s), ρ(s)] . (3.2.19)

By inserting the integral solution in Eq.(3.2.18), taking the trace over the reservoirand using the assumption TrE [HI(t), ρ(0)] = 0 we get

d

dtρS(t) = −

∫ t

0dsTrE

[HI(t), [HI(s), ρ(s)]

]. (3.2.20)

Now the first approximation appears: the Born approximation. We assume thatthe coupling between the system and the environment is weak such that we maywrite the total state as a product state

ρ(t) ≈ ρS(t)⊗ ρE . (3.2.21)

This leaves us with a closed integro-differential equation for ρS

d

dtρS(t) = −

∫ t

0dsTrE

[HI(t), [HI(s), ρS(s)⊗ ρE ]

]. (3.2.22)

Another simplification is obtained by the Markov approximation which makesthe master equation local in time by replacing ρS(s) at the retarded time with ρS(t)at the present time

d

dtρS(t) = −

∫ t

0dsTrE

[HI(t), [HI(s), ρS(t)⊗ ρE ]

]. (3.2.23)

This equation depends only on ρS(t) and is called the Redfield equation. Toget a Markovian quantum master equation we have to substitute s by t − s in theintegral and let the upper limit of the integral go to infinity in order to guaranteethe semigroup property, Eq.(3.2.10), which gives

d

dtρS(t) = −

∫ ∞

0dsTrE

[HI(t), [HI(t− s), ρS(t)⊗ ρE ]

]. (3.2.24)

The above two simplifications are called Born-Markov approximation. In generalthey do not guarantee to arrive at positive definite density matrices. Therefore we

31


have to apply another approximation, known as rotating wave approximation.We write the interaction Hamiltonian in the most general form as

HI =∑

k

Ak ⊗Bk . (3.2.25)

The approximation is carried out if the interaction Hamiltonian is decomposed intoeigenoperators |e〉〈e| of HS . Thus we can write

Ak(ω) =∑

e−e′=ω

|e〉〈e|Ak|e′〉〈e′| (3.2.26)

such that [HS , Ak(ω)] = ωAk(ω) and A†k(ω) = Ak(−ω). The operators Ak(ω) are

called eigenoperators of HS with the frequency ω.The interaction Hamiltonian in the Schrodinger picture can be written as

HI =∑k,ω

Ak(ω)⊗Bk , (3.2.27)

and for the interaction picture we ge

HI(t) =∑k,ω

e−iωtAk(ω)⊗Bk(t) . (3.2.28)

Inserting this in Eq.(3.2.24) and neglecting terms of different frequencies ω we getthe following equation

d

dtρS(t) =

∑ω,k,l

Γkl(ω)(Al(ω)ρS(t)A†

k(ω)−A†k(ω)Al(ω)ρS(t)

)+Γ∗

lk(ω)(Al(ω)ρS(t)A†

k(ω)−A†k(ω)Al(ω)ρS(t)

),

(3.2.29)

where the function Γ is the fourier-transformed of the reservoir correlation function

Γkl(ω) =∫ ∞

0dseiωs〈B†

k(t)Bl(t− s)〉 . (3.2.30)

By using the decomposition

Γkl(ω) =12γkl(ω) + iSkl(ω) , (3.2.31)

we can express Eq.(3.2.29) in a more familiar way (cf. Eq.(3.2.12))d

dtρS(t) = −i[HLS , ρS(t)]−D(ρS(t)) , (3.2.32)

where

HLS =∑ω,k,l

Skl(ω)A†k(ω)Al(ω)

D(ρS(t)) =12

∑ω,k,l

γkl(ω)(A†

k(ω)Al(ω)ρS(t) + ρS(t)A†k(ω)Al(ω)− 2Al(ω)ρS(t)A†

k(ω))

.

(3.2.33)

The term HLS provides a Hamiltonian contribution to the evolution and is calledthe Lamb shift Hamiltonian because it leads to a Lamb-type renomalization of theunperturbed energy levels induced by the interaction. HLS commutes with HS . Theform (3.2.13) of the dissipator is reached by diagonalizing the matrices γkl(ω).

32

3.3. Representations of the dissipator CHAPTER 3. Decoherence

3.3 Representations of the dissipator

The Lindblad master equation

∂

∂tρ = −i[H, ρ]−D(ρ) , (3.3.1)

describes the behavior of the time evolution of an open quantum system. The mostgeneral structure of the dissipator

D(ρ) =12

n2−1∑k=1

γk

(A†

kAkρ + ρA†kAk − 2AkρA†

k

), (3.3.2)

is determined by complete positivity which means that the quantum dynamical map

V (t) : ρ(0) �→ ρ(t) = V (t)ρ(0) , (3.3.3)

has to be positive for all possible extensions to higher dimensional spaces (see alsoSect.2.3.5).The dissipator D(ρ) describes 2 phenomena occurring in an open quantum system:decoherence and dissipation. When the system S interacts with the environment Ethe initial product state evolves into an entangled state of S+E in the course of time.This leads to mixed states in S and is called decoherence. Furthermore we getan energy exchange between S and E what is called dissipation. The decoherencedestroys the occurrence of longrange quantum correlations by suppressing the off-diagonal elements of the density matrix in a given basis and leads to an informationtransfer from S to E. In general, both effects are present, but decoherence actson a much shorter time scale than dissipation and therefore is the more importantprocess in quantum information theory.If we assume hermitian Lindblad operators A†

k = Ak then the von Neumann entropyS(ρ) = −Tr(ρ ln ρ) does not not decrease as a function of time due to a theoremby Benatti and Narnhofer [17]. For Lindblad operators that commute with theHamiltonian operator [Ak, H] = 0 we get conserved energy in case of a hermitianHamiltonian (see e.g. Ref.[2]). Consequently the dissipator can be written as

D(ρ) =12

∑γk

[Ak, [Ak, ρS ]

]. (3.3.4)

We can also write the dissipator in terms of projection operators Pk

D(ρ) =12

∑k

λk

(Pkρ + ρPk − 2PkρPk

), (3.3.5)

where we have used the replacement√

γkAk =√

λkPk and the fact that P 2k = Pk

[32]. The real and positive parameters λk are called decoherence parameters.If we assume that

∑k Pk = � and we have only one parameter λ which parameterizes

the strength of the interaction we can write the dissipator in the form

D(ρ) = λ(ρ−

∑PkρPk

). (3.3.6)

33

CHAPTER 3. Decoherence 3.4. Quantum operations

For example for two projection operators satisfying P1 + P2 = � we get

D(ρ) = λ(ρ− P1ρP1 − P2ρP2

)= λ

(P1ρP2 + P2ρP1

). (3.3.7)

Suppose we are working in a two dimensional Hilbert space H = C2. Then it ispossible to write all used hermitian 2× 2 matrices in terms of Pauli-matrices as

ρ =12(� + r · σ)

H =12(� + h · σ)

√γkAk =

12(ak� + nk · σ)

(3.3.8)

where ak, nk, r and h are real and positive. The time evolution can be reformulatedas

∂

∂tr = h× r − Lr , (3.3.9)

and the dissipator L is of the form

Lr =12

∑λk(r − nknk · r) . (3.3.10)

The matrix L is a positive linear combination of projectors which project onto a planeorthogonal to the vector nk. Eq.(3.3.9) determines the geometric version of the timeevolution which can be illustrated in the Bloch sphere picture (see Sects.2.1.3 and3.4).

3.4 Quantum operations

3.4.1 Operator-sum representation

We have seen in Section 3.2.1 that a quantum dynamical map can be represented interms of Kraus operators

ρ(0) �→ V (t)ρ(0) =∑ij

Eij(t)ρ(0)E†ij(t) , (3.4.1)

which can be written by replacing the double sum3

ρ(0) �→ ρ(t) =∑

i

Miρ(0)M †i , (3.4.2)

where the new Kraus operators Mi satisfy∑

i M†i Mi = �. The quantum dynamical

map described by Eq.(3.4.2) is called a quantum operation and the Kraus oper-ator decomposition is also called operator-sum representation of the quantumoperation. Any complete positive map can be written in terms of Kraus operators.

3The relation is given by Mi = Ei0 =√

λi〈φi|U(t, 0)|φ0〉.

34

3.4. Quantum operations CHAPTER 3. Decoherence

The physical interpretation of the operator-sum representation of a dynamical mapcan be related to the concept of noisy channels in quantum information theory.The dynamical map is completely characterized by the Kraus operators {Mi}. Theaction of the dynamical map corresponds to subjecting the state ρ randomly withprobabilities p(i) = Tr(Miρ(0)M †

i ) to a unitary operation Mi which gives for the final

state ρi = Miρ(0)M†i

Tr(Miρ(0)M†i )

. If the environment is measured we know which operation

was applied. So the final state of the system is a weighted sum over all possibleoperations

ρ(t) =∑

i

p(i)ρi =∑

Miρ(0)M †i , (3.4.3)

which reveals exactly the operator-sum representation. The above described actioncorresponds to the action of a noisy quantum channel, depicted in Fig.3.1.

ρ(0) ρ(t)

|φ0〉〈φ0|U

Figure 3.1: The action of a noisy quantum channel on the state ρ(0).

3.4.2 Correspondence Lindblad operators ↔ Kraus operators

Due to the fact that Lindblad operators Ak and Kraus operators Mi are differentways to describe nonunitary evolution there exists a certain relation between them(see Ref.[120]). The Lindblad approach gives a continuous time-dependence whereasthe Kraus operators models the evolution via discrete state changes. Both views areequivalent at least to first order in time.By making the expansion

ρ(dt) = ρ(0) + O(dt) = ρ(0) + ρ(0)dt , (3.4.4)

and comparing with Eq.(3.4.2) we see that one of the Kraus operators must be ofthe form � + O(dt) and the others are of the order

√dt. We can chose them in the

following way

M0 = �− (iH −K)dt

Mk =√

dtAk ,(3.4.5)

where H and K are hermitian operators and K is determined by the normalizationcondition which gives

K = −12

∑k

A†kAk . (3.4.6)

We can substitute Eqs.(3.4.4) and (3.4.5) into Eq.(3.4.2). Keeping only terms of theorder of dt we get

d

dtρ = −i[H, ρ]− 1

2

∑k

(A†

kAkρ + ρA†kAk − 2AkρA†

k

), (3.4.7)

35


which is exactly the Lindblad master equation. Each term AkρA†k describes a possi-

ble quantum jump and the terms A†kAkρ + ρA†

kAk serve for the right normalizationin the case where no jump occurs.Recall that the Kraus operators are not uniquely determined and can be transformedby unitary operations

Nl = MiUil . (3.4.8)

If the number of Kraus operators Nα and Mi is not the same then the smaller sethas to be filled up with zero operators.

In the following sections we describe the action of three quantum channels for singlequbits via their Kraus operators and illustrate their action on the Bloch sphere. Westart with a density matrix ρ0 and the according Bloch vector r0 given by

ρ0 =(

ρ11 ρ12

ρ21 ρ22

)and r0 =

⎛⎝ ρ12 + ρ21

i(ρ12 + ρ21)ρ11 − ρ22

⎞⎠ . (3.4.9)

3.4.3 Depolarizing channel

The depolarizing channel is a model for decoherence where with a certain probability3p4 the qubit gets totally mixed due to the occurrence of an error and with 1 − 3p

4the qubit remains unchanged. Three errors are possible.The spin flip error is described by the action of σ1 and interchanges the base statesof the qubit. The phase flip error σ3 just induces a relative phase between the basestates and the combination of spin flip and phase flip error is described by σ2.Therefore the Kraus operators for the depolarizing channel are given by

M0 =

√1− 3p

4� M1 =

√p

4σ1

M2 =√

p

4σ2 M3 =

√p

4σ3 ,

(3.4.10)

or in another representation where p = 4p′3

M0 =√

1− p′ � M1 =

√p′

3σ1

M2 =

√p′

3σ2 M3 =

√p′

3σ3 .

(3.4.11)

The time evolved density matrix

ρ =(

(1− p2)ρ11 + p

2ρ22 (1− p)ρ12

(1− p)ρ21 (1− p2)ρ22 + p

2ρ11

), (3.4.12)

determines the Bloch vector to

r = (1− p)

⎛⎝ ρ12 + ρ21

i(ρ12 + ρ21)ρ11 − ρ22

⎞⎠ . (3.4.13)

36

3.4. Quantum operations CHAPTER 3. Decoherence

Thus the effect of the depolarizing channel depicted on the Bloch sphere is a uniformcontraction of the entire sphere about a factor of 1− p.

3.4.4 Phase damping channel

The phase damping channel has no classical analog because it describes the loss ofquantum information without loss of energy. The quantum information correspondsto the ability of a system to produce quantum interferences hence is described bythe off-diagonal elements of a density matrix. Phase damping can occur for exampledue to random phase kicks or scattering processes.We can model this kind of channel by the following Kraus operators

M0 =√

1− p � M1 =√

p

4(1 + σ3) M2 =

√p

4(1− σ3) , (3.4.14)

but there are also representations with two Kraus operators

M0 =√

1− p

2� M1 =

√p

2σ3 , (3.4.15)

or

M0 =(

1 00√

1− p′

)M1 =

(0 00√

p′

), (3.4.16)

with the relation 1− p =√

1− p′. The final state is given by

ρ =(

ρ11 (1− p)ρ12

(1− p)ρ21 ρ22

), (3.4.17)

where the off-diagonal elements are damped by a factor of 1 − p. We get for theBloch vector

r =

⎛⎝ (1− p)(ρ12 + ρ21)i(1− p)(ρ12 + ρ21)

ρ11 − ρ22

⎞⎠ , (3.4.18)

which indicates that the phase damping channel can be illustrated on the Blochsphere by a contraction of the entire sphere to a prolate spheroid about the z-axisby a factor of 1− p. The preferential treatment of the z-direction indicates that thephase damping channel acts in a preferred basis.

3.4.5 Amplitude damping channel

The amplitude damping channel allows us to describe the decay of a two-level systemdue to spontaneous emission of a photon. This process is coupled with the loss ofenergy (dissipation). We suppose the excited state to be denoted by the state |1〉and the ground state by |0〉.We need the following Kraus operators to describe this channel

M0 =(

1 00√

1− p

)M1 =

(0√

p0 0

), (3.4.19)

37


where the decay happens with a probability p. The final state is given by

ρ =(

ρ11 + p ρ22√

1− pρ12√1− pρ21 (1− p)ρ22

), (3.4.20)

where we see clearly the decrease of the excited state and the suppression of theoff-diagonal elements. The Bloch vector

r =

⎛⎝√1− p(ρ12 + ρ21)i√

1− p(ρ12 + ρ21)ρ11 − (1− 2p)ρ22

⎞⎠ , (3.4.21)

shows that in the Bloch sphere representation the channel shrinks the entire Blochsphere towards the north pole where the state |0〉 is located.

38

Chapter 4

Geometric Phase

In 1984 the concept of geometric phases was introduced by Berry [24]. He showedthat under cyclic and adiabatic evolution of a quantum system an additional geo-metric phase factor occurs in contrast to the well known dynamical phase factor. Upto now there have been many generalizations, for instance the adiabatic conditionwas removed by Aharonov and Anandan [3] and Samuel and Bhandari considerednoncyclic evolutions [123] which is in deep connection to the work of Pancharatnam[112]. The geometric phase can also be defined for mixed states in a mathematicalway due to Uhlmann [140] or in a more phenomenological way due to Sjoqvist et al.[131] and it is possible to deal with off-diagonal geometric phases [58, 102].The geometric origin of the phase can be understood within differential geometry asSimon pointed out [128] where the phase is given by a holonomy of a line bundle.The holonomy emerges from the integral of the connection (or curvature) of thebundle over the parameter space.The application of geometric phases in quantum computation has been suggestedby several authors [54, 89, 155]. Experimentally, geometric phases have been testedin various cases, e.g., with photons [44, 96, 136], with neutrons [69, 73] and withatoms [147].In the following section we introduce the adiabatic and cyclic Berry phase to show theconcept of a pure states geometric phase and discuss the phase for the qubit system.For a more detailed introduction see Refs.[37, 55]. In Sect.4.3 the experimentalapproach to mixed states geometric phases due to Sjoqvist et al. is presented andanalyzed for qubits.

4.1 Berry phase

We consider a quantum system embedded in an external environment described bya multidimensional real parameter R(t), which is varied adiabatically1 and cyclicin time. The time dependent Hamiltonian is given by H(t) = H(R(t)) and the

1Adiabatic means that the changes happen slowly in time compared to the characteristic time

scale of the system.

39

CHAPTER 4. Geometric Phase 4.2. Spin- 12

particle – qubit

time-dependent Schrodinger equation reads

H(R(t))|ψ(t)〉 = i�∂

∂t|ψ(t)〉 . (4.1.1)

Furthermore we will use the simplifying notation R ≡ R(t). We can choose at anyinstant a basis of eigenstates |n(R)〉 for the Hamiltonian labelled by the quantumnumber n such that the eigenvalue equation is fulfilled

H(R)|n(R)〉 = En(R)|n(R)〉 . (4.1.2)

We assume that the energy spectrum of H is discrete, that the eigenvalues are notdegenerated and that no level crossing occurs during the evolution. The systemstarts in the n-th energy eigenstate |ψ(0)〉 = |n(R0)〉 and according to the AdiabaticTheorem [125] during the adiabatic evolution the system stays in the n-th eigenstate.After a cyclic evolution (that means R(0) = R(T )) the state of the system returnsto its initial state up to a phase factor. The final state can be written as

|ψ(T )〉 = eiθn(T )eiγn(C)|n(R0)〉 , (4.1.3)

where the dynamical phase is given by

θn(T ) = −1�

∫ T

0En(t)dt , (4.1.4)

and the geometric or Berry phase turns out to be

γn(C) = i

∮C〈n(R)|∇R|n(R)〉dR . (4.1.5)

The nonitegrable Berry phase depends on the circuit C traced out in the parameterspace {R}. Although we come back to the starting point in parameter space via theclosed circuit, the state vector does not and therefore the Berry phase is unequal tozero.The geometric phase is independent of the direction and of the velocity the circuitC is traversed. It is a purely geometric property of the system or the underlyingparameter space and can be described within differential geometry as a holonomyof the parallel transported eigenstates [128] (see also [109]).

4.2 Spin-12 particle – qubit

We consider a spin-12 particle (qubit) in an external magnetic field B which rotates

adiabatically (slowly) under an angle ϑ around the z-axis (see Fig.4.1).The magnetic field is given by

B(t) = B0

⎛⎝sinϑ cos(ωt)sinϑ sin(ωt)

cos ϑ

⎞⎠ = B0 n , (4.2.1)

40

4.2. Spin- 12

particle – qubit CHAPTER 4. Geometric Phase

y

x

z

B

ω

θ

Figure 4.1: Qubit in the magnetic field described by Eq.(4.2.1).

where ω is the angular frequency of the rotation and B0 = | B(t)|. When the fieldrotates slowly enough B

ω � 1, then the spin of the particle will follow the directionof the field. The interaction is described by the Hamiltonian

H(t) =μ

2B(t) · σ =

μ

2B0

(cos ϑ e−iωt sinϑ

eiωt sinϑ − cos ϑ

), (4.2.2)

where the coupling constant is given by μ = gμB, the Lande factor g times the Bohrmagneton μB = 1

2em� (e denotes the electric charge and m the mass of the particle).

The eigenstates are given by

|n+(t)〉 =(

cos ϑ2

eiωt sin ϑ2

), |n−(t)〉 =

(− sin ϑ

2

eiωt cos ϑ2

), (4.2.3)

with the corresponding energy eigenvalues E± = ±μ2 B0. We can interpret the

eigenstates as spin-up (+) and spin-down (−) along the respective B(t)-direction.The parameter space is defined by the allowed values of B(t) which forms an S2. Inour configuration B(t) traces out a curve C which is given by C : r = B0 = const.,ϑ = const., φ ∈ [0, 2π].We get for the Berry phase

γ±(C) = −π(1∓ cos ϑ) = ∓12Ω(C) , (4.2.4)

which can be related to the solid angle Ω traced out by the curve C in the parameterspace. The dynamical phase for one rotation with the period T = 2π

ω is given by

θ±(T ) = −1�

∫ T

0E±(t)dt = ∓ μ

2�B0T . (4.2.5)

The total state after one rotation reads

|n±(T )〉 = e−iπ(1∓cos ϑ)e∓i μ2�

B0T |n±(0)〉 . (4.2.6)

The dynamical phase depends on the period T of the rotation but the geometricalphase depends only on the geometry – in this case the opening angle ϑ of themagnetic field.

41

CHAPTER 4. Geometric Phase 4.3. Mixed states

4.3 Mixed states

In practice we have to deal with mixed states rather than with pure states. Thismotivates the extension of the concept of geometric phases to mixed states. Thereare two possible approaches: the experimental one proposed by Sjoqvist et al. [131]and the mathematical approach by Uhlmann [139, 140]. Both methods of extensionlead in principle to different results apart from some special cases such as pure states[57].Uhlmann [139, 140] was probably the first bringing up a geometric phase formalismdealing with mixed states in a rather mathematical way. In his approach (see alsoRefs.[57, 130]) the system of interest which is in a mixed state is embedded as asubsystem in a larger system which is in a pure state. For a given mixed state thereare infinitely many corresponding pure states, or purifications as they are called andfor a cyclic evolution of the smaller system there are many possible evolutions forthe big system. Uhlmann singles out the evolution where the pure state is paralleltransported in a maximally parallel manner which defines the geometric phase forthe small system.Now we want to deal with the more intuitive definition of Sjoqvist in a detailed way.

4.3.1 Experimental approach

The definition of a geometric phase for mixed states [131] arises quite naturally byconsidering a Mach-Zehnder interferometer where the input state ρ0 is a mixed statewith respect to an internal degree of freedom (for example the spin)

ρ0 =∑

k

λk|k0〉〈k0| , (4.3.1)

where |k0〉 denotes orthogonal (spin) eigenstates. In one arm of the interferometer(see Fig.4.2) there is a phase shift of χ acting on the spatial degree of freedom andthe unitary evolution operator U in the other arm acts only on the internal degreeof freedom by

ρt = Uρ0U† =

∑k

λkU |k0〉〈k0|U † =∑

k

λk|kt〉〈kt| . (4.3.2)

The interference pattern after the interferometer is given by

I =∑

k

λk

∣∣∣eiχ|k0〉+U |k0〉∣∣∣2 = 2+2

∑k

λk

∣∣〈k0|U |k0〉∣∣ cos

(χ−arg 〈k0|U |k0〉

), (4.3.3)

which can be also written in terms of the density operator as

I = 2 + 2|Tr(Uρ0)| cos(χ− arg Tr(Uρ0)

). (4.3.4)

In addition to the intrinsic phase shift χ of the interferometer the interference patterngets a phase shift

φ = arg Tr(Uρ0) , (4.3.5)

42

4.3. Mixed states CHAPTER 4. Geometric Phase

ρ0

eiχ

U

Figure 4.2: Mach-Zehnder interferometer to deduce the geometric phase for mixed states.

which depends strongly on the mixed state and its evolution and defines a geometricphase shift according to the idea of Pancharatnam [112] where two states are called“in phase”, that means φ = 0, if their superposition gives a maximum for theintensity.The quantity Tr(Uρ0) is a complex number and determines the interference profile:the absolute value specifies the visibility ν and the argument gives the phase φ. Wecan write

Tr(Uρ0) = νeiφ =∑

k

λk〈k0|U |k0〉 =∑

k

λkνkeiφk , (4.3.6)

which can be interpreted as the weighted sum of single phase factors φk with singlestate visibilities νk. The phase φk = γk + δk can be split into a geometrical γk anda dynamical δk component. The dynamical phase can be removed if the so-calledparallel transport condition is satisfied

Tr(ρtUU †) = 0 , (4.3.7)

which is necessary but not sufficient since the phases of the states |k〉 also have tobe fixed by parallel transport condition

〈kt|UU †|kt〉 = 0 ∀k . (4.3.8)

For the pure case case this reduces to to 〈ψ(t)|ψ(t)〉 = 0 which is both necessaryand sufficient.That means in the case where each eigenstate is parallel transported independentlyacquiring a pure geometric phase factor the mixed geometric phase γ can be definedas the weighted sum over individual geometric phase factors γk and is given by

γ = arg(∑

k

λkνkeiγk

)= arg Tr(Uρ0) , (4.3.9)

with the visibility ofν = |Tr(Uρ0)| , (4.3.10)

where U satisfies the parallel transport conditions, Eqs.(4.3.7) and (4.3.8).

43

CHAPTER 4. Geometric Phase 4.3. Mixed states

4.3.2 Example: qubit

To illustrate the mixed geometric phase we consider the mixed state of a qubitsystem

ρ(t) =12(� + r(t) · σ) (4.3.11)

where |r| = r defines the purity or mixedness of the state. The eigenvalues of thismatrix are given by λ+ = 1

2(1+r) and λ− = 12(1−r) and the eigenstates acquire the

pure state geometric phases of γ± = ∓Ω2 where Ω denotes the solid angle the vector

r traces out by the evolution. We assume the pure state visibilities to be identicalν+ = ν− = η and the mixed state geometric phase, Eq.(4.3.9), can be calculated to

γ = − arctan(r tan

Ω2

), (4.3.12)

and the visibility gives

ν = |Tr(Uρ0)| = η

√cos2

Ω2

+ r2 sin2 Ω2

. (4.3.13)

For cyclic evolutions we have η = 1. The formulas reduce precisely to Eq.(4.2.4)for pure states where r = 1 and for maximally mixed states r = 0 we obtainγ = arg cos Ω

2 and ν = cos Ω2 .

This formula (4.3.12) has been experimentally confirmed within NMR interferometryby Du et al. [53].

44

Chapter 5

Neutron-Interferometry

In an interferometer the incoming waves are split and afterwards recombined whichgives the interferometer its unique ability to experimentally access the phase ofwaves. People have built interferometers not only for photons but also for matter-waves consisting of neutrons [121], electrons [50], atoms [22] and even molecules[7].Neutron interferometry is a well established experimental technique where phasemeasurements are used to study the magnetic, nuclear, and structural properties ofmaterials, as well as fundamental questions in quantum physics. The concept of en-tanglement can be realized within neutron interferometry by using different degreesof freedom of a single neutron. This leads to the concept of contextuality which isdifferent from the common non-locality used in connection with usual Bell inequal-ities. The notion of contextuality and the experimental realization of entanglementwithin neutron interferometry are discussed in Sect.5.3.In Sect.5.1 we give a brief view on neutrons and neutron interferometry (see alsothe books of Rauch and Werner [121], Bonse and Rauch [38], Badurek, Rauch andZeilinger [10] or Bergmann and Schaefer [21]). We describe the basics of neutronoptics which is very similar to classical optics in Sect.5.2.

5.1 Properties of neutrons

5.1.1 Particle properties

The neutron as an elementary particle was discovered in 1932 by Chadwick.Neutrons are massive particles m = 939.6MeV = 1.67×10−27kg forming a compositesystem consisting of 3 quarks u – d – d. They have a finite lifetime of about 15minutes (885 seconds) and decay via β-decay into a proton, an electron and anantineutrino n −→ p + e− + νe. In neutron interferometry experiments the decaycan be neglected because the time the neutrons spend in the apparata is much shorterthan their lifetime. They have spin-1

2 and therefore obey the Fermi-Dirac statistic.The magnetic dipole moment of the neutron is given by μn = −1.91 × μB where

45

CHAPTER 5. Neutron-Interferometry 5.1. Properties of neutrons

μB = 5.05 × 10−27JT−1 denotes the Bohr magneton. Thus neutrons experience all4 fundamental forces: gravitation, electromagnetic, weak and strong interaction.

5.1.2 Wave properties

The wave properties of neutron matter waves can be described analogously to thetheory of light. The wavelength is given by the de Broglie relation λ = h

mc . We getfor thermal neutrons with a velocity of v = 2200m/s and an energy of E = 20meVa wavelength of λ = 1.8A. The time of flight through the interferometer is typically≈ 100μs which is rather long and the coherence length is typically 100 times thewavelength.

5.1.3 Production, moderation, detection

Neutrons are produced via nuclear reactions. Possible production mechanisms are:

• Nuclear reactions induced by accelerated particles:e.g. deuterons are accelerated towards a tritium target which yields about10−4 neutrons per deuteron

• Nuclear fission in research reactors:U-235 is hit by thermal neutrons which yields about 2.4 neutrons per reaction

• Spallation reactions:heavy nuclei (e.g. Pb) are demolished by high-energetic protons (∼ 1GeV)which yields up to 50 neutrons per reaction

All these reactions produce fast neutrons with energies in the MeV region. Thereforethey are slowed down (moderated) to thermal energies through elastic collisions withlight moderator nuclei. To reach the thermal region (meV) one needs for hydrogen18 collisions and a time of 10μs, for deuterium 25 collisions and 48μs and for graphite114 collisions and 150μs.Neutrons are characterized by a low density in phase space (about 10−14 comparedto thermal light sources of 10−3 or lasers 1014). Therefore one can conclude thaton average at any given time there is less than one neutron present in the wholeinterferometer. This phenomenon is called self-interference.

The detection of neutrons takes place via nuclear reactions. Neutrons have massbut no electrical charge. Because of this they cannot directly produce ionizationin a detector, and therefore cannot be directly detected. This means that neutrondetectors must rely upon a conversion process where an incident neutron interactswith a nucleus to produce a secondary charged particle. These charged particlesare then directly detected and the presence of neutrons can be deduced. The mostcommon reaction used in neutron detection today is n + 3He → p + 3H + 765keVwhere both the proton and the hydrogen are detected by gas filled proportionalcounters.

46

5.2. Neutron optics CHAPTER 5. Neutron-Interferometry

5.2 Neutron optics

The investigation of interference effects is one of the most interesting parts of QM.Neutrons are very suited for these kinds of experiments because they are ratherrobust against disturbances. In the following we discuss so called split beam inter-ferometers, which are based either on the division of wavefronts (analog to Young’sdoubleslit interferometer) or on the division of amplitudes (analog to the Mach-Zehnder or Michelson interferometer).

5.2.1 Various types of interferometers

We can distinguish at least 4 types of neutron interferometers shown in Fig.5.1.

(a)

(b)(c)

(d)

Figure 5.1: Various types on neutron interferometers. (a) Single slit diffraction, (b) Diffrac-

tion from gratings, (c) Perfect-crystal interferometer, (d) Spin-echo system.

(Source: [21])

Single slit diffraction and bi-prism deflection This was the first neutron in-terferometer based on wavefront division invented by Maier-Leibnitz and Springerin 1962. The beam separation was too small (∼ 50− 100μm) for carrying outsubstantial interference investigations.

Interferometry based on diffraction from gratings The gratings can be pro-duced mechanically or light induced. This method is used for very slow neu-trons with long wavelengths and was invented by Joffe, Zabiyakan and Drabkinin 1985.

47

CHAPTER 5. Neutron-Interferometry 5.2. Neutron optics

Perfect-crystal interferometer This interferometer represents the standard methodin neutron optics. It was invented in Vienna by Rauch, Bonse and Treimer in1974 based on the perfect silicon crystal interferometer for X-rays (Bonse andHart, 1964). For a further discussion see section 5.2.2.

Spin-echo systems, polarimetry, Larmor interferometry Invented in 1972 byMezei this represents an interferometer in momentum space where it is notnecessary to have spatially separated beams. The beam consists of a coherentsuperposition of spin-up and spin-down states, which have slightly differentmomenta along the beam path.

5.2.2 Perfect crystal interferometry

This type of interferometer is topological identical to the Mach-Zehnder Interfer-ometer for photons. The beam splitting is based on dynamical Bragg diffractioneffects at perfect crystals. The interferometer is made out of a perfect single siliconcrystal which must be absolutely free of dislocations and distortions. The reflectinglattice planes are arranged throughout the crystal with a precision comparable tothe lattice parameter. The beam separation is about 5cm.Two examples of such perfect crystal interferometers are shown in Fig.5.2. Hereonly types with 3 crystal plates are shown but there are also types with 4 or moreplates common.

Figure 5.2: Two examples of perfect crystal neutron interferometers. (Source: Y.

Hasegawa)

We consider a schematic picture (Fig.5.3) of the interferometer and deduce someessential features.The wave functions of the two paths I and II are identical |ψ0

I 〉 = |ψ0II〉 in amplitude

and phase in the forward direction (O-beam). This can be understood from sym-metry considerations because the beams are either transmitted-reflected-reflected(TRR) or reflected-reflected-transmitted (RRT). The intensity in the O-beam isgiven by

IO = |ψ0I + ψ0

II|2= 1 , (5.2.1)

if we assume |ψ0I |= |ψ0

I |= 12 .

When we insert a phase shifter in one beam path1 the phase difference between the1The same effect can be reached by two phase shifters in each path differing in thickness.

48

5.3. Entanglement in neutron-interferometry CHAPTER 5. Neutron-Interferometry

Figure 5.3: Schematic picture of a 3-plate neutron interferometer.

two paths is given byχ = −NbcλD , (5.2.2)

where N denotes the particle density, D the thickness and bc the coherent scatteringlength of the phase shifter material, λ denotes the wavelength of the neutrons.The wave functions are modified |ψI〉 = eiχ|ψ0

I 〉 and |ψII〉 = |ψ0II〉 therefore we expect

intensity modulations in the O-beam according to

IO = |ψI + ψII|2= 1 + cosχ . (5.2.3)

The modulations in the H-beam are inverse IH = 1− cos χ such that due to particleconservation we get IO + IH = 1 (see Fig.5.4).

25

20

15

10

5

Inte

nsity

(10

00n/

10s)

3002001000-100-200-300

ΔD (μm)

Figure 5.4: Typical intensity oscillations in the neutron interferometer plotted against the

thickness ΔD of the phase shifter. The lower curve shows the oscillations for

the O-beam, the curve in the middle shows the oscillations for the H-beam and

the sum of both beams is depicted in the upper curve. (Source: Y. Hasegawa)

5.3 Entanglement in neutron-interferometry

The properties of entanglement (see Chapt.2) are studied in many different systems.In this section we want to study entanglement and in particular Bell inequalitieswhich serve as a detection tool for entanglement within neutron interferometry.

49

CHAPTER 5. Neutron-Interferometry 5.3. Entanglement in neutron-interferometry

5.3.1 Definition of entanglement

Entanglement is a property of quantum states which are defined over multipartiteHilbertspaces realized by tensor products. For the mathematical studies it does notmatter which physical kind the Hilbert spaces are of. For the physical realizationsand interpretations this is somehow the crucial point.Therefore we can establish “mathematical” entanglement between two Hilbertspacesbelonging to the same particle but describing different degrees of freedom (in thisparticular case spin and position). The physical consequences of this kind of en-tanglement are different from that of a system consisting of two particles where thespinor degrees of freedom are entangled. In the later case we can talk for instanceabout the nonlocal features of entanglement including “spooky action at distance”and the EPR-Paradox (see Sect.2.5). The former case is a little bit involved and notso well known: we talk about the contextuality of QM (see Sect.5.3.2). Contextualityis a more general concept than nonlocality.

5.3.2 Contextuality

The Kochen-Specker (KS) theorem [16, 92, 104] arises naturally due to the question:May we think that the dynamical variables have a definite (but unknown) value beforethey are measured? The answer states that in a given situation the values of oneobservable depends on which commuting set of observables is being measured alongwith it. That means QM is contextual (see e.g. the book of Ballentine [13]).The assumption of non-contextual values for commutative observables leads to acontradiction which is states by a variety of “no-go theorems”, see e.g. Refs. [67,92, 115].In principle contextuality does not rely upon statistical predictions thus for theexperimental test of the theorem one has to measure the outcomes of dynamicalvariables with infinite precession which is impossible. Thus it was claimed that anymodel based on non-contextuality can not be discriminated from QM.The solution to this problem is to find statistical prediction of non-contextual theo-ries and compare them with QM predictions, where we can account for experimentalimperfections. This is done by Basu et al. [14].

5.3.3 Experiment

Bell-like inequality

We consider a single spin-12 particle with the bipartite Hilbertspace H = Hspin ⊗

Hpath where the first part corresponds to the spinor degrees of freedom and thesecond one to the spatial degrees of freedom. Both subspaces are assumed to be 2-dimensional and disjoint. We use two pairs of (dichotomic) observables As

1, As2 and

Bp1 , Bp

2 such that Asi ∈ B(Hspin) and Bp

i ∈ B(Hpath) and the commutator relation issatisfied [As

i , Bpj ] = 0. We can establish a Bell-like inequality according to Ref.[14]

|〈As1B

p1〉+ 〈As

1Bp2〉+ 〈As

2Bp1〉+ 〈As

2Bp2〉| = S ≤ 2 , (5.3.1)

50

5.3. Entanglement in neutron-interferometry CHAPTER 5. Neutron-Interferometry

where the S-function consists of a combination of expectation values. In non-contextual theories the inequality holds, whereas in contextual theories it is violatedfor special configurations.The observables are specified by certain parameters: in case of the spinor degree offreedom it is the angle α of the spin analyzer, in case of the spatial degree of freedomthe phase shift χ. We can write the observables in the spectral decomposition

Asi (αi) = P s

+(αi)− P s−(αi) and Bp

i (χ) = P p+(χi)− P p

−(χi) , (5.3.2)

where P s±(αi) and P p±(χi) denote the projection operators onto the spin and the

path states, 12(|⇒〉 ± eiα|⇐〉) and 1

2(|I〉 ± eiχ|II〉), respectively.The theoretical expectation value in a certain state |Ψ〉

E(α, χ) = 〈As(α)Bp(χ)〉 = 〈Ψ|As(α)Bp(χ)|Ψ〉 , (5.3.3)

is experimentally represented by

E(α, χ) =N(α, χ)−N(α, χ + π)−N(α + π, χ) + N(α + π, χ + π)N(α, χ) + N(α, χ + π) + N(α + π, χ) + N(α + π, χ + π)

, (5.3.4)

where N(α, χ) denote count rates which are related to joint probabilities

N(α, χ) = 〈Ψ|P s+(α)P p

+(χ)|Ψ〉 . (5.3.5)

Entangled state

The maximal discrepancy between non-contextual theories and QM arises for amaximally entangled state. A special kind of spin turner allows to construct amaximally entangled Bell state

|Ψ〉 =1√2(|⇒〉 ⊗ |I〉+ |⇐〉 ⊗ |II〉) , (5.3.6)

within the neutron interferometer such that the contrast2 of the interference patternis high enough to violate the Bell-like inequality (5.3.1).Joint measurements as described above are performed on this entangled state. Theexplicit quantum mechanical expectation value for the state (5.3.6) is given by

E(α, χ) = cos(α + χ) , (5.3.7)

and predicts a maximal violation of S = 2√

2 of the Bell-like inequality (5.3.1) forthe values

α1 = 0 , α2 =π

2, χ1 =

π

4, χ2 =

3π

4. (5.3.8)

Experimental realization

The experimental details are described in Refs.[70, 71]. The schematic experimentalsetup is shown in Fig.5.5.

2The contrast in the experiment, defined as the difference of the maximal and the minimal

intensity C = Imax−IminImax+Imin

, is lower than 100% due to experimental imperfections. A contrast of at

least 70.7%=√

22

is essential to show a violation of the Bell-like inequality.

51

CHAPTER 5. Neutron-Interferometry 5.3. Entanglement in neutron-interferometry

⇑ ⇒

⇐

Figure 5.5: Schematic experimental setup. (Source: [70])

The state preparation is accomplished by the first plate of the interferometer (cre-ation of the path superposition) and the spin turner (creation of the entangled state,Eq.(5.3.6)). The state manipulation is done by the phase shifter χ and the spin ro-tator α. Afterwards certain properties of the state are detected.The wavelength of the neutrons is λ = 1.92Aand the incoming beam is polarizedalong z-direction. The spin turning device turns the incident polarized spin state|⇑〉 to |⇒〉 in one path and to |⇐〉 in the other beam. The detector efficiency ismore than 99%.For the determination of each expectation value we need 4 different measurementsituations and the respective count rates N(α, χ), N(α, χ + π), N(α + π, χ) andN(α+π, χ+π). The combination of 4 values of expectation values gives the desiredvalue of the S-function.The value of the S-function was calculated to be [70]

Sexp = 2.051± 0.019 > 2 , (5.3.9)

for α1 = 0, α2 = 0.5π, χ1 = 0.79π and χ2 = 1.29π. This shows a clear viola-tion of the Bell-like inequality (5.3.1) of more than 2σ which results from quantumcontextuality.

52

Chapter 6

Neutral K–mesons

Entanglement serves as a resource for quantum information. Therefore it is of greatinterest to study entanglement not only in massless photon systems but also in mas-sive systems like meson – antimeson systems in particle physics. It turns out thatthese meson systems, like the neutral K–mesons or the neutral B–mesons, are verysuitable to study various aspects of particle physics (e.g. CP violation, CPT sym-metries) as well as fundamental questions of quantum theory (e.g. Bell inequalities,decoherence, quantum marking and erasure concepts [42], complementarity [41]).

Pairs of the elementary K–meson particles K0 and K0 are produced at the Φ reso-nance, for instance, in the e+e−–machine DAΦNE at Frascati. As it turns out thesepairs are created exactly in the Bell state |Ψ−〉 and therefore should reveal nonlocalfeatures. The presence of entanglement can indeed be demonstrated by a kind ofBell inequality related to CP violation which is a characteristic feature of the neutralK–meson system, for an overview see Ref.[76].

Similar systems are the entangled beauty mesons, B0B0 pairs, produced at theΥ(4S) resonance (see e.g., Refs. [31, 48, 49]).

In the following we introduce the quantum mechanical description of K–mesons inSect.6.1, compare the K–meson to the qubit system in Sect.6.2 and consider the fullunitary time evolution for K–mesons, Sect.6.3 which is essential due to the fact thatkaons have the property of decay.

6.1 QM description of K–mesons

Mesons are elementary particles containing of quark and antiquark. The K–meson,denoted by |K0〉, consists of a down d quark and a strange s antiquark, the anti-K–meson |K0〉 of a down d antiquark and a strange s quark. Although these particlesare normally described within quantum field theory they can also be described onthe quantum mechanical level. In the following we introduce several properties ofneutral K–mesons.

53

CHAPTER 6. Neutral K–mesons 6.1. QM description of K–mesons

6.1.1 Quantum operations

Strangeness

The strangeness quantum number “counts” the amount of strange quarks. We definethe strangeness operator S acting on K–mesons in the following way

S|K0〉 = +|K0〉 S|K0〉 = −|K0〉 . (6.1.1)

CPT theorem

Three discrete symmetries are known in physics: parity transformations P, chargeconjugation C and time inversion T . All physical processes are invariant under thecombined CPT transformations. As it turns out the neutral kaon system shows CPviolation.

Charge conjugation

The operation of charge conjugation C means that we interchange particle and an-tiparticle which can be expressed by

C|K0〉 = |K0〉 C|K0〉 = |K0〉 . (6.1.2)

Parity

The K–mesons are pseudoscalars. This can be expressed by the parity operation Pacting as

P|K0〉 = −|K0〉 P|K0〉 = −|K0〉 . (6.1.3)

CP transformation

The combined CP transformation therefore gives

CP|K0〉 = −|K0〉 CP|K0〉 = −|K0〉 . (6.1.4)

The eigenstates of the CP operator

CP|K01 〉 = +|K0

1 〉 CP|K02 〉 = −|K0

2 〉 , (6.1.5)

are superpositions of the states |K0〉 and |K0〉

|K01 〉 =

1√2

(|K0〉 − |K0〉

)|K0

2 〉 =1√2

(|K0〉+ |K0〉

). (6.1.6)

The CP quantum number is conserved in strong interactions.

54

6.1. QM description of K–mesons CHAPTER 6. Neutral K–mesons

CP violation

In weak interactions CP violation is observed. We can identify two physical statesdiffering in mass and lifetime1, called short and long-lived states |KS〉 and |KL〉

|KS〉 =1N

(p|K0〉 − q|K0〉

)|KL〉 =

1N

(p|K0〉+ q|K0〉

). (6.1.7)

They are defined with certain weights p and q which are related to the complexCP violating parameter ε via

p = 1 + ε , q = 1− ε , (6.1.8)

and the normalization N2 = |p|2 + |q|2. The amount of CP violation is small|ε| ≈ 10−3.K–mesons decay into π–mesons. As it turns out the short–lived K–meson decaysdominantly into KS −→ 2π and the long–lived K–meson decays dominantly intoKL −→ 3π. But it is observed that a small amount of decays is KL −→ 2π which isdue to CP violation.

6.1.2 Strangeness oscillation

The time evolution of K–mesons can be described by a non-hermitian Hamiltonian

H = M − i

2Γ , (6.1.9)

where the hermitian part M describes the ordinary evolution generated by the mass-term and the non-hermitian part i

2 Γ accounts for the decay. The eigenstates of theso-called “effective mass” Hamiltonian are given by

H |KS,L〉 = λS,L |KS,L〉 , (6.1.10)

where the eigenvalues λS,L contain the mass mS,L and the decay width ΓS,L = 1τS,L

of the short and the long-lived kaons

λS,L = mS,L −i

2ΓS,L . (6.1.11)

The time evolution of the “mass eigenstates” is given by the Wigner–Weisskopfapproximation

|KS,L(t)〉 = e−iλS,Lt|KS,L〉 . (6.1.12)

It turns out that the kaon oscillates between |K0〉 and |K0〉 before it decays. Weconsider the time evolution of the “strangeness eigenstates” |K0〉 and |K0〉 whichis given by

|K0(t)〉 = g+(t)|K0〉+ q

pg−(t)|K0〉

|K0(t)〉 =p

qg−(t)|K0〉+ g+(t)|K0〉 ,

(6.1.13)

1Mass: Δm = mL − mS = 3.49 × 10−6 eV, lifetime: τS = 0.89 × 10−10s, τL = 5.17 × 10−8 s.

55

CHAPTER 6. Neutral K–mesons 6.2. Quasi-spin formalism

where

g±(t) =12

[±e−iλSt + e−iλLt

]. (6.1.14)

Suppose at t = 0 we produce a K0 beam. Then the probability for finding a K0 orK0 in the beam after a time t is calculated by

∣∣〈K0|K0(t)〉∣∣2 =

14{e−ΓSt + e−ΓLt + 2 e−Γt cos(Δmt)

}∣∣〈K0|K0(t)〉

∣∣2 =14|q|2|p|2

{e−ΓSt + e−ΓLt − 2 e−Γt cos(Δmt)

},

(6.1.15)

where Δm = mL −mS and Γ = 12(ΓL + ΓS) . We see that the K0 beam oscillates

with a frequency Δm/2π. The oscillations are visible at times of the order of afew τS , before all KS have died out leaving only the KL in the beam. In a beamwhich contains only K0 mesons at time t = 0 the K0 will appear far from theproduction source with equal probability as the K0 meson. A similar feature occurswhen starting with a K0 beam.

6.2 Quasi-spin formalism

6.2.1 Single kaons

Due to the fact that the neutral kaon system is a two-level system we can draw aconnection to the qubit system introduced in Sect.2.1.3. This we will call “quasi-spin” picture [34], originally introduced by Lee and Wu [97] and Lipkin [100].Eigenstates of the σ3 operator, spin up |⇑〉 and spin down |⇓〉 along the z-direction,correspond to the states |K0〉 and |K0〉. We can identify the strangeness operatorS with the Pauli matrix σ3

S

(K0

K0

)= σ3

(K0

K0

)=(

K0

−K0

), (6.2.1)

and the CP operator with −σ1

CP(

K0

K0

)= −σ1

(K0

K0

)=(−K0

−K0

). (6.2.2)

Thus CP violation is proportional to the operator σ2. The Hamiltonian (6.1.9) canbe written as

H = a · 1 +b · σ =(

a + b3 b1 − ib2

b1 + ib2 a− b3

), (6.2.3)

where a, bi ∈ C and are given by

b1 = b cos α, b2 = b sinα, b3 = 0

a =12(λL + λS), b =

12(λL − λS) .

(6.2.4)

56

6.3. Unitary time evolution CHAPTER 6. Neutral K–mesons

The phase α is related to the CP parameter ε by

eiα =1− ε

1 + ε=

q

p. (6.2.5)

The analogy to the qubit system is summarized in the following table:

strangeness eigenstates mass eigenstates CP eigenstatesK–meson |K0〉, |K0〉 |KS〉, |KL〉 |K0

1 〉, |K02 〉

qubit, spin-12 |⇑〉z, |⇓〉z |⇑〉y, |⇓〉y |⇑〉x, |⇓〉x

Note, that the analogy is not perfect because firstly it is not possible to mea-sure arbitrary superpositions of kaon states like α|K0〉 + β|K0〉 and secondly theCP eigenstates are not physical states.

6.2.2 Entangled kaons

It is possible to construct entangled states with neutral kaons which are really pro-duced in physical experiments, such as the e+e−–collider at DAΦNE, Frascati or thepp–collider at LEAR, CERN.The entangled neutral kaon state can be described at time t = 0 by

|ψ(t = 0)〉 = |Ψ−〉 =1√2

(|K0〉l ⊗|K0〉r − |K0〉l ⊗ |K0〉r

), (6.2.6)

which is the Bell singlet states and can be re-written in the KSKL-basis

|ψ(t = 0)〉 = |Ψ−〉 =N2

2√

2pq

(|KS〉l ⊗|KL〉r − |KL〉l ⊗ |KS〉r

). (6.2.7)

Like in common entanglement experiments the neutral kaons fly apart and are de-tected on the left (l) and right (r) side of the source. During their propagation theproperty of strangeness represented by the eigenstates K0K0 causes the oscillationsand the property of mass via the mass eigenstates KS , KL is responsible for thedecay. This is an important difference to the case of qubits which are stable.

6.3 Unitary time evolution

The time evolution of a quantum system has to be unitary that means

|ψ(t)〉 = U(t, t0)|ψ(t0)〉 . (6.3.1)

The Wigner-Weisskopf approximation, Eq.(6.1.12), is not unitary because it doesnot include the fact that kaons can decay but only the strangeness oscillations.At any instant t the state of a kaon decays to a specific final state with a probabilityproportional to the absolute square of the transition matrix element. Because ofunitarity of the time evolution the norm of the total state must be conserved. Thismeans that the decrease in the norm of the kaon state must be compensated by theincrease in the norm of the final states.

57

CHAPTER 6. Neutral K–mesons 6.3. Unitary time evolution

We consider a formalism introduced by Ghirardi, Grassi and Weber [62] and gen-eralized to arbitrary quasi-spins by Bertlmann and Hiesmayr [34] which includesalso the decay products. We describe a complete evolution of mass eigenstates by aunitary operator U(t, 0) whose effect can be written as

|KS,L(t)〉 = U(t, 0) |KS,L〉 = e−iλS,Lt |KS,L〉+ |ΩS,L(t)〉 , (6.3.2)

where |ΩS,L(t)〉 denotes the state of all decay products. For the transition amplitudesof the decay product states the following relations hold

〈ΩS(t)|ΩS(t)〉 = 1− e−ΓSt (6.3.3)〈ΩL(t)|ΩL(t)〉 = 1− e−ΓLt (6.3.4)〈ΩL(t)|ΩS(t)〉 = 〈KL|KS〉(1− eiΔmte−Γt) (6.3.5)〈KS,L|ΩS(t)〉 = 〈KS,L|ΩL(t)〉 = 0 . (6.3.6)

The mass eigenstates |KS,L〉 are normalized but due to CP violation not orthogonal

〈KL|KS〉 =2Re{ε}1 + |ε|2 . (6.3.7)

For an entangled state of kaons we start at time t = 0 from the entangled state givenin the KSKL basis (6.2.7)

|ψ(t = 0)〉 =N2

2√

2pq

(|KS〉l ⊗ |KL〉r − |KL〉l ⊗ |KS〉r

). (6.3.8)

We get the time evolved state at time t by applying the unitary operator

U(t, 0) = Ul(t, 0)⊗ Ur(t, 0) , (6.3.9)

where the operators Ul(t, 0) and Ur(t, 0) act on the space of the left and of the rightkaon according to the time evolution (6.3.2).

58

Chapter 7

Spin-Geometry

Geometric considerations of the state space can provide further insight into someunsolved problems beside the fact that geometric pictures are attracting and illus-trative. For example the volume of the set of separable states turns out to decreasewith increasing dimension of the system [161] but depends on the used measure inthe space of density matrices [160]. Kus and Zyczkowski introduced a classificationof density matrices [95] which is in close relation to the Hopf fibration discussed inSect.14.3.3.It is tempting to find a geometric picture for the whole space of density matrices.But this is nearly impossible because even for the simplest case of a qubit the wholestate space is 4 dimensional and therefore hard to imagine. But by restricting toseveral properties one can find appealing pictures and visualizations for the statespace. The most prominent picture for sure is the Bloch sphere representation wherethe phase freedom of states is ignored. A kind of generalization of the Bloch spherepicture for two qubits is discussed in Sect.14.3. In the following sections we presentthe so-called spin geometric picture which allows for another visualization of the twoqubit system which we are going to use further on in Sect.13.We are investigating the concept of the Hilbert-Schmidt space in Sect.7.1 and discussit in detail for two qubits (Sect.7.2). In the last Sect.7.3 we introduce the picture ofspin geometry for two qubits which is an instrument of visualizing the correlationsfor two qubit states.

7.1 Hilbert-Schmidt space

In Sect.2.1.1 we have touched the notion of the Hilbert-Schmidt space which wewant to investigate further in this section.The algebra of bounded operators B(H) acting on the Hilbert space H contains theidentity operator � ∈ B(H) and is closed under products and adjoints

A, B ∈ B(H) =⇒ (A ·B) ∈ B(H) , A ∈ B(H) =⇒ A† ∈ B(H) . (7.1.1)

59

CHAPTER 7. Spin-Geometry 7.1. Hilbert-Schmidt space

The induced operator or spectral norm, given by

‖A‖∞ = sup|ψ|=1|Aψ| , (7.1.2)

is inherited form the vector norm |ψ|2 = 〈ψ|ψ〉 defined on the Hilbert space H. Thisalso defines the ordering of operators by

A ≥ B ⇐⇒ 〈ψ, Aψ〉 ≥ 〈ψ, Bψ〉 ∀ψ ∈ H . (7.1.3)

The algebra of bounded operators B(H) forms a Hilbert space which is calledHilbert-Schmidt space. For finite dimensional systems, where the Hilbert space isgiven by H = Cd, we get for the Hilbert-Schmidt space B(H) the algebra of complexd× d matrices.The Hilbert-Schmidt space also forms a Hilbert space concerning the scalar product

〈A, B〉 = TrA†B , (7.1.4)

for A, B ∈ B(H). We can define the trace or Hilbert-Schmidt norm (also calledFrobenius norm) which is different from the operator norm and given by

‖A‖ =√〈A, A〉 . (7.1.5)

Density operators ρ are selfadjoint operators ρ† = ρ which are normalized Trρ = 1.The trace norm provides us with a distance measure, the Hilbert-Schmidt distance

DHS(A ‖ B) = ‖B −A‖2 , (7.1.6)

which can be seen as an entanglement measure (see Sect.2.4.6 and Refs.[111, 150]).The so-called Hilbert-Schmidt measure given by

EHS(ρ) = minσ∈S

DHS(ρ ‖ σ) = minσ∈S‖σ − ρ‖2 , (7.1.7)

specifies the minimal distance between an entangled state ρ and the convex set ofseparable states S (see Fig.7.1).

σ0

ρ EHS(ρ) = DHS(ρ ‖ σ0)

S

Figure 7.1: The Hilbert-Schmidt distance between the closest separable state σ0 ∈ S and

the entangled state ρ defines a measure of entanglement, the Hilbert-Schmidt

measure.

The Hilbert-Schmidt space for finite dimensional systems can be equipped with abasis in the following way. The space of trace free matrices is the orthocomplement

60

7.2. Hilbert-Schmidt space for two qubits CHAPTER 7. Spin-Geometry

�⊥ of the unit operator �. We choose a basis of matrices {ei}d2−1

i=1 in �⊥ such that〈ei, ej〉 = δij . Then any self adjoint operator can be written as

A = a0� +d2−1∑i=1

aiei , (7.1.8)

and density operators in addition satisfy the normalization condition Trρ = 1.For example, we can consider the Hilbert space H = C2 for a single qubit. The setof 4 basis operators for the Hilbert-Schmidt space is given by {�, σi}, the identityand the Pauli operators. Therefore we can write any operator as

A = a0� + a · σ , (7.1.9)

where a = (a1, a2, a3)T ∈ R3. If the operator A represents a state ρ (density matrix)then we have to consider the normalization condition Trρ = 1 and therefore get

ρ =12� + a · σ , (7.1.10)

which gives exactly the Bloch sphere representation for a = 12r (see Sect.2.1.3).

7.2 Hilbert-Schmidt space for two qubits

For two qubits the Hilbert space is given by H = C2⊗C2. Thus the Hilbert-Schmidtspace is the algebra of 4× 4 complex matrices and the basis consists of 16 elements.These 16 basis operators can be chosen as product operators of the single qubit basisoperators {�⊗ �, σi ⊗ �,�⊗ σj , σi ⊗ σj}. A general operator is then given by

A = a�⊗ � + aσ ⊗ � +b�⊗ σ + oij σi ⊗ σj , (7.2.1)

which gives for the density matrix

ρ =14(�⊗ � + m σ ⊗ � + n �⊗ σ + cij σi ⊗ σj) . (7.2.2)

The operator A represents a density matrix if a = 1/4 and∑

i(a2i +b2

i )+∑

i,j c2ij ≤ 1

16[35]. The parameters of the density matrix can be obtained by

m = Tr(σ ⊗ � ρ) n = Tr(�⊗ σ ρ) cij = Tr(σi ⊗ σj ρ) , (7.2.3)

where the local parameters m,n ∈ R3 determine the reduced density matrices

ρ1 = Tr2ρ =12(� + m · σ) ρ2 = Tr1ρ =

12(� + n · σ) , (7.2.4)

and the real matrix c = (cij) determines the nonlocal correlations. The expectationvalue

E(α, β) = Tr(ρ α · σ ⊗ β · σ) = (α, c β) , (7.2.5)

61

CHAPTER 7. Spin-Geometry 7.3. Spin geometry

is fully determined by the correlation matrix c and given by the scalar product ofthe vector α with the “rotated” vector c β.

In Sect.14.3 we present a method to visualize the state space for pure two qubitstates. The Hilbert space C2 ⊗ C2 = C4 ∼= R8 reduces due to normalization to an

S7. The analog to the Bloch sphere representation S7 S1

−−−→ CP3 � Sn, where theglobal phase freedom is fibred out, is not possible because CP3 is not isomorphic

to a sphere. As it turns out the Hopf map S7 S3

−−−→ S4 allows to represent thestate space in an entanglement sensitive way on the sphere S3 but unfortunatelythis “visualization” is rather hard to imagine.Therefore in the following section we present another way of visualizing the full twoqubit state space – the so-called spin geometry.

7.3 Spin geometry

7.3.1 Equivalence classes of states

Fist we have to reduce the number of parameters describing a state ρ. We notethat the property of separability is invariant under unitary transformations U1⊗U2

(cf. Eq.(2.2.2)). This provides us with an equivalence relation for states with equalproperties concerning separability and entanglement. We can reduce the number ofparameters needed to describe the equivalence class of states by choosing a properrepresentative.To get the transformations of the parameters m, n and c of ρ under the unitarytransformation U1 ⊗ U2 we have to take into account that the groups SU(2) andSO(3) are linked together by a homomorphism. For each unitary transformation Uthere is a unique rotation O such that

Un · σU † = (On) · σ , (7.3.1)

where U ∈ SU(2) and O ∈ SO(3). Therefore the state parameters transform ac-cording to

m′ = O1 m , n′ = O2n , c′ = O1cOT2 . (7.3.2)

It turns out that the orthogonal transformations can be chosen such that the matrixc′ is diagonal. Thus it is sufficient to consider only such states as representativeswhere the c-matrix is diagonal.

7.3.2 Singular value decomposition

For a general state ρ with correlation matrix c we can determine the representative ofthe corresponding equivalence class via the singular value decomposition (SVD).This is a kind of diagonalization mechanism also valid for non-square matrices anddifferent to the eigenvalue decomposition (EVD). In the following we consider theSVD for square matrices which is given by

c′ = O1cOT2 , (7.3.3)

62

7.3. Spin geometry CHAPTER 7. Spin-Geometry

where O1OT1 = OT

1 O1 = � = O2OT2 = OT

2 O2. The singular values of the matrix c,which form the diagonal matrix c′ = diag(c1, c2, c3), are given by the square rootsof the eigenvalues of the matrix ccT or cT c. The singular values are always realand can be arranged to form a 3 dimensional vector c = (c1, c2, c3)T which is calledcorrelation vector. The spin geometry picture consists of all possible correlationvectors c.

Difference SVD ←→ EVD

Note that the EVD is different from the SVD. Suppose we have a state

|ψ(t)〉 =1√2(eiθ|01〉 − e−iθ|10〉) , (7.3.4)

where the parameters are given by

m = n = 0 c =

⎛⎝− cos θ sin θ 0− sin θ − cos θ 0

0 0 1

⎞⎠ , (7.3.5)

and c is a real matrix. The EVD gives complex eigenvaluesλ = (−eiθ,−e−iθ,−1)T , (7.3.6)

whereas the SVD gives reals singular values

c = (−1,−1,−1)T . (7.3.7)

Thus we see that dynamical contributions do not appear in the spin geometry pic-ture.

7.3.3 Geometric picture

The geometric picture is introduced and discussed by the Horodeckis [81, 85] andalso used by Vollbrecht and Werner [145] and Bertlmann, Narnhofer and Thirring[35].As we have seen it is possible to create the correlation vector c out of the correlationmatrix c and draw a 3 dimensional picture. The different stages from the densitymatrix to the spin geometry picture are summarized below.

ρ ←→ m,n, cij ←→ c = (cij) ←→ c = (ck)

For example the coordinates ck of the 4 Bell states are given by

|Ψ−〉 =1√2(|01〉 − |10〉) c = (−1,−1,−1)T

|Ψ+〉 =1√2(|01〉+ |10〉) c = (+1, +1,−1)T

|Φ−〉 =1√2(|00〉 − |11〉) c = (−1, +1, +1)T

|Φ+〉 =1√2(|00〉+ |11〉) c = (+1,−1, +1)T

(7.3.8)

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CHAPTER 7. Spin-Geometry 7.3. Spin geometry

and for the identity we get c = (0, 0, 0)T .The correlation vectors of the Bell states can be drawn into the spin geometrypicture. They form the corners of a tetrahedron which characterizes all possiblestates [85] (see Fig.7.2).

��

��

��

��

Figure 7.2: The tetrahedron of all possible states.

The shape of the tetrahedron is obtained by the positivity condition ρ ≥ 0 whichgives 4 inequalities that define the tetrahedron

1− c1 − c2 − c3 ≥ 0 1− c1 + c2 + c3 ≥ 01 + c1 − c2 + c3 ≥ 0 1 + c1 + c2 − c3 ≥ 0 .

(7.3.9)

The partial transposition condition (cf. Sect.2.3.2) leads to a reflection of the tetra-hedron. Thus the set of separable state is given by the intersection of both tetrahe-drons which results in an octahedron [85] (see Fig.7.3).

Figure 7.3: The inverted tetrahedron and the octahedron of the separable states.

64

Chapter 8

Quaternions and Hopf-fibration

Quaternions are the extension of complex numbers to higher dimensions. In 1843W.R. Hamilton discovered that the extension to 3 dimensions is not possible butin 4 dimensions we arrive at the division algebra of quaternions which is non-commutative. This is the price for going to higher dimensions.Today quaternions are used in computer graphics, control theory, signal processing,attitude control and orbital mechanics mainly for representing rotations in threedimensions (see Ref.[94]). The reason is that combining many quaternion transfor-mations is numerically stabler than combining many matrix transformations.As it turns out it is possible to reformulate QM in terms of quaternions insteadof complex numbers (see for instance the book of Adler[1]). The mathematicalreason is obviously to gain more insight into the underlying structures. The physicalmotivation can be seen in the quaternionic version of quantum field theory and anattempt to formulate a unified theory of all fundamental forces.The Hopf fibration is an important tool in the mathematical theory of homotopiessince it provides a connection between two spheres of different dimension and thusrelates the properties of the two spaces. Generalizations of the Hopf fibration areessential in algebraic topology.After introducing quaternions and discussing possible representations we introducethe description of spheres in Sect.8.2. We proceed on to the description of rotationswhich reveals a close relation between quaternions and rotations (Sect.8.3) whichbrings us to the general quaternionic conjugation map in Sect.8.4. The Hopf maparises naturally out of the conjugation map. The last Sect.8.5 is devoted to thestereographic projection which is an important tool for visualizing the Hopf map(see Sect.14.2).

8.1 Quaternions

8.1.1 Properties

The quaternions can be seen as a linear vector space over R with dimension 4 whichforms a skew field or division algebra by introducing a multiplication according to

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CHAPTER 8. Quaternions and Hopf-fibration 8.1. Quaternions

the rule stated in (8.1.1).The set of quaternions is denoted by H ∼= R4 and a general element can be writtenas q = a+bi+cj+dk where a, b, c, d ∈ R. Thereby we have introduced the imaginaryunits i, j and k which satisfy the multiplication rules

i2 = j2 = k2 =− 1ij = −ji = k , jk = −kj =i , ki = −ik = j .

(8.1.1)

We see that order matters due to the non-commutativity of the algebra concerningmultiplication q · q′ �= q′ · q. A quaternion consists of a real part a, an i-imaginarypart b, a j-imaginary part c and a k-imaginary part d.We can define the

• conjugate q = a− bi− cj − dk

• norm ‖q‖2 = qq = qq = a2 + b2 + c2 + d2

• inverse q−1 = q‖q‖2 .

The following equalities hold:‖q‖ = ‖q‖, ‖q1 q2‖ = ‖q1‖ · ‖q2‖, q = q, (q1 q2) = q2 q1, (q1 + q2) = q1 + q2.A unit quaternion has norm ‖q‖2 = 1 that means q−1 = q. The set of all unitquaternions forms a subset of R4, the 3-dimensional sphereS3 = {(a, b, c, d) ∈ R4 | a2 + b2 + c2 + d2 = 1} (see Sect.8.2).

8.1.2 Representations

Quaternions can be represented in several ways. In the following we show some kindsof representation. In contrast to the (first 3) vector-like representations we have the(last 2) matrix representations where quaternionic addition and multiplication corre-sponds to matrix addition and multiplication (quaternion-matrix homomorphism).

Standard representation

We have already introduced the standard-representation consisting of 4 real param-eters

q = a + bi + cj + dk = (a, b, c, d) , (8.1.2)

where a, b, c, d ∈ R. Addition and multiplication are defined pointwise

q + q′ = ((a + a′), (b + b′), (c + c′), (d + d′))q · q′ = ((aa′ − bb′ − cc′ − dd′), (ab′ + ba′ + cd′ − dc′),

(ac′ + ca′ − bd′ + db′), (ad′ + da′ + bc′ − cb′)) ,

(8.1.3)

for q = (a, b, c, d) and q′ = (a′, b′, c′, d′).

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8.1. Quaternions CHAPTER 8. Quaternions and Hopf-fibration

Vector representation

A quaternion can be written as a sum of a real number and a 3 vector

q = a + u , (8.1.4)

where u = (b, c, d)T . We call S(q) = a the scalar part and V (q) = u = (b, c, d)T thevectorial part. The conjugate is given by q = a−u and the norm by ‖q‖2 = a2+ |u|2.It holds that S(q) = 1

2(q + q) and V (q) = 12(q − q). We can have real quaternions

where V (q) = 0 or pure imaginary quaternions where S(q) = 0.In this representation addition and multiplication are defined as

q + q′ = (a + u) + (a′ + u′) = (a + a′) + (u + u′)q · q′ = (a + u) · (a′ + u′) = (aa′ − u · u′)︸︷︷︸

scalar

+ (au′ + a′u + u× u′)︸︷︷︸vector

, (8.1.5)

where the scalar product u · u′ and the cross product u× u′ of vector algebra arisenaturally.

Complex representation

In this representation we need 2 complex parameters to describe a general quaternion

q = z + wj = (z, w) , (8.1.6)

where z, w ∈ C. The two parameters (z, w) are related to the standard representation(a, b, c, d) by z = a + bi and w = c + di. We define the conjugate by q = z∗ − wj =(z∗,−w) where z∗ = a − bi denotes complex conjugation. The norm is given by‖q‖2 = |z|2 + |w|2 where |z|2 = zz∗.Addition and multiplication can be defined

q + q′ = (z + wj) + (z′ + w′j) = (z + z′) + (w + w′)jq · q′ = (z + wj) · (z′ + w′j) = (zz′ − ww′∗) + (zw′ + wz′∗)j ,

(8.1.7)

where addition is pointwise but multiplication not.

2× 2 matrix representation

Now we consider the first kind of matrix representation. We construct 4 complex2× 2 basic matrices

�2×2 =(

1 00 1

)� =

(i 00 −i

)� =

(0 1−1 0

)� =

(0 ii 0

), (8.1.8)

and express a quaternion as a linear combination

q = a�2×2 + b� + c� + d� . (8.1.9)

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CHAPTER 8. Quaternions and Hopf-fibration 8.2. Spheres

In matrix form this gives

q =(

a + bi c + di−c + di a− bi

)=: A

=(

z w−w∗ z∗

)=: B ,

(8.1.10)

where we immediately realize the connection to the complex representation of quater-nions with z and w.The conjugate is given by the adjoint of the corresponding matrix q = A† = B†

and the norm is the determinant of the matrix ‖q‖2 = det A = det B. Addition andmultiplication correspond to the ordinary matrix operations.If we restrict ourselves to unit quaternions, this representation provides the isomor-phism between S3 and SU(2) (cf. Sect.8.2).

4× 4 matrix representation

There also exists a representation that uses 4 real 4× 4 matrices given by

� =

⎛⎜⎜⎜⎜⎝0 −1 0 0

1 0 0 0

0 0 0 −1

0 0 1 0

⎞⎟⎟⎟⎟⎠ � =

⎛⎜⎜⎜⎜⎝0 0 0 −1

0 0 −1 0

0 1 0 0

1 0 0 0

⎞⎟⎟⎟⎟⎠ � =

⎛⎜⎜⎜⎜⎝0 0 1 0

0 0 0 −1

−1 0 0 0

0 1 0 0

⎞⎟⎟⎟⎟⎠ . (8.1.11)

A general quaternion can be written as

q = a�4×4 + b� + c� + d� , (8.1.12)

which gives in matrix notation

q =

⎛⎜⎜⎝a −b d −cb a −c −d−d c a −bc d b a

⎞⎟⎟⎠ =: C . (8.1.13)

The conjugate corresponds to transposition q = CT and addition and multiplicationare equivalent to the ordinary matrix operations.

8.2 Spheres

The n-dimensional sphere Sn is defined as the set of all points in n+1-dimensionalreal space with a radius r from a center point c

Sn ={

x = (x1, . . . , xn+1) ∈ Rn+1 |n+1∑i=1

(xi − ci)2 = r2

}⊂ Rn+1 . (8.2.1)

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8.3. Quaternions and rotations CHAPTER 8. Quaternions and Hopf-fibration

Most of the time we work with the specifications r = 1 and c = 0 which gives theunit sphere

Sn ={

x = (x1, . . . , xn+1) ∈ Rn+1 |n+1∑i=1

x2i = 1

}⊂ Rn+1 . (8.2.2)

For example, S0 consists of the two points {−1, 1} in R1, S1 is the unit circle in R2

and S2 is the usually known sphere in R3.One can also define the interior of a sphere, which is called disk or ball. We definethe n-dimensional ball Bn to be the set of all points inside the (n− 1)-sphere

Bn ={

x = (x1, . . . , xn) ∈ Rn |n∑

i=1

x2i ≤ 1

}⊂ Rn . (8.2.3)

The 3-dimensional unit sphere S3, given by

S3 ={

x = (x1, x2, x3, x4) ∈ R4 : x21 + x2

2 + x23 + x2

4 = 1}

, (8.2.4)

is a subset of R4. We can define the 3-sphere also as a subset of C2 by

S3 ={

(c1, c2) ∈ C2 : |c1|2 + |c2|2 = 1}

, (8.2.5)

or as a subset of H, the set of all unit quaternions

S3 ={

q ∈ H : ‖q‖2 = 1}

. (8.2.6)

We can establish an isomorphism between S3 and the special unitary matrix groupSU(2) = {A ∈ (2× 2)(C) | A†A = AA† = �, det A = 1} by the mapping

S3 ∼= SU(2) : (x1, x2, x3, x4) �→(

x1 + ix2 x3 + ix4

−x3 + ix4 x1 − ix2

). (8.2.7)

Note that this isomorphism is used for the 2×2 matrix representation of quaternionsintroduced in Sect.8.1.2.

8.3 Quaternions and rotations

Rotations in n-dimensional space are linear transformations that preserve i) thelength of vectors, ii) the angle between vectors and iii) the orientation. Furthermorethey are nonabelian that means the order of the transformations is important.Rotations can be represented by n × n orthogonal real matrices R. The orthogo-nality RT R = � assures that the inner product is preserved Ru · Rv = u · v for∀ u,v ∈ Rn which implies the first two properties. These matrices form a groupcalled orthogonal group O(n) = {R ∈ (n× n)(R) | RT R = �}. The orientation ispreserved by assuming det R = 1, which defines a subgroup of O(n) called specialorthogonal group SO(n) = {R ∈ O(n) | det R = 1}.

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CHAPTER 8. Quaternions and Hopf-fibration 8.3. Quaternions and rotations

8.3.1 Rotations in 2 dimensions

The group of matrices R that describe rotations in 2 dimensions is SO(2). We needone parameter θ to characterize the matrix R which describes a counterclockwiserotations of the vector x = (x1, x2)T around the origin by an angle θ given by(

x′1

x′2

)= R(θ)

(x1

x2

)=(

cos θ − sin θsin θ cos θ

)(x1

x2

). (8.3.1)

The rotation can be visualized in the complex plane where we interpret the vectorx in terms of a complex number xc = x1 + ix2. The rotated vector x′

c is a functionf of the initial vector xc which is defined as multiplication with a complex (unit)number c = eiθ = cos θ + i sin θ

x′c = f(xc) = c · xc = eiθxc = (cos θ + i sin θ)(x1 + ix2) = x′

1 + ix′2 . (8.3.2)

8.3.2 Rotations in 3 dimensions

In 3 dimensions rotations are described by the group SO(3). We need 3 parametersto parameterize a general rotation R of a vector

x′ =

⎛⎝x′1

x′2

x′3

⎞⎠ = Rx = R

⎛⎝x1

x2

x3

⎞⎠ . (8.3.3)

There are several kinds of representing a rotation.

Euler angles

Rotations in 3 dimensions can be described by the product of three successive 2dimensional counterclockwise rotations. The standard rotations about the x-, y-and z-axis are given by

Rx =

⎛⎜⎝1 0 0

0 cos θ − sin θ

0 sin θ cos θ

⎞⎟⎠ , Ry =

⎛⎜⎝ cos θ 0 sin θ

0 1 0

− sin θ 0 cos θ

⎞⎟⎠ , Rz =

⎛⎜⎝cos θ − sin θ 0

sin θ cos θ 0

0 0 1

⎞⎟⎠ .

(8.3.4)Commonly, the rotations are carried out first around z-axis (angle φ), then aroundthe x-axis (angle θ) and finally again around the z-axis (angle ψ) where the finalrotation matrix is given by

R = R(φ, θ, ψ) = Rz(ψ)Rx(θ)Rz(φ) =⎛⎜⎝ cos ψ cos φ− sin ψ cos θ sinφ cos ψ sin φ + sin ψ cos θ cos φ sinψ sin θ

− sin ψ cos φ− cos ψ cos θ sin φ − sin ψ sinφ + cos ψ cos θ cos φ cos ψ sin θ

sin θ sin φ − sin θ cos φ cos θ

⎞⎟⎠ ,

(8.3.5)

but numerous other definitions are in use. The angels φ, θ and ψ are called Eulerangles.

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8.3. Quaternions and rotations CHAPTER 8. Quaternions and Hopf-fibration

Axis plus angle

One can also specify a rotation-axis n = (n1, n2, n3)T (which should be normalizedn2

1 + n22 + n2

3 = 1) and an angle of rotation θ which gives for the matrix of rotation

R = R(n, θ) =⎛⎜⎝ cos θ + (1− cos θ)n21 (1− cos θ)n1n2 − sin θn3 (1− cos θ)n1n3 + sin θn2

(1− cos θ)n1n2 + sin θn3 cos θ + (1− cos θ)n22 (1− cos θ)n2n3 − sin θn1

(1− cos θ)n1n3 − sin θn2 (1− cos θ)n2n3 + sin θn1 cos θ + (1− cos θ)n23

⎞⎟⎠ .

(8.3.6)

Quaternions

In an analog way to 2-dimensional rotations we can describe 3-dimensional rotationsby the use of quaternions.We use the vector representation of quaternions (Sect.8.1.2) and the “axis plus angle”representation of rotations (Sect.8.3.2) such that the quaternion q

q = a + u = cosθ

2+ sin

θ

2n , (8.3.7)

represents the rotation defined by the angle θ and the axis n. The vector we want torotate is encoded into a pure imaginary quaternion with zero scalar part xq = 0+x.Then the rotation is described by a function f acting on the quaternion xq by thefollowing formula1

x′q = f(xq) = qxqq

−1 . (8.3.8)

The resulting quaternion x′q is again a quaternion with zero scalar part and rep-

resents the rotated vector x′q = 0 + x′. The transformation (8.3.8) is known as

conjugation of quaternion xq by q. If we restrict ourselves to use unit quaternionsfor the conjugation the transformation reads

x′q = f(xq) = qxq q . (8.3.9)

We have seen, Eq.(8.2.7), that the set of unit quaternions is isomorphic to the groupSU(2) and to the sphere S3 (seen as a topological group). Now we want to establishanother important isomorphism [122].It turns out that the rotation about an angle θ encoded in the quaternion q and therotation about an angle θ + 2π encoded in the quaternion −q both give rise to thesame rotated vector x′

q. Thus a given rotation corresponds to two quaternions, qand −q. This is a consequence of the homomorphism between the unit quaternionsand the rotation group

SO(3) = SU(2)/Z2 = S3/Z2 , (8.3.10)

where Z2 denotes the two-element group consisting of {�,−�}.

1Note the similarities to section 8.3.1.

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CHAPTER 8. Quaternions and Hopf-fibration 8.4. Quaternionic conjugation map

The connection to the matrix representation of a rotation is easily calculated byusing the decomposition q = a + bi + cj + dk which gives

R =

⎛⎝a2 + b2 − c2 − d2 2(bc− ad) 2(ac + bd)2(ad + bc) a2 − b2 + c2 − d2 2(cd− ab)2(bd− ac) 2(ab + cd) a2 − b2 − c2 + d2

⎞⎠ . (8.3.11)

The above considerations are only valid if the axis of rotation passes through theorigin. Otherwise one has to translate the vectors into the origin, rotate them andtranslate them back.

Quaternions are used in computer graphics, control theory, signal processing, andorbital mechanics mainly for representing rotations and orientations in three di-mensional space. The reason is that combining many quaternion transformationsis numerically stabler than combining many matrix transformations. Furthermorethey need smaller computational power than matrices and operations on them suchas composition can be computed more efficiently [94].

8.4 Quaternionic conjugation map

Let us consider the following quaternionic map

c : H×H −→ H

(x, y) �−→ c(x, y) = yxy−1 = yxy(8.4.1)

known as conjugation of the quaternion x by the unit quaternion y (compare Eqs.(8.3.8)and (8.3.9)). We get an explicit form of the conjugation map by setting y =α + βi + γj + δk and x = a + bi + cj + dk which gives

c(x, y) = yxy = (α + βi + γj + δk)(a + bi + cj + dk)(α− βi− γj − δk) =

a +(b(α2 + β2 − γ2 − δ2) + 2c(βγ − αδ) + 2d(αγ + βδ)

)i

+(2b(βγ + αδ) + c(α2 − β2 + γ2 − δ2) + 2d(γδ − αβ)

)j

+(2b(βδ − αγ) + 2c(αβ + γδ) + d(α2 − β2 − γ2 + δ2)

)k

(8.4.2)

There are two possibilities to interpret this map. The first one concerns rotations(see also Sect.8.3.2) and the second one is the famous Hopf map.

8.4.1 Rotations

As we have seen in Sect.8.3.2 we can describe rotations by the conjugation map.Then the quaternion x which represents the vector we want to transform has tobe purely imaginary x = bi + cj + dk. The rotation is encoded in the quaterniony = cos θ

2 +sin θ2(n1i+n2j+n3k) where the scalar part corresponds to the angle θ and

the imaginary part stands for the axes n (|n| = 1). In this case the conjugation map

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8.4. Quaternionic conjugation map CHAPTER 8. Quaternions and Hopf-fibration

c(x, y) gives a pure imaginary quaternion which corresponds to the rotated vector.We consider the quaternion y to be fixed which provides us with the following map

cy(x) : R3 −→ R3

x �−→ cy(x) = yxy

(b, c, d) �−→(b(α2 + β2 − γ2 − δ2) + 2c(βγ − αδ) + 2d(αγ + βδ)

)i

+(2b(βγ + αδ) + c(α2 − β2 + γ2 − δ2) + 2d(γδ − αβ)

)j

+(2b(βδ − αγ) + 2c(αβ + γδ) + d(α2 − β2 − γ2 + δ2)

)k ,

(8.4.3)

where we see clearly that “vectors” (or pure imaginary quaternions) are transformedinto “vectors”. This corresponds exactly to the matrix representation of a rotation,equation (8.3.11)⎛⎝b′

c′

d′

⎞⎠ =

⎛⎝α2 + β2 − γ2 − δ2 2(βγ − αδ) 2(αγ + βδ)2(βγ + αδ) α2 − β2 + γ2 − δ2 2(γδ − αβ)2(βδ − αγ) 2(αβ + γδ) α2 − β2 − γ2 + δ2

⎞⎠⎛⎝bcd

⎞⎠ .

(8.4.4)

8.4.2 Hopf map

The Hopf map is an important tool in pure mathematics as well as in mathematicalphysics [119]. It maps the sphere S3 to the sphere S2 in a certain way and can bedescribed with the help of the conjugation map.Points on the 3-sphere are encoded in the unit quaternion y and the quaternion xis considered to be fixed. The final form of the Hopf map depends on which specificx is used which has to be purely imaginary. The map c(x, y) gives again a purelyimaginary quaternion, which can be identified with a point on S2 ⊂ R3.Suppose we fix the quaternion x in such a way that a = c = d = 0 and b = 1 whichgives x = i. Then we get for the conjugation map

cx(y) : S3 −→ S2

y �−→ cx(y) = yxy

(α, β, γ, δ) �−→ (α2 + β2 − γ2 − δ2)i + 2(βγ + αδ)j + 2(βδ − αγ)k

(8.4.5)

where the final imaginary quaternion represents an element of S2 as a short calcula-tion shows (α2+β2−γ2−δ2)2+(2(βγ+αδ))2+2(βδ−αγ))2 = (α2+β2+γ2+δ2)2 = 1.This map represents exactly the standard Hopf map found by H. Hopf [78] in 1931which is given by

h : S3 −→ S2⎛⎜⎜⎝αβγδ

⎞⎟⎟⎠ �−→⎛⎝α2 + β2 − γ2 − δ2

2(βγ + αδ)2(βδ − αγ)

⎞⎠ ,(8.4.6)

where α2 + β2 + γ2 + δ2 = 1. The Hopf map is continuous and surjective.

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CHAPTER 8. Quaternions and Hopf-fibration 8.4. Quaternionic conjugation map

8.4.3 Hopf fibration

For the Hopf map

h : S3 S1

−−−−−→ S2 , (8.4.7)

it turns out that the pre-image h−1 of a point in S2 is a circle S1 in S3.We can investigate the map in terms of fibre bundle theory (see e.g.[25, 108, 109]).Loosely speaking the fibration of a space E is a map

Π : EF−−−−→M , (8.4.8)

consisting of the base space M , the total space E, a surjective map, the projectionΠ, and the fibre F . All elements of a fibre F are mapped to the same base elementx ∈M such that Π−1(x) = F . Globally a fibre bundle can have a complex structure(e.g. it can be twisted) but locally it always has simple product structure M × F .The Hopf fibration (8.4.7) is nontrivial since globally it is not homeomorphic toS2 × S1. The fibres S1 are called Hopf circles or Hopf fibres.

8.4.4 Pre-image of the Hopf fibration

The Hopf map assigns each (unit) quaternion y a point h(y) = P on the sphere S2

h : S3 −→ S2

y �−→ h(y) = P .(8.4.9)

As we have seen the conjugation map c(x, y) describes on the one hand rotationscy(x) by fixing y corresponding to the rotation matrix and varying the vector x andon the other hand it describes the Hopf map cx(y) = h(y) by fixing the “vector” xand varying the quaternion y. But we can also consider the Hopf map as a kind ofrotation: via the Hopf map we try to find out which points P can be reached byany possible rotation y from the starting point P0 = x.To find out the explicit form of the pre-image of the Hopf fibration we have to searchfor the rotation y that rotates a general point P = (p1, p2, p3) ∈ S2 into the pointP0 = (1, 0, 0) ∈ S2. The simplest way is to find the axis n and the angle θ of rotation(compare with Ref.[101]).Consider Fig.8.1 where the rotation of a point A into a point B on the sphere S2 isshown. The direction of the axis of rotation is given by the vector M = 1

2( A + B)and the angle of rotation θ = π.

Figure 8.1: Rotation of a point A into a point B on the sphere S2. (Source: [101])

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8.5. Stereographic projection CHAPTER 8. Quaternions and Hopf-fibration

Applying this knowledge to our points P0 and P the quaternion which describes therotation is given by

y =1√

2(1 + p1)((1 + p1)i + p2j + p3k) , (8.4.10)

and the pre-image of a point P ∈ S2 is given by

h−1(P ) = {y · eit}0≤t≤2π for p1 �= −1 ,

h−1((−1, 0, 0)) = {k · eit}0≤t≤2π for p1 = −1 ,(8.4.11)

where we have used the fact that the fibres of h are circles S1 described by eit. Theexplicit form of the pre-image is given by

h−1 :

⎛⎝p1

p2

p3

⎞⎠ �−→ {1√

2(1 + p1)

⎛⎜⎜⎝− sin t(1 + p1)cos t(1 + p1)

p2 cos t + p3 sin tp3 cos t− p2 sin t

⎞⎟⎟⎠}0≤t≤2π

, (8.4.12)

for p1 �= −1 and for p1 = −1 we get

h−1 :

⎛⎝−100

⎞⎠ �−→ {⎛⎜⎜⎝00

sin tcos t

⎞⎟⎟⎠}0≤t≤2π

. (8.4.13)

8.5 Stereographic projection

The stereographic projection is a bijective map that preserves angle measures (con-formal map) where circles are mapped to circles or in the limiting case to straightlines (see e.g. Ref.[51]).This method can be used to visualize for example the sphere S3 (see Sect.14.2.2).

8.5.1 Simple case S2 −→ R2

First we describe the projection map for the 2-sphere, which is rather easy to imag-ine. We project all points of the 2-sphere from the north pole N = (0, 0, 1) to theequatorial plane. Each point on the sphere is connected with the north pole Nand the intersection of these rays with the equatorial plane defines the projectedpoints. The projection can be done for all points on the sphere except for the nothpole itself, but it is also possible to project from the south pole S = (0, 0,−1)2(seeFig.8.2).The intersection of the sphere and the equatorial plane defines a circle. We note thatpoints on the upper hemisphere are mapped outside this circle on the plane, pointson the lower hemisphere are mapped inside the circle and points on the equator areidentical with their image.

2Remember that one needs at least 2 charts (=projections) in differential geometry to cover the

sphere S2, e.g. one from the north and the other one from the south pole.

75

CHAPTER 8. Quaternions and Hopf-fibration 8.5. Stereographic projection

N

R2

S2

p

P

Figure 8.2: Schematic stereographic projection S2 −→ R2, where the projection center is

the north pole N and we map the point p = (p1, p2, p3) to the point P =

(P1, P2, 0).

In the following we want to find the transformations for the stereographic projectionP from the point N

P : S2 \ (0, 0, 1) −→ R2 . (8.5.1)

The coordinates of a point p on the sphere are given by p = (p1, p2, p3). The straightline g connecting the north pole N = (0, 0, 1) and the point p is given by

g : x =−−→ON + λ

−→Np =

⎛⎝001

⎞⎠+ λ

⎛⎝ p1

p2

p3 − 1

⎞⎠ =

⎛⎝ λp1

λp2

1 + λ(p3 − 1)

⎞⎠ , (8.5.2)

and defines the point P on the plane via the condition x3 = 0. The Point P onthe plane therefore is reached with the parameter-value of λ = 1

1−p3which gives

P = (P1, P2, 0) = ( p1

1−p3, p2

1−p3, 0) for the coordinates.

The inverse projection, where the point P on the plane is mapped to the point p onthe sphere is defined by

P−1 : R2 −→ S2 \ (0, 0, 1) , (8.5.3)

and we get for the connecting line g through the north pole N = (0, 0, 1) and thepoint P = (P1, P2, 0)

g : x =−−→ON + λ

−−→NP =

⎛⎝001

⎞⎠+ μ

⎛⎝P1

P2

−1

⎞⎠ =

⎛⎝ μP1

μP2

1− μ

⎞⎠ . (8.5.4)

We intersect the line with the sphere defined by x21 + x2

2 + x23 = 1 and get μ1 = 0

(which corresponds to the north pole) and μ2 = 2P 2

1 +P 22 +1

=: 2P 2+1

for the parameter-

values. The coordinates of the point p are therefore p = ( 2P1P 2+1

, 2P2P 2+1

, P 2−1P 2+1

).

To summarize we have the stereographic projection given by

P : S2 \ (0, 0, 1) −→ R2

(p1, p2, p3) �−→ (P1, P2, 0) =(

p1

1− p3,

p2

1− p3, 0)

,(8.5.5)

76

8.5. Stereographic projection CHAPTER 8. Quaternions and Hopf-fibration

and the inverse stereographic projection

P−1 : R2 −→ S2 \ (0, 0, 1)

(P1, P2, 0) �−→ (p1, p2, p3) =(

2P1

P 2 + 1,

2P2

P 2 + 1,P 2 − 1P 2 + 1

).

(8.5.6)

One can easily verify that P−1 ◦ P = IdS2 and P ◦ P−1 = IdR2 .

8.5.2 Generalization Sn −→ Rn

The general stereographic projection map from north or south pole (0, 0,±1) to theequatorial plane is defined by

P : Sn \ (0, . . . , 0,±1) −→ Rn

(p1, . . . , pn+1) �−→ (P1, . . . , Pn, 0) =(

p1

1∓ pn+1, . . . ,

pn

1∓ pn+1, 0)

,

(8.5.7)

and the inverse map

P−1 : Rn −→ Sn \ (0, . . . , 0,±1)

(P1, . . . , Pn, 0) �−→ (p1, . . . , pn+1) =(

2P1

P 2 + 1, . . . ,

2Pn

P 2 + 1,±P 2 − 1

P 2 + 1

),

(8.5.8)

where P 2 = P 21 + . . . + P 2

n .

77

CHAPTER 8. Quaternions and Hopf-fibration 8.5. Stereographic projection

78

Part II

Applications

79

Chapter 9

Berry Phase and Entanglement

Whereas the geometric phase in a single particle system is already studied very well,both theoretically and experimentally, its effect on entangled quantum systems isless known. Thus there is increasing interest to combine both quantum phenomena,the geometric phase and the entanglement of a system [105, 129, 137, 138].In the following we study the influence of the Berry phase on the entanglement ofa spin-1

2 system by considering a Bell inequality. The Berry phase is generated byimplementing an adiabatically rotating magnetic field into one of the paths of theparticles (see Sect.4.2). Our goal is to propose an explicit experimental setup whicheliminates the dynamical phase, which would spoil the geometric effect, so thatwe are sensitive just to the geometric phase. We can achieve this within neutroninterferometry (see Chapt.5) where we have entanglement between different degreesof freedom, i.e., the spin and the path of the neutron. In this case it is physicallyrather non-contextuality than locality which is tested experimentally (see Sect.5.3).In our case the observables, which belong to mutually disjoint Hilbert spaces, are thespin and the path of the neutron in the interferometer and we use a BI containingthese observables to test non-contextual hidden variable theories versus QM.The following chapter is based on the investigations of Ref.[28].

9.1 Theoretical Setup

9.1.1 Berry phase for qubits

In Section 4.2 we have seen that the eigenstates of a qubit in an adiabatically rotatingmagnetic field with opening angle ϑ, rotation frequency ω and time period T = 2π

ωpick up a geometric (Berry) phase γ±, and a dynamic phase θ±

|n+(0)〉 ≡ |⇑n; t = 0〉 −→ |⇑n; t = T 〉 = eiγ+(ϑ)eiθ+ |⇑n; t = 0〉|n−(0)〉 ≡ |⇓n; t = 0〉 −→ |⇓n; t = T 〉 = eiγ−(ϑ)eiθ− |⇓n; t = 0〉 ,

(9.1.1)

81

CHAPTER 9. Berry Phase and Entanglement 9.1. Theoretical Setup

with

γ+(ϑ) = −π(1− cos ϑ) γ−(ϑ) = −π(1 + cos ϑ) = −γ+(ϑ)− 2π (9.1.2)

θ+ = −E+T

�θ− = +

E−T

�= −θ+ . (9.1.3)

9.1.2 Spin-echo method

Dynamical effects are always larger than geometrical ones. Therefore it is convenientto eliminate the dynamical phase by the so called “spin-echo” method. This meansthe particle is subjected to two consecutively magnetic fields which have oppositeorientations:

• first rotation: magnetic field direction n (opening angle ϑ)

|⇑n〉 −→ eiγ+(ϑ)eiθ+ |⇑n〉 |⇓n〉 −→ eiγ−(ϑ)eiθ− |⇓n〉

• second rotation: magnetic field direction −n (opening angle π − ϑ)

|⇑n〉 ≡ |⇓−n〉 −→ eiγ−(π−ϑ)eiθ− |⇓−n〉 ≡ eiγ+(ϑ)eiθ− |⇑n〉|⇓n〉 ≡ |⇑−n〉 −→ eiγ+(π−ϑ)eiθ+ |⇑−n〉 ≡ eiγ−(ϑ)eiθ+ |⇓n〉

The eigenstates are exchanged which has an effect on both phases. The geo-metric phase of both rotations is related as a short calculation showsγ±(π − ϑ) = −π(1∓ cos(π − ϑ)) = −π(1± cos ϑ) = γ∓(ϑ).

The net-effect after two rotation-periods cancels the dynamical phase totally

|⇑n〉 → e2iγ+(ϑ)|⇑n〉 |⇓n〉 → e2iγ−(ϑ)|⇓n〉 . (9.1.4)

To make the following calculations clearer we assume two half-periods of rotationand suppress the argument of the geometric phase

|⇑n〉 → eiγ+ |⇑n〉 |⇓n〉 → eiγ− |⇓n〉 . (9.1.5)

9.1.3 Berry phase and entangled qubits

We consider two entangled qubits where one of them interacts with a magnetic fieldas described in Section 9.1.1. Thus only one subspace of the total Hilbert spaceH = H1 ⊗ H2, e.g. H1, is influenced by the phases. A schematical setup is shownin Fig.9.1. Suppose we start with the Bell singlet state |Ψ−〉

|Ψ(t = 0)〉 = |Ψ−〉 =1√2

(|⇑n⇓n〉 − |⇓n⇑n〉

), (9.1.6)

written down in the eigenbasis of the interaction Hamiltonian Hint = −μB2 n · σ.

According to the “spin-echo” construction after one cycle we pick up precisely thephases given in Eq.(9.1.5)

|Ψ(t = T )〉 =1√2

(eiγ+ |⇑n⇓n〉 − eiγ− |⇓n⇑n〉

), (9.1.7)

82

9.1. Theoretical Setup CHAPTER 9. Berry Phase and Entanglement

which can be rewritten by neglecting an overall phase factor (from now on γ+ ≡ γ)

|Ψ(t = T )〉 =1√2

(|⇑n⇓n〉 − e−2iγ |⇓n⇑n〉

). (9.1.8)

This state is maximally entangled for all values of γ. By varying the magnetic fieldangle ϑ : 0→ 60◦ → 90◦ we change the Berry phase |γ| : 0→ π

2 → π therefore movecontinuously from the Bell state |Ψ−〉 to the Bell state |Ψ+〉 and back to |Ψ−〉.

�B

− �B

ω0

ω0

ϑ

π − ϑ

�α �β

�n�n

α1 β1

β2

2γ

ϑ

particle 1 particle 2

z-axis

x-axis

source

Figure 9.1: Schematic view of the setup. The vector n denotes the quantization direction,

α and β are the measurement directions which determine the measurement

planes. (Source: [28])

9.1.4 Joint measurements

We want to perform joint measurements on the state and calculate the expectationvalue of the joint observable A(α)⊗B(β) given by

E(α, β) = 〈Ψ(t = T )|A(α)⊗B(β)|Ψ(t = T )〉 . (9.1.9)

We use the spectral decomposition of the observables A(α) = P+(α) − P−(α) andsimilarly for B(β) with the projectors P±(α) = |±α〉〈±α| and the projection statesfor spin up (+) and spin down (−) along the direction α

|+α〉 = cosα1

2|⇑n〉+ sin

α1

2eiα2 |⇓n〉

|−α〉 = − sinα1

2|⇑n〉+ cos

α1

2eiα2 |⇓n〉 .

(9.1.10)

The measurement direction α is specified by the polar angle α1 and the azimuthalangle α2 measured from the n-direction (cf. Sect.2.5.1).

83

CHAPTER 9. Berry Phase and Entanglement 9.1. Theoretical Setup

The expectation value of the joint measurement yields

E(α, β) = − cos α1 cos β1 − cos(α2 − β2 + 2γ) sin α1 sinβ1

−→ − cos(α1 − β1) for (α2 − β2)→ −2γ .(9.1.11)

The last line in Eq.(9.1.11) shows that we can compensate the effect of the Berryphase by changing the difference of the azimuthal angles (α2−β2) = −2γ of the twomeasuring directions α and β and regain the familiar expressions Eq.(2.5.3) withoutBerry phase.

9.1.5 Bell-CHSH inequality

Now we can establish a Bell-CHSH inequality, Eq.(2.5.1), to test the local featuresof the states. The S-function is given by

S(α, α′, β, β′; γ) =∣∣E(α, β)− E(α, β′)

∣∣+ ∣∣E(α′, β) + E(α′, β′)∣∣

=∣∣∣− sin α1

(cos(α2 − β2 + 2γ) sin β1 − cos(α2 − β′

2 + 2γ) sin β′1

)− cos α1

(cos β1 − cos β′

1

)∣∣∣+∣∣∣− sin α′

1

(cos(α′

2 − β2 + 2γ) sin β1 + cos(α′2 − β′

2 + 2γ) sin β′1

)− cos α′

1

(cos β1 + cos β′

1

)∣∣∣ .

(9.1.12)

Without loss of generality we can eliminate one angle by setting, e.g., α = 0 (α1 =α2 = 0), which gives

S(α′, β, β′; γ) =∣∣∣− sinα′

1

(cos(α′

2 − β2 + 2γ) sinβ1 + cos(α′2 − β′

2 + 2γ) sin β′1

)− cos α′

1(cos β1 + cos β′1)∣∣∣+ ∣∣∣− cos β1 + cos β′

1

∣∣∣ .

(9.1.13)

9.1.6 Analysis of the S-function

Possibility I

Due to the fact that the state (9.1.8) is maximally entangled irrespective of theBerry phase γ we can use a Theorem of the Horodeckis [86] and Gisin [64].

Theorem 6. For a maximally entangled state we always can find angles that violate

the Bell-CHSH inequality maximally Smax = 2√

2.

We can achieve this by rotating the Bell angles with respect to the Berry phase γby the azimuthal amount (α2 − β2) = −2γ, which is demonstrated in Fig.9.1.That means we keep the polar angles α′

1, β1 and β′1 constant at the Bell angles

α′1 = π

2 , β1 = π4 , β′

1 = 3π4 (cf. Eq.(2.5.4)) which gives for S

S(α′2, β2, β

′2; γ) =

√2 +

∣∣∣−√22

(cos(α′

2 − β2 + 2γ) + cos(α′2 − β′

2 + 2γ))∣∣∣ , (9.1.14)

84

9.2. Experimental setup CHAPTER 9. Berry Phase and Entanglement

and then adjust the azimuthal parts to reach the maximum. This happens at theangles β2 = β′

2, α′2 − β′

2 = −2γ (mod π) and for convenience α′2 = 0. Now the

measurement planes on both sides enclose an angle of 2γ.

Possibility II

But what happens if it is not possible, e.g. due to experimental limitations, to rotatethe measurement planes accordingly to reach Smax = 2

√2?

Assume we have to keep the azimuthal angles fixed, e.g., α2 = α′2 = β2 = β′

2 = 0.Then the only freedom in maximizing the S-function is to vary the polar angles.The S-function

S(α′1, β1, β

′1; γ) =

∣∣∣− sinα′1 cos(2γ)

(sinβ1 + sinβ′

1

)− cos α′

1(cos β1 + cos β′1)∣∣∣

+∣∣∣− cos β1 + cos β′

1

∣∣∣ ,

(9.1.15)

consists of two absolute values S = |f1| + |f2|. To determine the maximum withrespect to the Berry phase γ we have to consider the extremal condition

∂S

∂β1= − sinβ1 ∓ cos α′

1 sinβ1 ± cos(2γ) sin α′1 cos β1 = 0

∂S

∂β′1

= sinβ′1 ∓ cos α′

1 sinβ′1 ± cos(2γ) sin α′

1 cos β′1 = 0 (9.1.16)

∂S

∂α′1

= ∓ sinα′1(cos β1 + cos β′

1)± cos(2γ) cos α′1(sin β1 + sin β′

1) = 0 ,

where the upper sign corresponds to the case f1 < 0 and f2 < 0 and the lower signto f1 < 0 and f2 > 0. The solutions are given by

β1 = ± arctan(cos(2γ)) β′1 = π − β1 α′

1 =π

2, (9.1.17)

and are plotted in Fig.9.2 and Fig.9.3. Inserting these solutions into the S-functionwe get the behavior plotted in Fig.9.4.The maximal value of S decreases for γ : 0→ π

4 and touches at γ = π4 even the limit

of the CHSH inequality S = 2, where we are unable to distinguish between QM andlocal realistic theories. It increases again to the familiar value S = 2

√2 at γ = π

2 ,which corresponds to the Bell state |Ψ+〉.This analysis hold not only for geometric phases but for all kind of phases. Onehas to keep in mind that phases can reduce the maximally expected value of theS-function if it is not possible to compensate the effect by a relative rotation of themeasurement planes.

9.2 Experimental setup

We want to test the predicted behavior of the S-function, Fig.9.4, within neutroninterferometry where it is possible to create entanglement between the spin and path

85

CHAPTER 9. Berry Phase and Entanglement 9.2. Experimental setup

β1-values

3π4

π2

π4

−π4

π4

π2

3π4 π

|γ|

β′1-values

5π4

π3π4π2π4

π4

π2

3π4 π

|γ|

Figure 9.2: The Bell angles β1 and β′1 with respect to the Berry phase γ for the case f1 < 0

and f2 < 0. (Source: [28])

β1-values

3π4

π2

π4

−π4

π4

π2

3π4 π

|γ|

β′1-values

5π4

π3π4π2π4

π4

π2

3π4 π

|γ|

Figure 9.3: The Bell angles β1 and β′1 with respect to the Berry phase γ for the case f1 < 0

and f2 > 0. (Source: [28])

maximal S-values

|γ|π4

π2

3π4 π

ϑ0◦ 41.4◦ 60◦ 75.5◦ 90◦

Ψ(−) Ψ(+) Ψ(−)

1

2

2√

2

Figure 9.4: The maximum of the S-function (9.1.13) with respect to the Berry phase γ

with the choice of zero azimuthal angles α′2 = β2 = β′

2 = 0. (Source: [28])

86

9.2. Experimental setup CHAPTER 9. Berry Phase and Entanglement

degree of freedom of a single neutron (see Sect.5.3). The Berry phase affects onlythe spinor part of the Hilbert space H = Hspin ⊗Hpath.A schematic view of the experimental setup is shown in Fig.9.5. The spin flipperproduces the entangled state. The Berry phase is implemented by two magneticfields B and B′. The analysis of the final state is done with a phase shifter χ and aspin rotator α.

spin flipper

|⇑〉 |⇑〉 ⊗ |I〉

|⇓〉 ⊗ |II〉

B B′

Phase shifter χ

Spin rotator δ

Figure 9.5: Schematic view of the experimental setup.

The entangled wave function of each neutron is given by

|Ψ〉 =1√2

(|⇑〉 ⊗ |I〉 − |⇓〉 ⊗ |II〉

), (9.2.1)

where |⇑〉 and |⇓〉 denote the spin states and |I〉 and |II〉 stand for the states of thetwo paths in the interferometer.The Berry phase is produced a little bit different than described in the theoreticalpart. The two magnetic fields act as RF spin flippers and enable the neutron spinorsto evolve along a particular curve on the Poincare sphere inducing only a geometricphase γB without any dynamical component, see e.g. [6, 146]. The solid angle ofthe evolution, shown in Fig.9.6, is related to the geometrical phase

γB =12Ω = φ1 − φ2 , (9.2.2)

where φ1 − φ2 is the relative phase of the RF spin flippers.After the spinor evolution the total wave function is represented by

|Ψ(γB)〉 =1√2

(|⇑〉 ⊗ |I〉 − eiγB |⇓〉 ⊗ |II〉

). (9.2.3)

Now we perform measurements of joint observables of the form As(δ)⊗Bp(χ) where

As(δ) = P s+(δ)− P s

−(δ) and Bp(χ) = P p+(χ)− P p

−(χ) , (9.2.4)

with the projection operators onto spin and path states

P s±(δ) = |±nδ〉〈±nδ| and P p

±(χ) = |±p〉〈±p | . (9.2.5)

87

CHAPTER 9. Berry Phase and Entanglement 9.2. Experimental setup

�1

�2

-

|⇑〉

|⇓〉

Figure 9.6: Schematic representation of the spinor evolution with the use of the Poincare

sphere. (Source: [28])

The projection states are given by

|+nδ〉 = cosδ1

2|⇑〉+ sin

δ1

2eiδ2 |⇓〉 |+p〉 = cos

χ

2|I〉+ sin

χ

2|II〉

|−nδ〉 = − sinδ1

2|⇑〉+ cos

δ1

2eiδ2 |⇓〉 |−p〉 = − sin

χ

2|I〉+ cos

χ

2|II〉 .

(9.2.6)

Note that χ and δ play the role of the angles α and β described in Sect.9.1.4.We can compare the experimentally determined expectation value with the theo-retically predicted E(χ,δ) = 〈Ψ(γB)|As(δ)⊗ Bp(χ)|Ψ(γB)〉 by varying the openingangle of the magnetic fields (the relative phase of the RF spin flippers).We are also able to test the influence of the geometric phase on the Bell-CHSHinequality and its maximal possible violation for the analysis described in Section9.1.6. For parallel measurement planes Smax is achieved for the settings χ ≡ α1 = 0(α2 = 0), which corresponds to the choice of one path, and χ′ ≡ α′

1 = π2 (α′

2 = 0),which means an equal superposition of the states |I〉 and |II〉, whereas the anglesδ ≡ β and δ′ ≡ β′ of the spinor analysis have to be chosen according to Eq.(9.1.17).The experimental implementation and analysis is work in progress.

88

Chapter 10

Decoherence for entangled

Kaons

Particle physics has become an interesting testing ground for fundamental questionsof QM, for an overview see [76].In the following we concentrate on possible decoherence effects arising due to someinteraction of the system with its environment. Sources for “standard” decoherenceeffects are strong interaction scatterings of kaons with nucleons, weak interactiondecays and noise of the experimental setup. “Nonstandard” decoherence effects re-sult from a fundamental modification of QM and can be traced back to the influenceof quantum gravity [74, 132] – quantum fluctuations in the space-time structure onthe Planck mass scale – or to dynamical state reduction theories [63, 65, 114], andarise on a different energy scale.In the following treatise, which is based on Ref.[29], we do not bother about the spe-cific reasons and sources of decoherence but we want to develop a specific model ofdecoherence based on the investigations of Sect.3.3. The model of decoherence allowsthe determination of bounds for the strength of decoherence via a comparison withdata of existing experiments (Sect.10.2). The model is connected to a phenomeno-logical model of decoherence, Sect.10.3, and the bridge to quantum information isdrawn in Sect.10.4 by the analysis of the model in terms of entanglement measures.

10.1 Theoretical model

Let us begin our decoherence discussion with the 1-particle kaon system as an in-troduction then we proceed to the case of two entangled neutral kaons.

10.1.1 Decoherence model for single kaons

The neutral kaon system is introduced in Chapt.6. The “effective mass” Hamilto-nian H is non-Hermitian and describes the decay properties and the strangenessoscillations of the kaons. The mass eigenstates, the short lived |KS〉 and long lived

89

CHAPTER 10. Decoherence for entangled Kaons 10.1. Theoretical model

|KL〉 states, are determined by

H |KS,L〉 = λS,L |KS,L〉 with λS,L = mS,L −i

2ΓS,L , (10.1.1)

with mS,L and ΓS,L being the corresponding masses and decay widths. For ourpurpose CP invariance1 is assumed, which means that the CP eigenstates |K0

1 〉, |K02 〉

are equal to the mass eigenstates

|K01 〉 ≡ |KS〉, |K0

2 〉 ≡ |KL〉, and 〈KS |KL〉 = 0 . (10.1.2)

We use the decoherence formalism described in Sect.3.3. The Lindblad master equa-tion reads

∂ρ

∂t= −iHρ + iρH† −D(ρ) , (10.1.3)

and for the dissipator D(ρ) we choose the following ansatz (compare Ref.[31])

D(ρ) = λ(PSρPL + PLρPS

), (10.1.4)

where Pj = |Kj〉〈Kj | (j = S, L) represent the projectors to the eigenstates of theHamiltonian and the decoherence parameter λ is positive, λ ≥ 0.The density matrix ρ in the KSKL basis is defined by

ρ(t) =∑

i,j=S,L

ρij(t) |Ki〉〈Kj | . (10.1.5)

For the above choice of the dissipator, Eq.(10.1.4), the time evolution (10.1.3) de-couples for the components of ρ and we obtain

ρSS(t) = ρSS(0) · e−ΓSt

ρLL(t) = ρLL(0) · e−ΓLt

ρLS(t) = ρLS(0) · e−iΔmt−Γt−λt (10.1.6)

with Δm = mL − mS and Γ = 12(ΓS + ΓL). We see that only the off-diagonal

elements are effected by the decoherence.

10.1.2 Decoherence model for entangled kaons

For two entangled kaons we make the following identification

|e1〉 = |KS〉l ⊗ |KL〉r and |e2〉 = |KL〉l ⊗ |KS〉r . (10.1.7)

Then the initial quasi-spin singlet state |Ψ−〉 = 1√2

(|e1〉 − |e2〉

)is equivalently given

by the density matrix

ρ(0) = |Ψ−〉〈Ψ−| =12

(|e1〉〈e1|+ |e2〉〈e2| − |e1〉〈e2| − |e2〉〈e1|

). (10.1.8)

1Note that corrections due to CP violations are of order 10−3. We compare this model of

decoherence with the data of the CPLEAR experiment [47] which are not sensitive to CP violating

effects.

90

10.2. Connection to experiment CHAPTER 10. Decoherence for entangled Kaons

The total Hamiltonian is a tensor product of the 1-particle Hamiltonians H = Hl ⊗�r + �l ⊗Hr, where l denotes the left-moving and r the right-moving particle.The Lindblad operators are now given by Pj = |ej〉〈ej | (j = 1, 2) and project tothe eigenstates of the 2-particle Hamiltonian H. Again the time evolution (10.1.3)decouples

ρ11(t) = ρ11(0) e−2Γt

ρ22(t) = ρ22(0) e−2Γt

ρ12(t) = ρ12(0) e−2Γt−λt , (10.1.9)

and we obtain for the time-dependent density matrix

ρ(t) =12e−2Γt

{|e1〉〈e1|+ |e2〉〈e2| − e−λt

(|e1〉〈e2|+ |e2〉〈e1|

)}. (10.1.10)

The decoherence affects only the off-diagonal elements (compare with Sect.11.2).Note that the assumption of CP invariance, Eq.(10.1.2), – which is sufficient for ourpurpose – implying 〈e1|e2〉 = 0, is crucial. Otherwise we would have a time evolutioninto the full 4-dimensional Hilbert space of states.

10.2 Connection to experiment

10.2.1 Experimental setup

We want to compare the theoretical model with experimental data from the so-calledCPLEAR experiment [47] performed at CERN in 1998. They produced K0K0 pairsin the Bell singlet state |Ψ−〉 in the pp collider. The initial state is

|ψ(t = 0)〉 = |Ψ−〉 =1√2

(|K0〉l ⊗|K0〉r − |K0〉l ⊗ |K0〉r

), (10.2.1)

which can also be written in the KSKL basis.

Figure 10.1: Example of CPLEAR event: like-strangeness (K−,Λ) (Source: [47]).

91

CHAPTER 10. Decoherence for entangled Kaons 10.2. Connection to experiment

The experimental setup is shown in Fig.10.1 and has two configurations. In thefirst configuration, C(0), the kaons on both sides propagate 2cm before they aremeasured in the absorbers, thus the time difference of both sides is zero (tr ≈ tl).This corresponds to the condition for an EPR-type experiment: detecting a kaon onthe right side implies to detect an anti-kaon on the left side and vice versa.In the second setup, C(5), the first kaon is detected after 2cm and the second oneafter 7cm, thus the flight-path difference is 5cm, corresponding to a proper timedifference |tr − tl| ≈ 1.2τS . Due to the strangeness oscillations we are confrontedwith a different situation than in the standard EPR-type experiment.In the absorbers made out of copper and carbon the strangeness of the kaons is de-tected via strong interactions. The kaons are detected indirectly by their electricallycharged reaction products as indicated in Table 10.1.

strangeness reaction detection particle

S = +1 K0(ds) + N −→ K+(us) + X K+

S = −1 K0(ds) + N −→ K−(us) + X K−

S = −1 K0(ds) + N −→ Λ(uds) + X and Λ −→ pπ− Λ

Table 10.1: Detection of strangeness property of neutral kaons via nuclear reactions.

That means the like–strangeness events can be traced back by detecting (K−, Λ)and for the unlike–strangeness events we can have either (K+, K−) or (K+, Λ).

10.2.2 Mathematical description of measurements

Two-particle decoherence

The 2-particle density matrix including decoherence effects is given by Eq.(10.1.10).Now we measure the strangeness content S of the left-moving particle at time tl andof the right-moving particle at time tr

2. After the first measurement and before thesecond measurement (tr ≤ t ≤ tl) we have a 1-particle state which does not pick upany further decoherence and evolves exactly according to QM. That means we areonly interested in two-particle decoherence effects.

Measurement and probabilities

The measurement of the strangeness content (K0 or K0) is described by a projectionoperator |S〉〈S| with strangeness |S〉 = |±〉 and |+〉 = |K0〉, |−〉 = |K0〉. After thefirst measurement at t = tr of the right-moving particle we are left with the reduceddensity matrix given by

ρl(t = tr; tr) = Trr

(�l ⊗ |S〉〈S|r ρ(tr)

), (10.2.2)

For times t ≥ tr the density matrix ρl(t; tr) describes the one-particle state of theleft-moving particle and evolves according to pure QM. At t = tl the strangeness

2For sake of definiteness we choose tr ≤ tl.

92

10.2. Connection to experiment CHAPTER 10. Decoherence for entangled Kaons

content of the second particle is measured and we end up with the probability

Pλ(S′, tl; S, tr) = Trl

(|S′〉〈S′|l ρl(tl; tr)

). (10.2.3)

Explicitly, we find the following results for the like- and unlike-strangeness proba-bilities

P likeλ (K0, tl; K0, tr) = P like

λ (K0, tl; K0, tr) =

=18

(e−ΓStl−ΓLtr + e−ΓLtl−ΓStr − e−λtr 2 cos(ΔmΔt) · e−Γ(tl+tr)

)P unlike

λ (K0, tl; K0, tr) = P unlikeλ (K0, tl; K0, tr) =

=18

(e−ΓStl−ΓLtr + e−ΓLtl−ΓStr + e−λtr 2 cos(ΔmΔt) · e−Γ(tl+tr)

),

(10.2.4)

with Δt = tl − tr.Note that at equal times tl = tr = t the like-strangeness probabilities, given by

P likeλ (K0, t; K0, t) = P like

λ (K0, t; K0, t) =14

e−2Γt (1− e−λt) , (10.2.5)

do not vanish, in contrast to common EPR-correlations.

Asymmetry

The CPLEAR experiment measured the difference between like-strangeness eventsand unlike-strangeness events. This quantity normalized to the sum of these twoprobabilities is called asymmetry term

A(tl, tr) =P like(tl; tr)− P unlike(tl; tr)P like(tl; tr) + P unlike(tl; tr)

. (10.2.6)

For pure QM we get

AQM (Δt) =cos(ΔmΔt)

cosh(12ΔΓΔt)

, (10.2.7)

with ΔΓ = ΓL − ΓS , and for the decoherence model under consideration we find byinserting the probabilities (10.2.4)

Aλ(tl, tr) = AQM (Δt) · e−λ min {tl,tr} . (10.2.8)

Thus the decoherence effect, simply given by the factor e−λ min {tl,tr}, depends onlyon the time of the first measured kaon, in our case: min {tl, tr} = tr.

10.2.3 Bounds from experimental data

Fitting the decoherence parameter λ by comparing the asymmetry (10.2.8) withthe experimental data [47] we find, when averaging over both configurations, thefollowing bounds on λ

λ = (1.84+2.50−2.17) · 10−12 MeV and Λ =

λ

ΓS= 0.25+0.34

−0.32 . (10.2.9)

The results (10.2.9) are certainly compatible with QM (λ = 0), nevertheless, theexperimental data allow an upper bound λup = 4.34 · 10−12 MeV for possible deco-herence in the entangled K0K0 system.

93

CHAPTER 10. Decoherence for entangled Kaons 10.3. Connection to phenomenological model

10.3 Connection to phenomenological model

10.3.1 Phenomenological model

There exists a model [33, 48, 75] which introduces decoherence “by hand”. Themodel is based on the spontaneous factorization of the wave function proposed bySchrodinger [126] and Furry [60] and therefore called Schrodinger-Furry hypothesis.The idea is that the wave function of the entangled kaon state

|Ψ−(t = 0)〉 =1√2

(|KS〉l ⊗|KL〉r − |KL〉l ⊗ |KS〉r

), (10.3.1)

factorizes in one half of the cases in the state |KS〉l ⊗|KL〉r and in the other half ofthe cases in the state |KL〉l ⊗ |KS〉r.We can modify the calculation of probabilities in such a way that we include onthe one side the quantum mechanical predictions and on the other side spontaneousfactorization by introducing an effective decoherence parameter ζ. Thus the like-signprobability, given by

P like(K0, tl; K0, tr) =∥∥∥〈K0|l ⊗ 〈K0|r |Ψ−(tl, tr)〉

∥∥∥2, (10.3.2)

is modified to give

P likeζ (K0, tl; K0, tr) =

=12

{e−ΓStl−ΓLtr |〈K0|KS〉l|2 |〈K0|KL〉r|2 + e−ΓLtl−ΓStr |〈K0|KL〉l|2 |〈K0|KS〉r|2

− 2 (1− ζ)︸︷︷︸ Re{〈K0|KS〉∗l 〈K0|KL〉∗r〈K0|KL〉l〈K0|KS〉r e−iΔmΔt} · e−Γ(tl+tr)

}modification

=18

{e−ΓStl−ΓLtr + e−ΓLtl−ΓStr − 2 (1− ζ)︸︷︷︸ cos(ΔmΔt) · e−Γ(tl+tr)

},

modification(10.3.3)

and the same for the unlike probability but with a sign change in front of theinterference term. We see that the interference term of the transition amplitude ismodified by a factor (1 − ζ) where the value ζ = 0 corresponds to pure QM andζ = 1 to total decoherence or spontaneous factorization of the wave function.The effective decoherence parameter ζ, introduced in this way “by hand”, interpo-lates continuously between these two limits and represents a measure for the amountof decoherence which results in a loss of entanglement of the total quantum state.It is clear that due to the spontaneous factorization hypothesis the decoherenceparameter ζ depends on the basis where the factorization happens.Calculating the asymmetry of the strangeness events with the probabilities (10.3.3)we obtain

Aζ(tl, tr) = AQM (Δt) ·(1− ζ(tl, tr)

). (10.3.4)

94

10.4. Connection to quantum information CHAPTER 10. Decoherence for entangled Kaons

10.3.2 Comparison

We have computed the asymmetry term in both models the fundamental one,Eq.(10.2.8), and the phenomenological one, Eq.(10.3.4). If we compare them witheach other we find the following simple relation

ζ(tl, tr) = 1− e−λ min {tl,tr} . (10.3.5)

The decoherence parameter λ is considered to be a fundamental constant, whereasthe value of the effective decoherence parameter ζ depends on the time when ameasurement is performed. In the time evolution of the state (10.1.10) we have therelation

ζ(t) = 1− e−λt , (10.3.6)

which after the measurement of the left- and right moving particles at tl and tr turnsinto formula (10.3.5), when decoherence is implemented at the two-particle level.

In Ref.[33] the authors derive bounds on the effective decoherence parameter ζ fromthe CPLEAR experiment given by ζ = 0.13+0.16

−0.15, which are in agreement with thevalues for λ (10.2.9).

10.4 Connection to quantum information

In the following section we investigate the decoherence model for the neutral kaonsystem with respect to entanglement measures, discussed in Section2.4.The initial state of the entangled K0K0 system is pure and maximally entangled.As time goes on the presence of decoherence causes a decrease of entanglement ofthe system. We can explicitly quantify and visualize the loss of entanglement.Since we are only interested in the effect of decoherence we can normalize the state(10.1.10) in order to compensate for the decay property of the non-Hermitian Hamil-tonian H and introduce the density matrix

ρN (t) =ρ(t)

Trρ(t). (10.4.1)

10.4.1 Quasi-spin picture

We recall the “quasi-spin” picture for the neutral kaon system, introduced in Sect.6.2.We can express the projection operators to the mass eigenstates in terms of the fol-lowing Pauli matrices

PS = |KS〉〈KS | = σ↑ =12

(1 + σz) =(

1 00 0

),

PL = |KL〉〈KL| = σ↓ =12

(1− σz) =(

0 00 1

),

(10.4.2)

95

CHAPTER 10. Decoherence for entangled Kaons 10.4. Connection to quantum information

and the transition operators are the “spin-ladder” operators

PSL = |KS〉〈KL| = σ+ =12

(σx + i σy) =(

0 10 0

),

PLS = |KL〉〈KS | = σ− =12

(σx − i σy) =(

0 01 0

).

(10.4.3)

Now we can express the density matrix (10.4.1) either as

ρN (t) =12(σ↑ ⊗ σ↓ + σ↓ ⊗ σ↑ − e−λt [σ+ ⊗ σ− + σ− ⊗ σ+]

), (10.4.4)

or asρN (t) =

14(1− σz ⊗ σz − e−λt [σx ⊗ σx + σy ⊗ σy]

). (10.4.5)

For t = 0 we get the expression for the Bell spin singlet state ρN (0) = 14 (1−σ⊗σ).

We can also write the density matrix as 4× 4 matrix

ρN (t) =12

⎛⎜⎜⎝0 0 0 00 1 −e−λt 00 −e−λt 1 00 0 0 0

⎞⎟⎟⎠ . (10.4.6)

By using the “Bell basis”, introduced in Sect.2.2.3, the state ρN (t) looks like

ρN (t) =12(1 + e−λt

)ρ− +

12(1− e−λt

)ρ+ . (10.4.7)

which is a mixture of the Bell states ρ− and ρ+ . The states ω± do not contribute.

10.4.2 Mixing and entanglement

Mixing

In Eq.(2.1.6) we have introduced a measure δ = Trρ2 for the mixedness of a state.We can compute

ρ 2N (t) =

14

⎛⎜⎜⎝0 0 0 00 1 + e−2λt −2e−λt 00 −2e−λt 1 + e−2λt 00 0 0 0

⎞⎟⎟⎠ , (10.4.8)

and get

δ =12(1 + e−2λt) . (10.4.9)

For t = 0 the state is pure but for t > 0 it starts to get mixed and approaches fort→∞ the value of δ = 1

2 . So the state gets mixed but not maximally mixed whichwould correspond to δ = 1

4 .

96


Entanglement

To show the entanglement of the state ρN (t) we use the ppt-criterion, discussed inSect.2.3.2, which in this case is necessary and sufficient to prove entanglement.The transposition of the Pauli matrices is given by T (σi)kl = (σi)lk. Applying thistransposition to the second subspace we end up with the following state

(�⊗ T ) ρN (t) =12(σ↑ ⊗ σ↓ + σ↓ ⊗ σ↑ − e−λt [σ+ ⊗ σ+ + σ− ⊗ σ−]

), (10.4.10)

or written in matrix form

(�⊗ T ) ρN (t) =12

⎛⎜⎜⎝0 0 0 −e−λt

0 1 0 00 0 1 0

−e−λt 0 0 0

⎞⎟⎟⎠ . (10.4.11)

This matrix has the eigenvalues{

12 , 1

2 , 12e−λt,−1

2e−λt}

which are not all positive.Thus we conclude that the state ρN (t) is entangled for 0 < t <∞ and gets separablein the asymptotic limit t→∞.Alternatively, we can use the reduction criterion, Sect.2.3.3,

�l ⊗ ρ rN (t)− ρN (t) =

12

⎛⎜⎜⎝1 0 0 00 0 e−λt 00 e−λt 0 00 0 0 1

⎞⎟⎟⎠ , (10.4.12)

where ρ rN (t) = Trl(ρN (t)) denotes the reduced density matrix. The eigenvalues are

again{

12 , 1

2 , 12e−λt,−1

2e−λt}

and we arrive at the same conclusions.

10.4.3 Measures of entanglement

Von Neumann entropy

The von Neumann entropy, Eq.(2.4.1), of the state (10.4.1) is given by

SvN

(ρN (t)

)= −1− e−λt

2log

1− e−λt

2− 1 + e−λt

2log

1 + e−λt

2. (10.4.13)

At the time t = 0 the entropy is zero, there is no uncertainty in the system, thequantum state is pure and maximally entangled. For t > 0 the entropy gets nonzero,increases and approaches the value 1 for t→∞. Hence the state becomes more andmore mixed. Mixed states provide only partial information about the system, andthe entropy measures how much of the maximal information is missing.In Fig.10.2 the von Neumann entropy S(ρN (t)) is plotted for the mean value andupper bound of the decoherence parameter λ, Eq.(10.2.9), as determined from theCPLEAR experiment (see Sect.10.2.3).The von Neumann entropy of the reduced density matrix is a good measure forentanglement but only for pure states, see Sect.2.4.2. We can show this by explicitly

97


calculating the quantity. The reduced density matrices of the subsystems, i.e. thepropagating kaons on the left l and right r hand side are given by

ρ lN (t) = Trr{ρN (t)} and ρ r

N (t) = Trl{ρN (t)} . (10.4.14)

For the von Neumann entropy S(ρ lN (t)) of the reduced density matrices we find

EvN (ρN ) = SvN

(ρ l

N (t))

= SvN

(ρ r

N (t))

= 1 ∀ t ≥ 0 . (10.4.15)

The reduced entropies are independent of λ and t. They do not show any variationin the amount of entanglement which is in contrast to the statements in Sect.10.4.2.So we see that the reduced entropy indeed is no good measure for entanglementfor mixed states. But we can deduce that the correlations stored in the compositesystem are not lost to the subsystems but to the environment, what is expectedintuitively.

Entanglement of formation and concurrence

For mixed states the concurrence and the entanglement of formation are good mea-sures for entanglement, compare Sects.2.4.3 and 2.4.4.The density matrix ρN (t) of our model is invariant under the spin-flip operation(2.4.6) ρN = ρN and thus ρN ρN = ρ 2

N . Therefore we get for the concurrence

C(ρN (t)

)= max

{0, e−λt

}= e−λt . (10.4.16)

This immediately gives a connection to the effective decoherence parameter viaEq.(10.3.6)

1− C(ρN (t)

)= ζ(t) . (10.4.17)

The entanglement of formation of the K0K0 system is given by

EF

(ρN (t)

)= −1 +

√1− e−2λt

2log

1 +√

1− e−2λt

2

−1−√

1− e−2λt

2log

1−√

1− e−2λt

2. (10.4.18)

We define the loss of entanglement to be one minus the entanglement of formation1 − EF which is plotted for the mean value and upper bound of the decoherenceparameter λ, Eq.(10.2.9), in Fig.10.2.We can expand the expression for the entanglement of formation (10.4.18) for smallvalues of λ and get the following relation

1− EF

(ρN (t)

) .=1

ln 2ζ(t) .=

λ

ln 2t . (10.4.19)

We find that entanglement loss in terms of the concurrence equals precisely thedecoherence parameter ζ, Eq.(10.4.17). In terms of entanglement of formation,Eq.(10.4.19), the decoherence parameter is weighted by a factor3 1

ln 2 = 1.44 and theentanglement loss increases linearly with time.Via these relations we can propose a way how to measure experimentally the entan-glement of the K0K0 system.

3The factor ln 2 reflects the dimension 2 of the K-meson quasispin space.

98


0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1absorber in the

CPLEAR experiment�� SvN (λup)

1− EF (λup)

SvN (λ)

1− EF (λ)

t/τS

Figure 10.2: The time dependence of the von Neumann entropy (dashed lines) and the

loss of entanglement of formation 1 − EF (solid lines) are plotted for the

experimental mean value (lower curve) and the upper bound (upper curve)

of the decoherence parameter λ. (Source: [29])

10.4.4 Discussion of the results

Fig.10.2 shows the loss of entanglement 1−EF , Eq.(10.4.18), and the von Neumannentropy function SvN , Eq.(10.4.13). The time t is scaled versus the lifetime τs of theshort lived kaon KS : t → t/τs. The curves are plotted for the experimental meanvalue λ = 1.84 · 10−12 MeV and the upper bound λup = 4.34 · 10−12 MeV of thedecoherence parameter λ, Eq.(10.2.9).

The von Neumann entropy function visualizes the loss of the information about thecorrelation stored in the composite system. Remember that the information is notflowing into the subsystems but into the environment, see Eq.(10.4.15).The loss of entanglement of formation increases slower with time and visualizes theresources needed to create a given entangled state. At t = 0 the pure Bell state ρ− iscreated and becomes mixed for t > 0. In the total state the amount of entanglementdecreases until separability is achieved asymptotically for t→∞.The vertical lines represent the propagation time t0/τs ≈ 0.55 of one kaon, includingthe experimental error bars, until it is measured by the absorber in the CPLEARexperiment after 2cm. The loss of entanglement is about 18% for the mean value andmaximal 38% for the upper bound of the decoherence parameter λ. These valuescould diminish considerably in future experiments.

We can relate the model to the case of a phenomenologically introduced decoherenceparameter ζ and find a one-to-one correspondence, Eq.(10.3.6). The existing dataare not yet sufficient to measure the time-dependence of ζ as predicted by ourmodel, Eq.(10.3.5). So further measurements of the time-dependent asymmetry

99


term, Eqs.(10.2.8), (10.3.4), would be of high interest in future experiments and willsharpen considerably the bounds of the parameters λ and ζ.

100

Chapter 11

Decoherence modes in a two

qubit system

The study of decoherence is of great importance. On the one hand it is of interest toknow the origin of decoherence and protect the quantum systems against unwanted“noise”. Such models of possible decoherence scenarios are discussed for instance inRefs.[156, 157]). On the other hand we intend to have models which characterizedecoherence present in a system.Our aim in this chapter, which is based on Refs.[26, 27], is to provide a completephenomenological treatment of decoherence models which describe the effects ofdecoherence regardless of their microscopic origin. These models or modes as we callthem can be implemented and studied in experiments under controlled conditionsas we will see in Chapt.12.We consider three possible scenarios, called mode A , B and C in Sects.11.2, 11.3 and11.4 which are discussed for the initial condition of a Bell singlet state in Sect.11.5to show the essential features of each mode.

11.1 Notations

11.1.1 Hilbert space

A two qubit system is described by the total Hilbert spaceH = H(1)⊗H(2) = C2⊗C2

where the Hamiltonian of the undisturbed system is given by H = H(1)⊗�+�⊗H(2)

which defines an eigenbasis {|ek〉}4k=1 of the system

H|ek〉 = Ek|ek〉 . (11.1.1)

A general state of the system is described by a density matrix ρ which can bedecomposed in the eigenbasis as

ρ =∑k,j

ρkj |ek〉〈ej | , (11.1.2)

and (ρkj) denotes the 4× 4 coefficient matrix.

101

CHAPTER 11. Decoherence modes in a two qubit system 11.1. Notations

11.1.2 Decoherence

We study decoherence introduced by various Lindblad generators Pk (see Sect.3.3).We consider 4 Lindblad generators Pk that project onto one-dimensional subspacesand fulfil

∑k Pk = �, furthermore we assume only one dissipation parameter λ

that parameterizes the strength of the interaction and therefore of the decoherence.Therefore the dissipator, Eq.(3.3.6), can be written as

D(ρ) = λ(ρ−

4∑k=1

PkρPk

). (11.1.3)

To make a further simplification we consider a reduced form of the Lindblad masterequation

∂

∂tρ(t) = −D(ρ(t)) , (11.1.4)

which takes into account only the effects caused by pure decoherence and not by thedynamical evolution described by the Hamiltonian H. In the following sections thisequation will be solved for different configurations of Lindblad generators which wecall decoherence modes.

11.1.3 Different bases

In the following we consider different kinds of decoherence modes indicated by dif-ferent projection operators. The notation will be the following.Suppose we split the eigenstates of the undisturbed Hamiltonian {ek} in eigenstatesof the subspace Hamiltonians {|a1〉, |a2〉} for the first subspace and {|b1〉, |b2〉} forthe second subspace in the following way

|e1,3〉 = |a1,2〉|b1〉 , |e2,4〉 = |a1,2〉|b2〉 . (11.1.5)

We consider a general rotation in one subspace, e.g. the first, given by

|α1〉 = cosθ

2|a1〉+ eiφ sin

θ

2|a2〉

|α2〉 = − sinθ

2|a1〉+ eiφ cos

θ

2|a2〉 .

(11.1.6)

This basis we call GR (general rotated basis) in contrast to the eigenbasis which iscalled E. There are two special cases. For θ = π

2 and φ = 0 we recover

|→〉 =1√2(|a1〉+ |a2〉)

|←〉 =1√2(|a1〉 − |a2〉) ,

(11.1.7)

which we call R basis (rotated basis) and for θ = π2 and φ = π

2 we get

|!〉 =1√2(|a1〉+ i|a2〉)

|⊗〉 =1√2(|a1〉 − i|a2〉) ,

(11.1.8)

102

11.2. Mode A: E ⊗ E CHAPTER 11. Decoherence modes in a two qubit system

the so called IR basis (imaginary rotated basis).The different decoherence modes are indicated for example by E⊗E or R⊗E whichspecifies the kind of basis-vectors used for the first and the second subspace.In the case of photons (see e.g. [39]) the eigenbasis E corresponds to horizontal |H〉and vertical |V 〉 polarization whereas the rotated basis R represents polarizationstates |+45◦〉 and |−45◦〉. In the case of neutral kaons we can identify the eigenbasisE with the states |KS〉 and |KL〉 and the rotated basis R with |K0〉 and |K0〉 (seeChapt.6 and Ref.[76]).

11.2 Mode A: E ⊗ E

The first mode depicts the simplest possible case. The Lindblad generators arechosen to be projectors Pk = |ek〉〈ek| onto the eigenbasis of the Hamiltonian whichgives for the dissipator in matrix notation

D(ρ) = λ

⎛⎜⎜⎝0 ρ12 ρ13 ρ14

ρ∗12 0 ρ23 ρ24

ρ∗13 ρ∗23 0 ρ34

ρ∗14 ρ∗24 ρ∗34 0

⎞⎟⎟⎠ . (11.2.1)

In this mode of decoherence the time evolution (11.1.4) for the coefficient matrix isgiven by decoupled differential equations

˙ρkj = −λA ρkj for k �= j

˙ρkk = 0 ,(11.2.2)

which can be easily solved

ρkj(t) = e−λAtρkj(0) for k �= j

ρkk(t) = ρkk(0) .(11.2.3)

The decoherence affects only the off-diagonal elements and leaves the diagonal ele-ments untouched.This kind of decoherence mode is already used in Refs.[29, 31] to describe possibledecoherence scenarios for neutral meson systems (see also Chapt.10, in particularEqs.(10.1.6) and (10.1.9)).

11.3 Mode B: GR⊗ E

The next mode combines the general rotated basis in one subspace with the eigen-basis in the other subspace. The projection operators are given by

Pk = |ek〉〈ek| , (11.3.1)

103

CHAPTER 11. Decoherence modes in a two qubit system 11.3. Mode B: GR ⊗ E

with the projection state

|e1〉 = |α1〉|b1〉 = cosθ

2|e1〉+ eiφ sin

θ

2|e3〉

|e2〉 = |α1〉|b2〉 = cosθ

2|e2〉+ eiφ sin

θ

2|e4〉

|e3〉 = |α2〉|b1〉 = − sinθ

2|e1〉+ eiφ cos

θ

2|e3〉

|e4〉 = |α2〉|b2〉 = − sinθ

2|e2〉+ eiφ cos

θ

2|e4〉 .

(11.3.2)

We get for the matrix of the dissipator

D(ρ) = λ

⎛⎜⎜⎝D11 ρ12 D13 ρ14

ρ∗12 D22 ρ23 D24

D31 ρ∗23 D33 ρ34

ρ∗14 D42 ρ∗34 D44

⎞⎟⎟⎠ , (11.3.3)

where

D11 = −12

(12

sin 2θ(ρ13eiφ + ρ31e

−iφ) + sin2 θ(ρ33 − ρ11))

D22 = −12

(12


−iφ) + sin2 θ(ρ44 − ρ22))

D33 =12

(12


−iφ) + sin2 θ(ρ33 − ρ11))

D44 =12

(12


−iφ) + sin2 θ(ρ44 − ρ22))

D13 =14((cos 2θ + 3)ρ13 + (cos 2θ − 1)e−2iφρ31 + (ρ33 − ρ11)e−iφ sin 2θ)

D31 =14((cos 2θ + 3)ρ31 + (cos 2θ − 1)e2iφρ13 + (ρ33 − ρ11)eiφ sin 2θ)

D24 =14((cos 2θ + 3)ρ24 + (cos 2θ − 1)e−2iφρ42 + (ρ44 − ρ22)e−iφ sin 2θ)

D42 =14((cos 2θ + 3)ρ42 + (cos 2θ − 1)e2iφρ24 + (ρ44 − ρ22)eiφ sin 2θ) .

(11.3.4)

This provides us with two types of differential equations. Type I,

˙ρ12 = −λ ρ12 (11.3.5)

is uncoupled, holds for the components ρ12, ρ14, ρ23, ρ34 and has the solution

ρ12(t) = e−λtρ12(0) . (11.3.6)

104

11.3. Mode B: GR ⊗ E CHAPTER 11. Decoherence modes in a two qubit system

Type II represents coupled differential equations of the form

˙ρ11 = −λ

2sin2 θρ11 +

λ

2sin2 θρ33 +

λ

4eiφ sin 2θρ13 +

λ

4e−iφ sin 2θρ31

˙ρ33 =λ

2sin2 θρ11 −

λ

2sin2 θρ33 −

λ

4eiφ sin 2θρ13 −

λ

4e−iφ sin 2θρ31

˙ρ13 =λ

4e−iφ sin 2θρ11 −

λ

4e−iφ sin 2θρ33 −

λ

4(cos 2θ + 3)ρ13 −

λ

4(cos 2θ − 1)e−2iφρ31

˙ρ31 =λ


λ


λ

4(cos 2θ − 1)e2iφρ13 −

λ

4(cos 2θ + 3)ρ31 ,

(11.3.7)

and the same system for the components {ρ22, ρ44, ρ24, ρ42}. The solutions for thissystem are given by

ρ11(t) =14

(((1− a)e−λt + 3 + a

)ρ11(0)− (a− 1)(1− e−λt)ρ33(0)

+√

1− a2(1− e−λt)(eiφρ13(0) + e−iφρ31(0)

))ρ33(t) =

14

(− (a− 1)(1− e−λt)ρ11(0) +

((1− a)e−λt + 3 + a

)ρ33(0)

−√

1− a2(1− e−λt)(eiφρ13(0) + e−iφρ31(0)

))ρ13(t) =

14

(√1− a2(1− e−λt)e−iφ

(ρ11(0)− ρ33(0)

)+((3 + a)e−λt + 1− a

)ρ13(0)− (a− 1)(1− e−λt)e−2iφρ31(0)

)ρ31(t) =

14

(√1− a2(1− e−λt)eiφ

(ρ11(0)− ρ33(0)

)− (a− 1)(1− e−λt)e2iφρ13(0) +

((3 + a)e−λt + 1− a

)ρ31(0)

),

(11.3.8)

where a = cos 2θ.Now let us consider two special cases to get a feeling for this mode of decoherence.

11.3.1 Mode B1: R⊗ E

One special case of the mode GR⊗E is the mode R⊗E where we have to set θ = π2

or a = −1 and φ = 0.The projection states reduce to

|e1,3〉 =1√2(|e1〉 ± |e3〉) , |e2,4〉 =

1√2(|e2〉 ± |e4〉) , (11.3.9)

where the upper (lower) sign corresponds to the fist (second) index and the dissipator

105

CHAPTER 11. Decoherence modes in a two qubit system 11.3. Mode B: GR ⊗ E

is given by

D(ρ) = λ

⎛⎜⎜⎝12(ρ11 − ρ33) ρ12

12(ρ13 − ρ31) ρ14

ρ∗1212(ρ22 − ρ44) ρ23

12(ρ24 − ρ42)

12(ρ31 − ρ13) ρ∗23

12(ρ33 − ρ11) ρ34

ρ∗1412(ρ42 − ρ24) ρ∗34

12(ρ44 − ρ22)

⎞⎟⎟⎠ . (11.3.10)

The equations of type I are left unchanged (see Eqs.(11.3.5) and (11.3.6)). Type IIreduced to pairwise coupled differential equations of the form

˙ρ11 = −λB

2ρ11 +

λB

2ρ33 , ˙ρ33 =

λB

2ρ11 −

λB

2ρ33 , (11.3.11)

which also hold for the components {ρ22, ρ44}, {ρ13, ρ31}, {ρ24, ρ42}. These equationscan be solved by

ρ11(t) =12(1 + e−λBt)ρ11(0) +

12(1− e−λBt)ρ33(0) ,

ρ33(t) =12(1− e−λBt)ρ11(0) +

12(1 + e−λBt)ρ33(0) .

(11.3.12)

The decoherence mode affects not only the off-diagonal elements of the density ma-trix but also the diagonal ones. Eq.(11.3.9) indicates the basis where the densitymatrix gets diagonal for t→∞.

11.3.2 Mode B2: IR⊗ E

The second special case is IR⊗E where we have to set θ = π2 or a = −1 and φ = π

2 .We get for the projection states

|e1,3〉 =1√2(|e1〉 ± i|e3〉) , |e2,4〉 =

1√2(|e2〉 ± i|e4〉) , (11.3.13)

and the dissipator

D(ρ) = λ

⎛⎜⎜⎝12(ρ11 − ρ33) ρ12

12(ρ13 + ρ31) ρ14

ρ∗1212(ρ22 − ρ44) ρ23

12(ρ24 + ρ42)

12(ρ31 + ρ13) ρ∗23

12(ρ33 − ρ11) ρ34

ρ∗1412(ρ42 + ρ24) ρ∗34

12(ρ44 − ρ22)

⎞⎟⎟⎠ . (11.3.14)

The equations of type I are again left unchanged (see Eqs.(11.3.5) and (11.3.6)) buttype II gives to kinds of pairwise coupled differential equations. Type IIa holds forthe diagonal element ({ρ11, ρ33}, {ρ22, ρ44}) and is given by

˙ρ11 = −λB

2ρ11 +

λB

2ρ33 , ˙ρ33 =

λB

2ρ11 −

λB

2ρ33 . (11.3.15)

Type IIb holds for {ρ13, ρ31}, {ρ24, ρ42} and reads

˙ρ13 = −λ

2ρ13 −

λ

2ρ31 , ˙ρ31 = −λ

2ρ31 −

λ

2ρ13 . (11.3.16)

106

11.4. Mode C: GR ⊗ GR CHAPTER 11. Decoherence modes in a two qubit system

The solutions of type IIa is given by

ρ11(t) =12(1 + e−λBt)ρ11(0) +

12(1− e−λBt)ρ33(0) ,

ρ33(t) =12(1− e−λBt)ρ11(0) +

12(1 + e−λBt)ρ33(0) .

(11.3.17)

and type IIb is solved by

ρ11(t) =12(1 + e−λBt)ρ11(0)− 1

2(1− e−λBt)ρ33(0) ,

ρ33(t) =12(1− e−λBt)ρ11(0)− 1

2(1 + e−λBt)ρ33(0) .

(11.3.18)

11.4 Mode C: GR⊗GR

For the third mode under consideration both subspaces are generally rotated. Theprojection operators are given by

Pk = |ek〉〈ek| , (11.4.1)

with the projection states

|e1〉 = |α1〉|β1〉 = cosθ1

2cos

θ2

2|e1〉+ eiφ2 cos

θ1

2sin

θ2

2|e2〉

+ eiφ1 sinθ1

2cos

θ2

2|e3〉+ ei(φ1+φ2) sin

θ1

2sin

θ2

2|e4〉

|e2〉 = |α1〉|β2〉 = − cosθ1

2sin

θ2

2|e1〉+ eiφ2 cos

θ1

2cos

θ2

2|e2〉

− eiφ1 sinθ1

2sin

θ2

2|e3〉+ ei(φ1+φ2) sin

θ1

2cos

θ2

2|e4〉

|e3〉 = |α2〉|β1〉 = − sinθ1

2cos

θ2

2|e1〉 − eiφ2 sin

θ1

2sin

θ2

2|e2〉

+ eiφ1 cosθ1

2cos

θ2

2|e3〉+ ei(φ1+φ2) cos

θ1

2sin

θ2

2|e4〉

|e4〉 = |α2〉|β2〉 = sinθ1

2sin

θ2

2|e1〉 − eiφ2 sin

θ1

2cos

θ2

2|e2〉

− eiφ1 cosθ1

2sin

θ2

2|e3〉+ ei(φ1+φ2) cos

θ1

2cos

θ2

2|e4〉 .

(11.4.2)

The general situation where both subspaces are rotated is rather difficult to solve andto write down in a closed analytical expression because of the number of parametersinvolved. Therefore we present two special cases in the following sections.

11.4.1 Mode C1: R⊗R

For θ1 = θ2 = π2 and φ1 = φ2 = 0 we get the fist special case R⊗R.

107

CHAPTER 11. Decoherence modes in a two qubit system 11.4. Mode C: GR ⊗ GR

The projection states are given by

|e1〉 =1√2(|e1〉+ |e2〉+ |e3〉+ |e4〉)

|e2〉 =1√2(−|e1〉+ |e2〉 − |e3〉+ |e4〉)

|e3〉 =1√2(−|e1〉 − |e2〉+ |e3〉+ |e4〉)

|e4〉 =1√2(|e1〉 − |e2〉 − |e3〉+ |e4〉) ,

(11.4.3)

and the dissipator reads

D(ρ) = λ

⎛⎜⎜⎝ρ11 − 1

4h0 ρ12 − 14h1 ρ13 − 1

4h2 ρ14 − 14h3

ρ21 − 14h1 ρ22 − 1

4h0 ρ23 − 14h3 ρ24 − 1

4h2

ρ31 − 14h2 ρ32 − 1

4h3 ρ33 − 14h0 ρ34 − 1

4h1

ρ41 − 14h3 ρ42 − 1

4h2 ρ43 − 14h1 ρ44 − 1

4h0

⎞⎟⎟⎠ , (11.4.4)

with the expressions

h0 = ρ11 + ρ22 + ρ33 + ρ44 h1 = ρ12 + ρ21 + ρ34 + ρ43

h2 = ρ13 + ρ24 + ρ31 + ρ42 h3 = ρ14 + ρ23 + ρ32 + ρ41 .(11.4.5)

This gives one type of coupled differential equations

˙ρ11 = −3λ

4ρ11 +

λ

4ρ22 +

λ

4ρ33 +

λ

4ρ44

˙ρ22 =λ

4ρ11 −

3λ

4ρ22 +

λ

4ρ33 +

λ

4ρ44

˙ρ33 =λ

4ρ11 +

λ

4ρ22 −

3λ

4ρ33 +

λ

4ρ44

˙ρ44 =λ

4ρ11 +

λ

4ρ22 +

λ

4ρ33 −

3λ

4ρ44 ,

(11.4.6)

and the same for {ρ12, ρ21, ρ34, ρ43} , {ρ13, ρ24, ρ31, ρ42} and {ρ14, ρ23, ρ32, ρ41}. Thesolutions are of the form

ρ11(t) =14

((1 + 3e−λt)ρ11(0) + (1− e−λt)

(ρ22(0) + ρ33(0) + ρ44(0)

))ρ22(t) =

14

((1− e−λt)ρ11(0) + (1 + 3e−λt)ρ22(0) + (1− e−λt)

(ρ33(0) + ρ44(0)

))ρ33(t) =

14

((1− e−λt)

(ρ11(0) + ρ22(0)

)+ (1 + 3e−λt)ρ33(0) + (1− e−λt)ρ44(0)

)ρ44(t) =

14

((1− e−λt)

(ρ11(0) + ρ22(0) + ρ33(0)

)+ (1 + 3e−λt)ρ44(0)

).

(11.4.7)

We see that mixing occurs between the matrix elements involved in one system ofcoupled differential equations in dependence of the initial conditions.

108

11.4. Mode C: GR ⊗ GR CHAPTER 11. Decoherence modes in a two qubit system

11.4.2 Mode C2: IR⊗ IR

For θ1 = θ2 = π2 and φ1 = φ2 = π

2 we recover the second special case IR⊗ IR wherethe projection states are given by

|e1〉 =1√2(|e1〉+ i|e2〉+ i|e3〉 − |e4〉)

|e2〉 =1√2(−|e1〉+ i|e2〉 − i|e3〉 − |e4〉)

|e3〉 =1√2(−|e1〉 − i|e2〉+ i|e3〉 − |e4〉)

|e4〉 =1√2(|e1〉 − i|e2〉 − i|e3〉 − |e4〉) .

(11.4.8)

We get for the dissipator

D(ρ) = λ

⎛⎜⎜⎝ρ11 − 1

4k0 ρ12 − 14k1 ρ13 − 1

4k2 ρ14 − 14k3

ρ21 + 14k1 ρ22 − 1

4k0 ρ23 − 14k3 ρ24 − 1

4k2

ρ31 + 14k2 ρ32 + 1

4k3 ρ33 − 14k0 ρ34 − 1

4k1

ρ41 + 14k3 ρ42 + 1

4k2 ρ43 + 14k1 ρ44 − 1

4k0

⎞⎟⎟⎠ (11.4.9)

with the expressions

k0 = ρ11 + ρ22 + ρ33 + ρ44 k1 = ρ12 − ρ21 + ρ34 − ρ43

k2 = ρ13 − ρ31 + ρ24 − ρ42 k3 = ρ14 − ρ41 + ρ23 − ρ32(11.4.10)

We have three types of coupled differential equations. The first type holds for thediagonal elements {ρ11, ρ22, ρ33, ρ44} and is identical to mode C1, Eqs.(11.4.6) andthe solution (11.4.7). The second type of equations is

˙ρ12 = −3λ

4ρ12 −

λ

4ρ21 +

λ

4ρ34 −

λ

4ρ43

˙ρ21 = −λ

4ρ12 −

3λ

4ρ21 −

λ

4ρ34 +

λ

4ρ43

˙ρ34 =λ

4ρ12 −

λ

4ρ21 −

3λ

4ρ34 −

λ

4ρ43

˙ρ43 = −λ

4ρ12 +

λ

4ρ21 −

λ

4ρ34 −

3λ

4ρ43 ,

(11.4.11)

and holds for {ρ12, ρ21, ρ34, ρ43} and {ρ13, ρ31, ρ24, ρ42}. The third type holds for{ρ14, ρ41, ρ23, ρ32} and reads

˙ρ14 = −3λ

4ρ14 +

λ

4ρ41 −

λ

4ρ23 −

λ

4ρ32

˙ρ41 =λ

4ρ14 −

3λ

4ρ41 −

λ

4ρ23 −

λ

4ρ32

˙ρ23 = −λ

4ρ14 −

λ

4ρ41 −

3λ

4ρ23 +

λ

4ρ32

˙ρ32 = −λ

4ρ14 −

λ

4ρ41 +

λ

4ρ23 −

3λ

4ρ32 .

(11.4.12)

109

CHAPTER 11. Decoherence modes in a two qubit system 11.5. Initial condition

The solutions for type II are given by

ρ12(t) =14

((1 + 3e−λt)ρ12(0) + (1− e−λt)

(−ρ21(0) + ρ34(0)− ρ43(0)

))ρ21(t) =

14

(−(1− e−λt)ρ12(0) + (1 + 3e−λt)ρ21(0) + (1− e−λt)

(−ρ34(0) + ρ43(0)

))ρ34(t) =

14

((1− e−λt)

(ρ12(0)− ρ21(0)

)+ (1 + 3e−λt)ρ34(0)− (1− e−λt)ρ43(0)

)ρ43(t) =

14

((1− e−λt)

(−ρ12(0) + ρ21(0)− ρ34(0)

)+ (1 + 3e−λt)ρ43(0)

),

(11.4.13)

and for type III we get

ρ14(t) =14

((1 + 3e−λt)ρ14(0) + (1− e−λt)

(ρ41(0)− ρ23(0)− ρ32(0)

))ρ41(t) =

14

((1− e−λt)ρ14(0) + (1 + 3e−λt)ρ41(0) + (1− e−λt)

(−ρ23(0)− ρ32(0)

))ρ23(t) =

14

((1− e−λt)

(−ρ14(0)− ρ41(0)

)+ (1 + 3e−λt)ρ23(0) + (1− e−λt)ρ32(0)

)ρ32(t) =

14

((1− e−λt)

(−ρ14(0)− ρ41(0) + ρ23(0)

)+ (1 + 3e−λt)ρ32(0)

).

(11.4.14)

Only components of the density matrix which are part of the same ensemble ofdifferential equations show mixing depending on the initial conditions.

11.5 Initial condition – Bell singlet state

We want to illustrate the above discussed decoherence modes by choosing the Bellsinglet state |Ψ−〉 as initial condition which is a pure δ = 1 and maximally entangledstate (concurrence C = 1).

11.5.1 Mode A: E ⊗ E

For mode A (see Sect.11.2) we end up with the state

ρ(t) =12

⎛⎜⎜⎝0 0 0 00 1 −e−λt 00 −e−λt 1 00 0 0 0

⎞⎟⎟⎠ . (11.5.1)

We get for the mixedness

δ =12(1 + e−2λt) , (11.5.2)

which ranges from a pure state (δ = 1) to a mixed but not maximally mixed state(δ t→∞−−−→ 1

2). The concurrenceC(ρ) = e−λt , (11.5.3)

110

11.5. Initial condition CHAPTER 11. Decoherence modes in a two qubit system

goes from a maximally entangled state (C = 1) to a separable state asymptotically(C t→∞−−−→ 0).The behavior of δ and C is plotted in Fig.11.1.

11.5.2 Mode B: GR⊗ E

The decoherence mode B, Sect.11.3, where one subspace is rotated gives with theinitial condition of the Bell singlet state the following density matrix

ρ(t) =18·⎛⎜⎜⎜⎝

2ζ sin2 θ 0 −ζe−iφ sin(2θ) 0

0 3 + e−λt + ζ cos(2θ) −4e−λt ζe−iφ sin(2θ)

−ζeiφ sin(2θ) −4e−λt 3 + e−λt + ζ cos(2θ) 0

0 eiφζ sin(2θ) 0 2ζ sin2 θ

⎞⎟⎟⎟⎠(11.5.4)

where ζ = 1 − e−λt and θ and φ specify the basis in the first subspace. We cancalculate the mixedness of this state

δ =18(3 + 5e−2λt + (1− e−2λt) cos(2θ)

)(11.5.5)

and the concurrence, which gives a rather complicated analytical formula we donot want to present here. We note that the mixedness and the concurrence do notdepend on the angle φ but only on the angle θ which leads us to the conclusion thatmode B1 and B2 are equivalent for the initial condition of a Bell singlet state dueto their equal θ-values of π

2 .Now let us have a look at two special cases. The first one arises for θ = 0 and revealsexactly mode A. The second case is represented by the modes discussed in Sect.11.3,mode B1 (R⊗E) and mode B2 (IR⊗E), which are equivalent. We get for the finalstate

ρ(t) =14

⎛⎜⎜⎝1− e−λt 0 0 0

0 1 + e−λt −2e−λt 00 −2e−λt 1 + e−λt 00 0 0 1− e−λt

⎞⎟⎟⎠ , (11.5.6)

where the mixedness gives

δ =14(1 + 3e−2λt) , (11.5.7)

which goes from a pure state (δ = 1) to a maximally mixed state (δ t→∞−−−→ 14). The

amount of entanglement

C(ρ) = max{0,

12(3e−λt − 1)

}, (11.5.8)

starts at the maximally entangled state (C = 1), decreases and reaches the border ofseparability (C = 0) at finite time t = ln 3

λ where the mixing has the value of δ = 13 .

In Fig.11.1 the dependence of δ and C with respect to λt is shown for the twoextremal cases of state (11.5.4). The upper curves represent mode A where θ = 0,

111

CHAPTER 11. Decoherence modes in a two qubit system 11.5. Initial condition

the lower curves represent mode B1 and B2 where θ = π2 . The curves for all other

modes with 0 ≤ θ ≤ π2 are in between these two extremal curves.

At the point λt = ln 3 mode B1 and B2 reach the border of separability whereas inmode A the state is still entangled by an amount of 33%. This is in close relationto the work of Yu and Eberly [154] who consider so called survival inequalities forentanglement. That means for certain parameterized states they investigate theinequality C > 0 for certain decoherence modes.

1

δ

λt

ln 3

12

14

1

C

λt

ln 3

Figure 11.1: Graphical comparison of the mixedness δ and the concurrence C in depen-

dence of λt for mode A (upper curve) and modes B1 and B2 (lower curve)

11.5.3 Mode C: GR⊗GR

In Sect.11.4 we did not get a general expression for the mode C where both subspacesare rotated in a general way but we investigated two special cases. For mode C1both spaces were rotated by an angle of θ1 = θ2 = π

2 , φ1 = φ2 = 0 and for mode C2we choose the angles θ1 = θ2 = π

2 and φ1 = φ2 = π2 .

With the initial condition of the Bell singlet state |Ψ−〉 we get the following differentfinal states

ρ(t) =14

⎛⎜⎜⎝1− e−λt 0 0 −1 + e−λt

0 1 + e−λt −1− e−λt 00 −1− e−λt 1 + e−λt 0

−1 + e−λt 0 0 1− e−λt

⎞⎟⎟⎠ , (11.5.9)

and

ρ(t) =14

⎛⎜⎜⎝1− e−λt 0 0 1− e−λt

0 1 + e−λt −1− e−λt 00 −1− e−λt 1 + e−λt 0

1− e−λt 0 0 1− e−λt

⎞⎟⎟⎠ . (11.5.10)

Although they look different the have the same behavior concerning mixing andconcurrence which gives

δ =12(1 + e−2λt

), and C(ρ) = e−λt . (11.5.11)

Note the similarities to mode A.

112

Chapter 12

Decoherence in neutron

interferometry

In Chapt.11 we discuss several types of decoherence modes and analyze them forthe Bell singlet state |Ψ−〉 (Sect.11.5). The desire is to implement the theoreticallyobtained models in experiments and to study them under controlled conditions.This can be done within neutron interferometry as demonstrated in the followingsections.We use the experimental setup discussed in Sect.5.3 where single neutrons showentanglement between an internal degrees of freedom (spin) and an external degreeof freedom (path) which is formally described by a bipartite Hilbertspace H =Hspin⊗Hpath. The only modification is that we do not make use of entangled statesbut of separable states. The reason is that they are easier to implement althoughthe conclusions for decoherence stay the same.The work presented in this chapter is based on Ref.[27]. We concentrate our ex-perimental considerations on the decoherence modes A, B1 and C1 of Sects.11.2,11.3.1 and 11.4.1 and show that they can be implemented by randomly fluctuat-ing magnetic fields. In Sect.12.2 we present the connection to the Kraus operatordecomposition and another experimental approach.

12.1 Implementation of decoherence modes

12.1.1 State preparation and detection

In the following treatment we consider separable states of the form

|Ψexp〉 =1√2(|⇑〉 ⊗ |I〉+ |⇑〉 ⊗ |II〉) ≡ 1√

2(|e1〉+ |e2〉) , (12.1.1)

where |⇑〉 and |⇓〉 represent ±z polarized spin states whereas |I〉 and |II〉 denotethe paths in the interferometer. Experimentally such states are easier to implementand can be manipulated more simply and accurately in the experiment because we

113

CHAPTER 12. Decoherence in neutron interferometry 12.1. Implementation

do not need the special kind of spin turner used to produce entangled states of theform

|Ψ−〉 =1√2(|⇑〉 ⊗ |II〉 − |⇓〉 ⊗ |I〉) ≡ 1√

2(|e2〉 − |e3〉) . (12.1.2)

The state (12.1.1) is also very suitable to implement and investigate the decoherencemodes discussed in Chapt.11 for neutrons which can be shown by the final states

ρA(t) =12

⎛⎜⎜⎝1 e−λAt 0 0

e−λAt 1 0 00 0 0 00 0 0 0

⎞⎟⎟⎠

ρB(t) =14

⎛⎜⎜⎝1 + e−λBt 2e−λBt 0 02e−λBt 1 + e−λBt 0 0

0 0 1− e−λBt 00 0 0 1− e−λBt

⎞⎟⎟⎠

ρC(t) =14

⎛⎜⎜⎝1 + e−λCt 1 + e−λCt 0 01 + e−λCt 1 + e−λCt 0 0

0 0 1− e−λCt 1− e−λCt

0 0 1− e−λCt 1− e−λCt

⎞⎟⎟⎠ .

(12.1.3)

These states are equivalent to Eqs.(11.5.1), (11.5.6) and (11.5.9).The state detection is done by quantum state tomography proposed in Ref.[87] andexperimentally confirmed for entangled states within neutron interferometry [72]. Inthis experiment the four main poles in the real-part of the density matrix, which isone of the characteristics of the Bell state, clearly showed up.

12.1.2 Decoherence via random magnetic fields

For the implementation of decoherence we use magnetic fields which act randomlyon an ensemble of neutrons produced in a specific state ρ.The action of a magnetic field B = Bn in the direction n on a neutron state isdescribed by the unitary operator U(α) = ei α

2�n·�σ where α = 2μBt = ωLt denotes

the rotation angle and ωL the Larmor frequency (μ magnetic moment, B magneticfield strength, σ Pauli matrices).For example a magnetic field oriented along the z-axis acts on the spin states as

|⇑〉 −→ ei α2

σz |⇑〉 = ei α2 |⇑〉

|⇓〉 −→ ei α2

σz |⇓〉 = e−i α2 |⇓〉 ,

(12.1.4)

and a field oriented along the x-axis gives

|⇑〉 −→ ei α2

σx |⇑〉 = cosα

2|⇑〉+ i sin

α

2|⇑〉

|⇓〉 −→ ei α2

σx |⇓〉 = i sinα

2|⇑〉+ cos

α

2|⇑〉 .

(12.1.5)

Suppose the neutron beam passes a fluctuating magnetic field B(t) in such a waythat each neutron which is part of the quantum mechanical ensemble described by

114

12.1. Implementation CHAPTER 12. Decoherence in neutron interferometry

ρ experiences a time independent field B|t = const. This corresponds to a unitaryoperator U(α) with constant rotation angle α. For the whole ensemble we have totake the integral over all possible rotation angles α

ρ −→ ρ′ =∫

U(α) ρ U †(α)︸︷︷︸ρ(α)

P (α)dα , (12.1.6)

where P (α) denotes a distribution function. Due to the integration the evolution ofthe whole ensemble state is described by a non-unitary operation. We use Gaussian

distribution functions P (α) = 1√2πσ

e−α2

2σ2 with standard deviation σ.

12.1.3 Mode A

The state |Ψexp〉 is prepared after going through the beam-splitter. This initial statesuffers from the dissipative magnetic fields oriented along the z-axis in each path ofthe interferometer, see Fig.12.1. The rotations U(α) and U(β) caused by the fieldsare independent but their distributions have the same deviation σ.

|I〉

|II〉

B(1)z

B(2)z

Phase shifter χ

Spin rotator ξ

Figure 12.1: Schematic experimental setup for the implementation of mode A. The mag-

netic field B(1)z and B

(2)z produce independent rotations U(α) and U(β) re-

spectively. With the phase shifter χ and the spin rotator ξ the final state can

be analyzed.

The action of the two magnetic fields can be described by a conditioned operation[110]. Depending on the state of the spatial degree of freedom either operation U(α)or U(β) is applied to the spin state

|ψspin〉 ⊗ |I〉 −→ U(α)|ψspin〉 ⊗ |I〉|ψspin〉 ⊗ |II〉 −→ U(β)|ψspin〉 ⊗ |II〉 .

(12.1.7)

For a single neutron the application of the conditioned operation on the initial state

115

CHAPTER 12. Decoherence in neutron interferometry 12.1. Implementation

|Ψexp〉 (12.1.1) gives

ρ(α, β) =12

⎛⎜⎜⎜⎝1 ei α+β

2 0 0e−i α+β

2 1 0 00 0 0 00 0 0 0

⎞⎟⎟⎟⎠ , (12.1.8)

which after integration over α and β turns into

ρ′ =∫

ρ(α, β)P (α)P (β)dα dβ =12

⎛⎜⎜⎜⎝1 e−

σ2

4 0 0

e−σ2

4 1 0 00 0 0 00 0 0 0

⎞⎟⎟⎟⎠ . (12.1.9)

By comparison of Eq.(12.1.3) and Eq.(12.1.9) we immediately see that

λAt =σ2

4(12.1.10)

the decoherence parameter λA is directly related to the deviation σ of the fluctuatingmagnetic fields.

Note, that for only one magnetic field located in one of the paths fluctuating withdeviation σ the above relation is given by λAt = σ2

8 , and for one field acting on bothpaths in the same kind we get λAt = σ2

2 .

12.1.4 Mode B1

For mode B1 the initial state preparation is the same but then slightly differentfluctuating magnetic fields are applied, as shown in Fig.12.2. The different unitaryoperations caused by the magnetic fields are assumed to be independent but thedistributions have the same deviation σ.One neutron is modified as follows

ρ(α, β, γ, δ) =

=12

⎛⎜⎜⎜⎝cos2 γ

2 cos γ2 cos δ

2ei α−β2 − 1

2 i sin γei α∓α2 −i cos γ

2 sin δ2ei α∓β

2

cos γ2 cos δ

2e−i α−β2 cos2 δ

2 −i sin γ2 cos δ

2e−i α∓β2 − 1

2 i sin δei β∓β2

12 i sin γe−i α∓α

2 i sin γ2 cos δ

2ei α∓β2 sin2 γ

2 sin γ2 sin δ

2ei α−β2

i cos γ2 sin δ

2e−i α∓β2 1

2 i sin δe−i β∓β2 sin γ

2 sin δ2e−i α−β

2 sin2 δ2

⎞⎟⎟⎟⎠ ,

(12.1.11)

where the upper sign corresponds to the situation: first the fields in z-direction andthen the fields in x-direction and the lower sign to the other possibility. Because ofthe integration procedure the order of the magnetic fields does not matter any more

116

12.1. Implementation CHAPTER 12. Decoherence in neutron interferometry

|I〉

|II〉

B(1)x

B(1)z

B(2)x

B(2)z

Phase shifter χ

Spin rotator ξ

Figure 12.2: Schematic experimental setup for the implementation of mode B1. The mag-

netic fields B(1)x , B

(2)x , B

(1)z and B

(2)z generate independent unitary rotations

U(γ), U(δ), U(α) and U(β), respectively. The order of the magnetic fields in

each path does not matter in this context. With the phase shifter χ and the

spin rotator ξ the final state can be analyzed.

and we get for the ensemble state after the integrations

ρ′ =∫

ρ(α, β, γ, δ)P (α)P (β)P (γ)P (δ) dα dβ dγ dδ

=14

⎛⎜⎜⎜⎜⎜⎝1 + e−

σ2

2 2e−σ2

2 0 0

2e−σ2

2 1 + e−σ2

2 0 0

0 0 1− e−σ2

2 0

0 0 0 1− e−σ2

2

⎞⎟⎟⎟⎟⎟⎠ ,

(12.1.12)

where we can establish via Eq.(12.1.3) and Eq.(12.1.12) the relation

λBt =σ2

2, (12.1.13)

between decoherence parameter λB and the deviation σ of the Gaussian distributionfunction.

12.1.5 Mode C1

In Fig.12.3 the experimental configuration for mode C1 is shown. We only use onefluctuating magnetic field in x-direction applied over both paths. The distributionof the field is given by the deviation σ.The unitary operation U(α) acts on one neutron by

ρ(α) =12

⎛⎜⎜⎝cos2 α

2 cos2 α2 −1

2 i sinα −12 i sinα

cos2 α2 cos2 α

2 −12 i sinα −1

2 i sinα12 i sinα 1

2 i sinα sin2 α2 sin2 α

212 i sinα 1

2 i sinα sin2 α2 sin2 α

2

⎞⎟⎟⎠ , (12.1.14)

117

CHAPTER 12. Decoherence in neutron interferometry 12.2. Kraus operator decomposition

|I〉

|II〉

Bx

Phase shifter χ

Spin rotator ξ

Figure 12.3: Schematic experimental setup for the implementation of mode C1. The mag-

netic field Bx generates a unitary rotations U(α). With the phase shifter χ

and the spin rotator ξ the final state can be analyzed.

which gives after integration the state for the ensemble

ρ′ =∫

ρ(α)P (α) dα =14

⎛⎜⎜⎜⎜⎜⎝1 + e−

σ2

2 1 + e−σ2

2 0 0

1 + e−σ2

2 1 + e−σ2

2 0 0

0 0 1− e−σ2

2 1− e−σ2

2

0 0 1− e−σ2

2 1− e−σ2

2

⎞⎟⎟⎟⎟⎟⎠ .

(12.1.15)Via Eq.(12.1.3) and Eq.(12.1.15) we are led to the relation

λCt =σ2

2, (12.1.16)

between the decoherence parameter λC and the deviation σ of the Gaussian distri-bution function.

12.2 Kraus operator decomposition

In the following section we give a connection to the Kraus operator decompositionintroduced in Sect.3.4.2. We want to demonstrate that by a simple experimentwithin neutron interferometry we can check if the theoretically predicted Krausoperators correspond to the implemented decoherence models discussed in Sect.12.1.

118

12.2. Kraus operator decomposition CHAPTER 12. Decoherence in neutron interferometry

12.2.1 Mode A

Mode A represents a kind of phase damping channel (compare with Sect.3.4.4) whichdestroys the coherence in the system. The Kraus operators are given by

M0 =

√1− 3p

4�s ⊗ �p M1 =

√p

4�s ⊗ σp

z

M2 =√

p

4σs

z ⊗ �p M3 =√

p

4σs

z ⊗ σpz ,

(12.2.1)

where p = λt is the probability for the decoherence taking place which leads in firstorder of t to the state ρ(t), Eq.(11.2.3).

Bz(π)

Bz(π) Phase shifter (π)

Phase shifter (χ)

Spin rotator (ξ)

|I〉

|II〉

Figure 12.4: Schematic experimental setup for the realization of the Kraus operator σsz⊗σp

z

for mode A.

Experimentally the Kraus operators can be realized in the following way. Theidentity-operators �s and �p do nothing to the spinor and spatial degree of free-dom. The operator σs

z on the spinor part operates only on the spin state |⇓〉 whereit induces a phase shift of π. This phase shift difference of spin up and spin downcan be implemented by a magnetic field Bz in z-direction (modulo an overall phaseshift). The operator σp

z on the spatial subspace can be realized by a phase shifterPS in the path |II〉 which acts with a fixed phase shift of π.The states produced by the four Kraus operators are measured and the weightedsum according to (12.2.1) will show the expected behavior.

12.2.2 Mode B

Mode B is a combination of a bit flip channel and a phase flip channel, discussed inSect.3.4.3. The corresponding Kraus operators are

M0 =

√1− 3p

4�s ⊗ �p M1 =

√p

4�s ⊗ σp

z

M2 =√

p

4σs

x ⊗ �p M3 =√

p

4σs

x ⊗ σpz ,

(12.2.2)

119

CHAPTER 12. Decoherence in neutron interferometry 12.3. Conclusion

and create in first order of t the state given by Eq.(11.3.12) with p = λt.

Bx

Bx Phase shifter (π)

Phase shifter (χ)

Spin rotator (ξ)

|I〉

|II〉

Figure 12.5: Schematic experimental setup for the realization of the Kraus operator σsx⊗σp

z

for mode B.

The Kraus operator for the spatial part σpz is the same as for mode A, the π-spin

flipper SF. The only difference is the σsx operator for the spin part. It can be realized

by two magnetic fields Bx in both arms pointing in the x-direction which cause aspin flip. The experimental setup can be seen in Fig.12.5.The weighted sum of the measured states produced by the Kraus operators accordingto (12.2.2) will show the expected behavior.

12.3 Conclusion

The implementation of the proposed decoherence modes uses the bipartite Hilbertspace construction H = Hspin ⊗Hpath of neutron interferometry.We create decoherence via random magnetic fields in the interferometer (Sect.12.1)where we can establish simple relations between the decoherence parameter λ andthe deviation of the random distribution of the fields σ, Eqs. (12.1.10), (12.1.13)and (12.1.16). This allows an experimental control of the implemented decoherencein each mode. The strength of decoherence does not depend on the actual rotationparameters of the magnetic fields but only on the width of the distribution. We areable to construct any state predicted by the decoherence modes just by varying σ.The final states are measured by state tomography [72, 87].In the second type of proposed neutron experiment we want to test experimentallythe validity of the Kraus operator decomposition which describes completely positivetime evolutions (Sect.12.2). The Kraus operators for each mode are determined andimplemented within neutron interferometry. We show that by 4 different Krausoperators for each decoherence mode the final states can be modelled where eachKraus operator must be implemented separately. The state determination is doneby state tomography measurements.

120

Chapter 13

Geometry of decoherence modes

Geometric pictures are very useful for our intuition and the basic understanding.Therefore it is of interest to study the decoherence modes introduced in Chapt.11in terms of a geometric picture. We find that the spin geometric picture presentedin Sect.7.3.3 is very suitable to visualize the loss of entanglement in the differentdecoherence modes. For the sake of definiteness we restrict ourselves to the initialcondition of the Bell singlet state |Ψ−〉 already studied in Sect.11.5.The considerations of this chapter are based on Ref.[26]. We present geometricpictures for the modes A, B and C in Sects.13.1, 13.2 and 13.3 and discuss theKraus operator representation of these modes in Sect.13.4.

13.1 Mode A: E ⊗ E

The final state given by Eq.(11.5.1) reveals for the correlation matrix, Eq.(7.2.3),

c =

⎛⎝−e−λt 0 00 −e−λt 00 0 −1

⎞⎠ , (13.1.1)

and the local parameters are m = n = 01. The vector c for the spin geometry pictureis given by

c =

⎛⎝−e−λt

−e−λt

−1

⎞⎠ . (13.1.2)

The standardized vector c for λ = 1 is plotted in Fig.13.1 with respect to varying t.We start in the corner indicated by |Ψ−〉 and for t→∞ approach the point bisectingthe line connecting the projectors of the states |Ψ−〉 and |Ψ+〉. This asymptotic stateis mixed but not maximally mixed and lies at the border of separability which isgiven by the blue octahedron.

1In the following considerations the local parameters �m and �n always vanish.

121

CHAPTER 13. Geometry of decoherence modes 13.2. Mode B: GR ⊗ E

��

��

��

��

Figure 13.1: The red line depicts decoherence mode A visualized in the spin geometry

picture.

13.2 Mode B: GR⊗ E

Mode B denotes a rotation in one subspace whereas the other subspace is left inthe eigenbasis. The final state for a general rotation, Eq.(11.5.4), gives for thecorrelation matrix

c =

⎛⎝−e−λt 0 (−1 + e−λt) sin θ cos θ0 −e−λt 00 0 − cos2 θ − e−λt sin2 θ

⎞⎠ . (13.2.1)

The vector c is obtained as

c =

⎛⎝−e−λt

−f(θ)−g(θ)

⎞⎠ , (13.2.2)

where the function g(θ) is nonlinear in θ and f(θ) ∼ e−λt.For the special case of mode B1 and B2 where θ = π

2 we get for the correlationmatrix

c =

⎛⎝−e−λt 0 00 −e−λt 00 0 −e−λt

⎞⎠ , (13.2.3)

which gives for the vector

c =

⎛⎝−e−λt

−e−λt

−e−λt

⎞⎠ . (13.2.4)

In Fig.13.2 the c-vectors are shown with respect to evolving time t. The first graphdepicts the path traced out by mode B1 and B2, respectively. The state approachesthe totally mixed state sitting at the origin of the coordinate system. Thereby itreaches the border of separability at t = ln 3 (we have set λ = 1). The second graphshows the paths of the decoherence mode for different values of θ. The paths forvarying θ are dense in the plane drawn in the third plot. Thus we have obtaineda graphical interpretation of the statements gained in Sect.11.5 and depicted in

122

13.3. Mode C: GR ⊗ GR CHAPTER 13. Geometry of decoherence modes

Fig.11.1. The time the octahedron of separability is reached varies with respect toθ and the extremal cases are θ = π

2 where the border is reached after the shortesttime and θ = 0 where it is reached at infinity.

��

��

��

��

��

��

��

��

��

��

��

��

Figure 13.2: The red lines represent possible decoherence paths of mode B. The first pic-

ture depicts the modes B1 and B2 which are identical. The second picture

visualizes mode B for different θ-values and the third picture shows that the

paths depending on θ are dense in the red plane.

13.3 Mode C: GR⊗GR

We do not have an analytical form of the general solution for mode C. Thereforewe only consider the two special cases C1 R⊗R and C2 IR⊗ IR. The correlationmatrices are

c1 =

⎛⎝−1 0 00 −e−λt 00 0 −e−λt

⎞⎠ , c2 =

⎛⎝−e−λt 0 00 −1 00 0 −e−λt

⎞⎠ , (13.3.1)

and we get for the vectors

c =

⎛⎝ −1−e−λt

−e−λt

⎞⎠ , c =

⎛⎝−e−λt

−1−e−λt

⎞⎠ . (13.3.2)

Via Fig.13.3 we can check that the properties of the two special cases of mode C areindeed related to mode A. They are just states that run on another edge, e.g. modeC1 uses the connection between the projectors of the states |Ψ−〉 and |Φ−〉 whereasmode C2 approaches an equal mixture of the projectors onto |Ψ−〉 and |Φ+〉.It is possible to check via numerical calculations the hypothesis that for specialkinds of modes like GR ⊗ R and GR ⊗ IR we get paths similar to mode B whichis GR⊗ E. The results are depicted in Fig.13.4 where the mode GR⊗R is shown.We can observe the same properties for the mode GR ⊗ IR but due to the chosenperspective in the pictures it is difficult to display.

123

CHAPTER 13. Geometry of decoherence modes 13.4. Kraus operators

��

��

��

��

��

��

��

��

Figure 13.3: The red lines represent the paths of mode C1 and C2.

Up to now we have only investigated the basic structures of all possible decoherencemodes. Further work which is already in progress is needed to determine and classifythe properties of general decoherence modes GR⊗GR and to visualize them in thespin geometry picture.

Y-\

F-\

F+\

Y+\

Y-\

F-\

F+\

Y+\

Figure 13.4: Decoherence paths for the mode GR⊗R.

13.4 Kraus operators

It is possible to relate the decoherence modes to Kraus operators (see Sect.3.4.2).This gives a further explanation how the different decoherence modes are related totheir paths in the spin geometry picture.Mode A E ⊗ E can be represented by the following Kraus operators

M0 =

√1− 3λt

4�⊗ � M1 =

√λt

4�⊗ σ3

M2 =

√λt

4σ3 ⊗ � M3 =

√λt

4σ3 ⊗ σ3 .

(13.4.1)

124

13.4. Kraus operators CHAPTER 13. Geometry of decoherence modes

For mode B1 R⊗ E we need

M0 =

√1− 3λt

4�⊗ � M1 =

√λt

4�⊗ σ3

M2 =

√λt

4σ1 ⊗ � M3 =

√λt

4σ1 ⊗ σ3 ,

(13.4.2)

and for mode B2 IR⊗ E

M0 =

√1− 3λt

4�⊗ � M1 =

√λt

4�⊗ σ3

M2 =

√λt

4σ2 ⊗ � M3 =

√λt

4σ2 ⊗ σ3 .

(13.4.3)

Mode C1 R⊗R can be obtained by

M0 =

√1− 3λt

4�⊗ � M1 =

√λt

4�⊗ σ1

M2 =

√λt

4σ1 ⊗ � M3 =

√λt

4σ1 ⊗ σ1 ,

(13.4.4)

and mode C2 IR⊗ IR by

M0 =

√1− 3λt

4�⊗ � M1 =

√λt

4�⊗ σ2

M2 =

√λt

4σ2 ⊗ � M3 =

√λt

4σ2 ⊗ σ2 .

(13.4.5)

We see that the action of the operators σx ⊗ � and σy ⊗ � which is essential formodes B1 and B2 has the same effect and therefore the states lie on the same line(cf. Sect.13.2). The modes A, C1 and C2 represent different aspects of the samegeometry only rotated in the spin geometry picture corresponding on which Krausoperator they are depending σ1, σ2 or σ3.

125

CHAPTER 13. Geometry of decoherence modes 13.4. Kraus operators

126

Chapter 14

Hopf geometry for pure qubit

states

The Hopf fibration introduced in Sect.8.4 is an important object in mathematics andphysics. The variety of physical applications goes from magnetic monopoles [108]and rigid body mechanics [103] right to the field of quantum information [23, 106,107, 141]. This last application to QM and the theory of entanglement is investigatedin the following sections.In Sect.14.1 we discuss the case of one qubit where the state space can be describedby the use of the first Hopf map and we show the visualization of the fibrationvia stereographic projection (Sect.14.2). In Sect.14.3 we proceed to the case of twoqubits. Here it is possible to introduce the second Hopf map such that the fullHilbert space can be factorized in an entanglement sensitive way. In Sect.14.4 wegive an outlook to the case of three qubits and its relation to the third Hopf map.

14.1 Hilbert space for one qubit

In the work of Urbantke [141] the state space of a two-level system – a qubit – isdescribed in terms of Hopf fibration.

14.1.1 State space

The Hilbert space for one qubit is given by C2. A general pure state can be writtenin terms of the standard basis {|0〉, |1〉} (e.g. eigenstates of σz)

|ψ〉 = α|0〉+ β|1〉 (14.1.1)

where α, β ∈ C and |α|2+|β|2 = 1.Due to the isomorphism C2 ∼= R4 the complex 2-dimensional Hilbert space canalso be considered as a real 4-dimensional vector space. But not all elements of R4

represent possible states in QM, we have two limitations.

127

CHAPTER 14. Hopf geometry for pure qubit states 14.1. Hilbert space for one qubit

The first one is the normalization of states which reduces the state spaces C2 andR4 in the following way

|α|2+|β|2 = α21 + α2

2 + β21 + β2

2 = 1 , (14.1.2)

where α, β ∈ C and αi, βi ∈ R. This is exactly the definition of the unit sphere S3.The second limitation concerns the freedom of phase. In QM states are only definedup to a complex phase factor

|ψ〉 ∼ eiη|ψ〉 , (14.1.3)

which means global phase factors are irrelevant for physical quantities such as ex-pectation values. The phase factors {eiη}0≤η<2π form the unitary group U(1) whichis isomorphic to the sphere S1. By factorizing out the global phase factor we end upin the (complex) projective Hilbert space CP1 (the complex projective line). Thenwe can apply another isomorphism CP1 ∼= S2 which allows us to represent state ofqubits on the Bloch sphere (see Sect.2.1.3).To summarize we applied the following procedure S3/U(1) = S3/S1 = CP1 = S2

from the normalized Hilbert space to the projective Hilbert space which can bedescribed by the map

h : S3 S1

−−−−−→ S2 (14.1.4)

which represents the so-called (fist) Hopf map, analyzed in Sect.8.4.2.

14.1.2 First Hopf map

The mapping from the Hilbert space to the projective Hilbert space coincides withthe first Hopf map and takes points (α1, α2, β1, β2) on a 3-sphere to points (r1, r2, r3)on a 2-sphere which can be described by [107]

h(1) : S3 −→ S2

(α1, α2, β1, β2) �−→ (r1, r2, r3)

(α, β) �−→ h(1)(α, β) = ri = 〈σi〉ψ =(α∗ β∗)σi

(αβ

),

(14.1.5)

where σi are the Pauli matrices. Explicitly ri are coordinates on the Bloch spheregiven by (cf. Eq.(2.1.9))

r1 = αβ∗ + α∗β = 2(α1α2 + β1β2)r2 = i(αβ∗ − α∗β) = 2(α1β2 − α2β1)

r3 = |α|2 − |β|2 = (α21 + α2

2)− (β21 + β2

2) .

(14.1.6)

The map h can also be composed out of the following two maps

h1 : S3 −→ R2 + {∞}(α, β) �−→ h1(α, β) = (αβ−1)∗

h2 : R2 + {∞} −→ S2

h1(α, β) �−→ h2(h1(α, β)) = (r1, r2, r3) ,

(14.1.7)

128

14.2. Visualizing the fibration CHAPTER 14. Hopf geometry for pure qubit states

where h(1) = h2 ◦ h1. The second map h2 is just an inverse stereographic projectionfrom the equatorial plane via the north pole (see Sect.8.5). The important part ofthe mapping is h1 which shows clearly that by multiplying the pair (α, β) with aU(1) phase factor eiη the images under h1 stay the same

h1(α′, β′) = h1(αeiη, βeiη) = (α′β′−1)∗ = (αeiηe−iηβ−1)∗ = (αβ−1)∗ = h1(α, β) .

Therefore every point on S2 corresponds to a circle S1, called the Hopf circle, on S3.

14.2 Visualizing the fibration

The Hopf fibration introduced in Sect.8.4.2 can be visualized by stereographic pro-jection from S3 to R3 (see Refs.[122, 141]).

14.2.1 U(1) fibration

The fibres of the Hopf fibration are determined by the phase freedom of QM,Eq.(14.1.3). The global U(1) phase factor eiη acts on the two complex parame-ters of a states as

α −→ α(η) = eiηα , β −→ β(η) = eiηβ , (14.2.1)

which results in

α1(η) = α1 cos η − α2 sin η α2(η) = α2 cos η + α2 sin η

β1(η) = β1 cos η − β2 sin η β2(η) = β2 cos η + β2 sin η .(14.2.2)

We introduce the two orthogonal 4-dimensional vectors x = (α1, α2, β1, β2)T andy = (−α2, α1,−β2, β1)T and describe the variation of the phase by

x(η) = cos η x + sin η y . (14.2.3)

By variation of η the point x traces out a great circle on S3 which can be seenby Eq.(14.2.3). This is exactly the U(1)-fibration of S3. The great circles are allparallel (Clifford parallels, see [20]) in contrast to great circles on the sphere S2

which intersect at two points.

14.2.2 Stereographic S3 picture

We can visualize the fibration of S3 by stereographic projection on the equatorialspace which is R3. First we have to introduce the parametrization

α = cosθ

2eiφ1 , β = sin

θ

2eiφ2 , (14.2.4)

in close relation to Eq.(2.1.10) where φ = φ2−φ1. In the following we show two pos-sibilities to remove the global phase eiη which reveals on the one hand the structureof nested tori and on the other hand the structure of the Villarceau circles.

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CHAPTER 14. Hopf geometry for pure qubit states 14.2. Visualizing the fibration

Nested tori

The first possibility to remove the phase is by considering the absolute square of

|α(η)|2 = |α|2 = cos2θ

2. (14.2.5)

The stereographic projection in the inverse representation, Eq.(8.5.8), gives the fol-lowing

P−1 : R3 −→ S3 \ (0, 0, 0,±1)

(r1, r2, r3, 0) �−→ (α1, α2, β1, β2) =(

2r1

r2 + 1,

2r2

r2 + 1,

2r3

r2 + 1,r2 − 1r2 + 1

).

(14.2.6)

where r2 := r21 + r2

2 + r23. Now we use the relation |α|2 = α2

1 + α22 and insert the

according expressions from Eq.(14.2.6) which gives

α21 + α2

2 =4(r2

1 + r22)

(r21 + r2

2 + r23 + 1)2

= cos2θ

2, (14.2.7)

and can be re-written as

r23 =

±2cos θ

2

√r21 + r2

2 − (r21 + r2

2 + 1) . (14.2.8)

This equation describes a surface in R3 which is invariant under rotations aroundthe 3-axis due to the fact that only the sum r2

1 + r22 appears. Suppose we make a

cut in the 1− 3 plane by setting r2 = 0. Thus we get for Eq.(14.2.7)

2r1

r21 + r2

3 + 1= cos

θ

2

(r1 −1

cos θ2

)2 + r23 =

1cos2 θ

2

− 1 = tan2 θ

2.

(14.2.9)

We clearly see that this defines the equation of a circle with radius tan θ2 centered

at ( 1cos θ

2

, 0, 0). The rotation around the 3-axis provides us with a torus. Therefore

the surface given by Eq.(14.2.8) gives a family of nested, coaxial, concentric toriparameterized by θ ∈ [0, π] (see Fig.14.1).

Villarceau circles

The second possibility to remove the global phase freedom is by considering the ratio

β(η)α(η)

=β

α= tan

θ

2eiφ . (14.2.10)

We analyze the real part of the equation

tanθ

2(cos φ α1 − sinφ α2) = β1 , (14.2.11)

130

14.2. Visualizing the fibration CHAPTER 14. Hopf geometry for pure qubit states

Figure 14.1: The nested tori are shown. There are two exceptional tori drawn with bold

lines: for θ = 0 we get the circle in the 1 − 2 plane and for θ = π we regain

the 3-axis. (Source: [122])

which gives under stereographic projection

tanθ

2(cos φ r1 − sinφ r2) = r3 . (14.2.12)

This equation defines a plane through the origin in R3 with an angle of inclinationagainst the 1−2 plane given by θ

2 . A variation of φ results in a rotation of the planearound the 3-axis. For φ = 0 the projection of the plane onto the 1 − 3 plane is aline which is described by

tanθ

2r1 = r3 . (14.2.13)

Each line specified by θ touches the according torus (14.2.9) at two points. InFig.14.2 the circles and the touching lines are shown for the positive 1-axis. Thecomplete intersection of the bitangent plane with the torus gives two circles whichconstitute the stereographic image of a U(1) fibre.As it turns out these circles intersect each other in two points which are exactly thetouching points between the torus and the bitangent plane. The intersection circlesfor different φ-values between the torus and the planes, Eq.(14.2.12), are linked witheach other.These circles, called Villarceau circles, form a special kind of circles on the torus.Apparently a tours contains two families of circles: the circles of latitude and themeridians. But one can also find another family of circles by intersecting the toruswith a bitangent plane, the Villarceau circles. Circles arising from the same planeintersect at two points and those coming from different planes are linked together(see Fig.14.3).All circles on the same torus have the same radius of 1

cos θ2

and correspond to points

on the same latitude in the S2 picture. Fig.14.4 provides an illustration of the nestedtori with the Villarceau circles on them.

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CHAPTER 14. Hopf geometry for pure qubit states 14.2. Visualizing the fibration

1 2r1

1

2r3

Figure 14.2: The projection of the tori onto the 1−3 plane gives circles which are touched

by the according bitangent planes represented by straight lines. The config-

uration is shown for θ = π8 , π

4 , π2 , 3π

4 . For θ = 0 the line is identical to the

1-axis and the circle has radius zero, for θ = π the line corresponds to the 3

axis and the circle has radius ∞.

Figure 14.3: The section of a torus by a bitangent plane projected to the 1− 2 plane. The

figures of intersection are two circles. Circles arising from different planes are

linked together (Villarceau circles). In the second figure a pair of intersecting

circles from the bitangent plane φ = 0 and one linked circle from the plane

φ = π2 are shown. (Source: [20] and [141])

132

14.3. Hilbert space for two qubits CHAPTER 14. Hopf geometry for pure qubit states

Figure 14.4: The S3 Hopf fibration after a stereographic map onto R3. Circular S1 fibres

are mapped onto circles in R3, except the exceptional fibre through the pro-

jection pole, which is mapped onto a vertical straight line. Fibres can be

grouped into a continuous family of nested tori, three of which are shown

here. This illustration is drawn by R. Penrose. (Source: [20])

14.3 Hilbert space for two qubits

This section is based on the work of R. Mosseri and R. Dandoloff [107, 106] whowere the first to realize the connection between the state space of two qubits andthe (second) Hopf map which turns out to be an entanglement sensitive fibration ofthe state space.

14.3.1 State space

The Hilbert space for two qubits is C2 ⊗ C2 = C4. A general pure state can bewritten in terms of the standard product basis {|00〉, |01〉, |10〉, |11〉}

|ψ〉 = α|00〉+ β|01〉+ γ|10〉+ δ|11〉 , (14.3.1)

where α, β, γ, δ ∈ C and |α|2 + |β|2 + |γ|2 + |δ|2 = 1.The concurrence for this state is given by C = 2|αδ−βγ| that means |ψ〉 is separablefor αδ = βγ otherwise it is entangled (cf. Sect.2.4.3).The normalization of the state vectors reduces the state space C4 ∼= R8 to the 7-dimensional sphere S7. The reduction of the phase freedom gives the projectiveHilbert space S7/U(1) = CP3, which has 6 real dimensions

S7 U(1)∼=S1

−−−−−→ CP3 . (14.3.2)

In contrast to the one qubit case the space CP3 is not isomorphic to a sphere.Therefore we have to look for a different way of representing the two qubit case. Itturns out that quaternions and the second Hopf map serve very well for our purpose.

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CHAPTER 14. Hopf geometry for pure qubit states 14.3. Hilbert space for two qubits

14.3.2 Quaternionic formulation

First we introduce another way of representing the two qubit state in terms of 2quaternions (q1, q2) instead of 4 complex numbers (α, β, γ, δ). The quaternions aredefined by

q1 = α + βj = α1 + α2i + β1j + β2k

q2 = γ + δj = γ1 + γ2i + δ1j + δ2k ,(14.3.3)

where qi ∈ H, α, β, γ, δ ∈ C and αi, βi, γi, δi ∈ R. Therefore the state |ψ〉, Eq.(14.3.1),is given by the pair (q1, q2) satisfying

|q1|2+|q2|2 = |α|2+|β|2+|γ|2+|δ|2 = α21+α2

2+β21+β2

2+γ21+γ2

2+δ21+δ2

2 = 1 , (14.3.4)

which is exactly the definition of the sphere S7.

14.3.3 Second Hopf map

Now we introduce the second Hopf map S7 S3

−−−−−→ S4 which has a very similarstructure to the first Hopf map

h(2) : S7 −→ S4

(q1, q2) �−→ h(2)(q1, q2) = xi = 〈τi〉ψ =(q1 q2

)τi

(q1

q2

)(α1, α2, β1, β2, γ1, γ2, δ1, δ2) �−→ (x1, x2, x3, x4, x5) .

(14.3.5)

The matrices τi are in a way a “quaternionic” extension of the standard Pauli-matrices σi

τ1 =(

0 11 0

)τ2,3,4 =

(0 −i, j, k

i, j, k 0

)τ5 =

(1 00 −1

). (14.3.6)

The Hopf map h(2) takes points (α1, α2, β1, β2, γ1, γ2, δ1, δ2) on a 7-sphere to points(x1, x2, x3, x4, x5) on a 4-sphere. The map h(2) can also be composed out of twomaps h = h2 ◦ h1 where

h1 : S7 −→ R4 + {∞}

(q1, q2) �−→ h1(q1, q2) = q1q−12

h2 : R4 + {∞} −→ S4

h1(q1, q2) �−→ h2(h1(q1, q2)) = (x1, x2, x3, x4, x5) .

(14.3.7)

The map h2 is an inverse stereographic projection.

The fibres of the second Hopf map S7 S3

−−−−−→ S4 are 3-spheres and the fibration isnon-trivial, i.e. S7 �= S4 × S3.

134


The mapping h1 shows that two points on S7 given by (q1, q2) and (q′1 = q1 · q, q′2 =q2 · q) and differing only by a unit quaternion q (equal to an SU(2) phase1) projectto the same image under h1

h1(q′1, q′2) = q′1q

′−12 = q1qqq

−12 = q1q

−12 = h1(q1, q2) . (14.3.8)

The explicit form of the mapping h1 is provided by

h1(α, β, γ, δ) =1

|γ|2 + |δ|2((α∗γ + β∗δ) + (αδ − βγ)j

), (14.3.9)

and shows that states with equal amount of entanglement C = 2|αδ−βγ| are mappedto the same points by h1. This provides us with an entanglement sensitive fibrationof the two-qubit Hilbert space.The final coordinates xi under the inverse stereographic projection h2 from theequatorial R4 to the S4 via the north pole are explicitly given by

x1 = 2e(α∗γ + β∗δ) x3 = 2e(αδ − βγ)x2 = 2�m(α∗γ + β∗δ) x4 = 2�m(αδ − βγ)

x5 =|q1|2 − |q2|2 .

(14.3.10)

In analogy to the Bloch sphere representation for one qubit (compare Eq.(2.1.9))three coordinates are related to expectation values of simple two-qubit operators by

x1 = 〈σx ⊗ �〉ψ x2 = 〈σy ⊗ �〉ψ x5 = 〈σz ⊗ �〉ψ , (14.3.11)

and can be used to define a generalized Bloch sphere (see Eq.(14.3.25)). The othertwo coordinates are related to the spin flipped states defined in Eq.(2.4.5) and there-fore to the concurrence

x3 = e 〈ψ| − σ2 ⊗ σ2|ψ∗〉 = e 〈ψ|−ψ〉x4 = �m 〈ψ| − σ2 ⊗ σ2|ψ∗〉 = �m 〈ψ|−ψ〉 ,

(14.3.12)

which is due to the fact that the fibration is entanglement sensitive.

14.3.4 Relation to state representation

Now we introduce a useful kind of parametrization in close analogy to the case ofone qubit described in Sect.14.2 by Eqs.(14.2.1) and (14.2.4).According to Eq.(14.3.8) quaternions that differ only by a unit quaternion q aremapped to the same point

q1 −→ q1 q , q2 −→ q2 q . (14.3.13)

1Geometrically the space of unit quaternions corresponds to the space S3 ∼= SU(2) (cf. Sect.8.2).

135

CHAPTER 14. Hopf geometry for pure qubit states 14.3. Hilbert space for two qubits

The quaternions can be written in a kind of spinor notation in the following way2

q1 = cosθ

2eφ1t q = cos

θ

2e−

φ2t e

φ1+φ22

t q ,

q2 = sinθ

2eφ2t q = sin

θ

2e

φ2t e

φ1+φ22

t q ,

(14.3.14)

with φ = φ2 − φ1. We find the relations

cos2θ

2+ sin2 θ

2= ‖q1‖2 + ‖q2‖2 , cos φ =

x1

sin θ. (14.3.15)

The state |ψ〉 = (q1, q2) can be written by neglecting an overall phase factor as

ψQ = (cosθ

2e−

φ2tq, sin

θ

2e

φ2t q) , (14.3.16)

where q denotes the unit quaternion which spans the S3 fibre and the base space isdescribed by the pure imaginary unit quaternion t.Another way of writing the general state including the quaternionic SU(2) phase isthe following

ψQ = (cosθ

2q, sin

θ

2Qq) , (14.3.17)

where q denotes the unit quaternion of the S3 fibre and the unit quaternion relatedto the base space is Q = tan θ

2 q1q−12 . Let us write q = a + bj and Q = u + vj

with |a|2 + |b|2 = 1 and |u|2 + |v|2 = 1. The relation between both kinds of staterepresentation (14.3.16) and (14.3.17) is given by

cos φ = e(u) , t =1

sinφ(�m(u) i + e(v) j + �m(v) k) . (14.3.18)

We can write the state (14.3.17) as a quadruplet of complex numbers by

ψQ = (cosθ

2a, cos

θ

2b, sin

θ

2(ua− vb∗), sin

θ

2(ub + va∗)) , (14.3.19)

which can be easily compared to Eq.(14.3.1) where the state is given by

ψ = (α, β, γ, δ) . (14.3.20)

The quantities θ, u and v correspond to the base space of the fibration and thefollowing relations hold

u =x1 + ix2

sin θv =

x3 + ix4

sin θ. (14.3.21)

2Quaternions can be represented in exponential form q = ‖q‖eφ t = ‖q‖(cos φ + sin φ t) where

t is a pure imaginary unit quaternion. If t = i we recover the exponential representation for the

complex numbers.

136


Separable states

For separable states we have x3 = x4 = 0 and therefore v = 0 and u = eφi (accordingto Eqs.(14.3.9) and (14.3.15)) which gives for Eq.(14.3.19)

ψQ = (cosθ

2a, cos

θ

2b, sin

θ

2a eφi, sin

θ

2b eφi) . (14.3.22)

The corresponding state written in the standard basis is given by (up to a globalphase of e−i φ

2 )

|ψQ〉 =(cos

θ

2e−i φ

2 |0〉+ sinθ

2ei φ

2 |1〉)⊗(a|0〉+ b|1〉

). (14.3.23)

We can see clearly that the S4 base described by u and v reduces to an S2 due tothe fact that x3 = x4 = 0. This represents the Bloch sphere for the first qubit.The second qubit is recovered from the fibre spanned by q = a + bj by applying the

first Hopf fibration S3 S1

−−→ S2. Thus we gain the Bloch sphere description for thesecond qubit. This last fibration is necessary to to take into account the global U(1)phase of the total system. We conclude that the projective Hilbert space for twonon-entangled qubits is simply S2 × S2.

Maximally entangled states

Now we proceed to maximally entangled states where the concurrence is equal tounity and thus v = 1 and u = 0 and θ = π

2 which gives for the state

ψQ =1√2(a, b,−b∗, a∗) (14.3.24)

We see that maximally entangled states are described by a pair (a, b) of complexnumbers satisfying |a|2 + |b|2 = 1 and therefore describing the sphere S3. The pair(−a,−b) describes the same state up to a global phase which leads to the conclusionthat opposite points on the S3 have to be identified due to the topological structureS3/Z2 = SO(3).

Generalized Bloch sphere

In Refs.[106, 107] the authors propose an extension of the usual Bloch sphere forthe case of two qubit-states.Suppose we keep the following coordinates of the Hopf fibration

(x1, x2, x5) = (〈σx ⊗ �〉ψ, 〈σy ⊗ �〉ψ, 〈σz ⊗ �〉ψ) . (14.3.25)

They define a ball B3 of radius 1 where the set of separable states forms the bound-ary given by an S2 and the center corresponds to the maximally entangled states.Concentric spherical shells around the center represent states with fixed concurrenceC where the radius of the shells is given by

√1− C2, see Fig.14.5.

137

CHAPTER 14. Hopf geometry for pure qubit states 14.4. Hilbert space for three qubits

max. entangled, C = 1

separable, C = 0

C = const.

Figure 14.5: Generalized Bloch sphere for two qubits.

The relation of the generalized Bloch sphere for two qubit states to the usual Blochsphere representation can be seen by calculating the reduced density matrix, e.g., ofthe first subsystem for the pure state (14.3.1). We obtain

ρ1 =(|α|2 + |β|2 αγ∗ + βδ∗

α∗γ + β∗δ |γ|2 + |δ|2)

=12

(1 + x5 x1 + ix2

x1 − ix2 1− x5

). (14.3.26)

Recall that the reduced density matrix of a separable state is pure and of a maximallyentangled state it is maximally mixed. We can show this by invoking the determinantof ρ1 given by

det ρ1 =14(1− x2

1 − x22 − x2

5) =14(x2

3 + x24) . (14.3.27)

The reduced density matrix represents a pure state (det ρ1 = 0) for the separable casex3 = x4 = 0 and thus sits on the surface of the ball and a mixed state (det ρ1 �= 0)for the entangled case and thus is in the interior of the ball.

14.4 Hilbert space for three qubits and higher Hopf

maps

14.4.1 Third Hopf map

The connection between Hilbertspace and Hopf fibration for three qubits was workedout by Bernevig and Chen [23] (see also [106]) where the fibration again turns outto be entanglement sensitive.The Hilbert space for 3 qubits is C2 ⊗ C2 ⊗ C2 = C8. A general pure state can bewritten in terms of a direct product basis

|ψ〉 = α1|000〉+α2|001〉+α3|010〉+α4|011〉+α5|100〉+α6|101〉+α7|110〉+α8|111〉 ,

with αi ∈ C and∑

i|αi|2 = 1.

138

14.4. Hilbert space for three qubits CHAPTER 14. Hopf geometry for pure qubit states

Via the normalization of the state we reduce the space from the C8 ∼= R16 to the15-dimensional sphere S15. The third Hopf map

h : S15 S8

−−−−−→ S7 , (14.4.1)

indicates a nontrivial fibration where the fibres are S8. It can be described with thehelp of octonions.The set of octonions O is the a non-associative extension of the quaternionic set(for an introduction see Refs.[11, 12]). They were discovered in 1843 by Graves andindependently by Cayley. Therefore they are also called Cayley numbers and formthe Cayley algebra.In Ref.[23] the explicit form of the third Hopf map is presented and analyzed interms of entanglement measures.

14.4.2 Families of Hopf maps

In the following section we give a short overview of higher Hopf maps using realdivision algebras with increasing dimensions. The set R has dimension 1, C has(real) dimension 2, the set of quaternions H is of (real) dimension 4 and finallythe octonions O have dimension 8. Each transition involves the loss of a certainproperty: from R to C we loose ordering, from C to H we loose commutativity andfrom H to O we loose associativity.The fist Hopf map, discussed in Sect.14.1.2,

h(1) : S3 S1

−−−−−→ CP1 ∼= S2 , (14.4.2)

is a special case of the family of complex Hopf maps given by

h(c) : S2n+1 S1

−−−−−→ CPn , (14.4.3)

where fibers are S1.The second Hopf map, discussed in Sect.14.3.3,

h(2) : S7 S3

−−−−−→ HP1 ∼= S4 , (14.4.4)

describes the pure states of a two qubit system and is a special case of the quater-nionic Hopf maps

h(q) : S4n+3 S3

−−−−−→ HPn , (14.4.5)

where the fibers are S3.In Sect.14.4.1 we presented the third Hopf map

h(3) : S15 S7

−−−−−→ OP2 ∼= S8 (14.4.6)

which is a special case of the octonionic Hopf map

h(o) : S6n+3 S7

−−−−−→ OPn (14.4.7)

where the fibers are S7.It is expected in general to find new Hopf maps from S2n−1 −→ S2n−1

by invokinghigher division algebras [119] which could serve as a tool for describing multi-qubitsystems.

139

CHAPTER 14. Hopf geometry for pure qubit states 14.4. Hilbert space for three qubits

140

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152

List of publications

� [55] K.D.:“Geometrical Phases in Quantum Theory”Diploma Thesis, University of Vienna (2002)

� [29] R.A. Bertlmann, K.D., B.C. Hiesmayr:“Decoherence of entangled kaons and its connection to entanglement measures”Phys. Rev. A 68, 012111 (2003) [quant-ph/0209017]also “Virtual Journal of Quantuminformation”, August 2003, Vol.3, Issue 8reprinted in this section

� [28] R.A. Bertlmann, K.D., Y. Hasegawa, B.C. Hiesmayr:“Berry phase in entangled systems: an experiment with single neutrons”Phys. Rev. A 69, 032112 (2004) [quant-ph/0309089]also “Virtual Journal of Quantuminformation”, April 2004, Vol.4, Issue 4reprinted in this section

� [30] R.A. Bertlmann, K.D., B.C. Hiesmayr, Ph. Krammer:“Optimal entanglement witnesses for qubits and qutrits”accepted by Phys. Rev. A (2005) [quant-ph/0508043]

� [27] R.A. Bertlmann, K.D., Y. Hasegawa:“Decoherence within neutron interferometry and testing of the Kraus operatordecomposition”in preparationpreprinted in this section

� [26] R.A. Bertlmann, K.D.:“Spin geometry of entangled qubits under decoherence”in preparation

153

Decoherence of entangled kaons and its connection to entanglement measures

Reinhold A. Bertlmann, Katharina Durstberger, and Beatrix C. Hiesmayr*Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria

�Received 4 September 2002; published 30 July 2003�

We study the time evolution of the entangled kaon system by considering the Liouville–von Neumannequation with an additional term which allows for decoherence. We choose, as generators of decoherence, theprojectors to the two-particle eigenstates of the Hamiltonian. Then we compare this model with the data of theCPLEAR experiment and find in this way an upper bound on strength � of the decoherence. We also relate �to an effective decoherence parameter � considered previously in literature. Finally we discuss our model in thelight of different measures of entanglement, i.e., von Neumann entropy S, entanglement of formation E, andconcurrence C, and we relate decoherence parameter � to the loss of entanglement: 1�E .

DOI: 10.1103/PhysRevA.68.012111 PACS number�s�: 03.65.Ud, 03.65.Ta, 03.67.�a, 14.40.Aq

I. INTRODUCTION

Particle physics has become an interesting testing groundfor fundamental questions of quantum mechanics �QM�. Forinstance, QM versus local realistic theories �1–4� and Bellinequalities �5–12� have been tested. Furthermore, possibledeviations from the quantum-mechanical time evolutionhave been studied, particularly in the neutral K-meson sys-tem �13–19� and B-meson system �20–24�. Recently, neu-trino oscillations have also become of interest in this connec-tion �25�.

In this paper we concentrate on possible decoherence ef-fects arising due to some interaction of the system with its‘‘environment.’’ Sources for ‘‘standard’’ decoherence effectsare the strong interaction scatterings of kaons with nucleons,the weak interaction decays, and the noise of the experimen-tal setup. ‘‘Nonstandard’’ decoherence effects result from afundamental modification of QM and can be traced back tothe influence of quantum gravity �26–28� �quantum fluctua-tions in the space-time structure on the Planck mass scale� orto dynamical state reduction theories �29–32�, and arise on adifferent energy scale. However, we do not pursue further thereasons for decoherence effects; rather we want to develop aspecific model of decoherence and quantify strength � ofsuch possible effects with the help of data of existing experi-ments.

For our model we focus on entangled massive particlesmoving apart in their center of mass system, in particular, onthe K0K0 system, where strangeness S�� ,� plays the roleof spin ‘‘up’’ and ‘‘down’’ �for details see Ref. �33��. Weconsider here the famous EPR-like �Einstein, Podolsky,Rosen� scenario, as described by Bell �34� for spin-1/2 par-ticles, where the initial spin singlet state evolves in time andafter macroscopic separation, the strangeness of the left- andright-moving particle is measured. In contrast to other con-cepts in literature, we introduce decoherence in the time evo-lution of the two-particle entangled state which becomesstronger with increasing distance between the two particles,whereas for the one-particle state we assume the usualquantum-mechanical time evolution.

Then we compare our model of decoherence with the ex-perimental data of the CPLEAR experiment performed atCERN �35� and find an upper bound on possible decoher-ence. We can also relate our model to an effective decoher-ence parameter � , introduced previously in literature, whichquantifies the spontaneous factorization of the wave functioninto product states �Furry-Schrodinger hypothesis �36,46��.

Finally we discuss our model within concepts of quantuminformation, where entanglement is quantified by certainmeasures. We can connect directly the amount of decoher-ence of the K0K0 system parametrized by � or � , with theloss of entanglement expressed in terms of the concurrenceor in terms of entanglement of formation. We calculate ex-plicitly the information loss, the von Neumann entropy, andthe entanglement loss of the evolving K0K0 system as afunction of time

II. THE MODEL

Let us begin our decoherence discussion with the one-particle kaon system as an introduction. Then we proceed tothe case of two entangled neutral kaons and compare it withexperimental data.

A. The one-particle case

We discuss the model of decoherence in a two-dimensional Hilbert space H�C2 and consider the usualnon-Hermitian ‘‘effective mass’’ Hamiltonian H which de-scribes the decay properties and the strangeness oscillationsof the kaons. The mass eigenstates, the short-lived �KS� andlong-lived �KL� states, are determined by

H�KS ,L��S ,L�KS ,L� with �S ,L�mS ,L�i

2S ,L , �2.1�

with mS ,L and S ,L being the corresponding masses and de-cay widths. For our purpose CP invariance1 is assumed, i.e.,CP eigenstates �K1

0� and �K20� are equal to mass eigenstates

*Electronic address: [email protected]

1Note that corrections due to CP violations are of order 10�3,however, we compare this model of decoherence with the data ofthe CPLEAR experiment �35�, which are not sensitive to CP vio-lating effects.

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�K10��KS� , �K2

0��KL� and �KS�KL��0. �2.2�

As a starting point for our model of decoherence we con-sider the Liouville–von Neumann equation with Hamiltonian�2.1� and allow for decoherence by adding a so-called dissi-pator D�� , so that the time evolution of density matrix � isgoverned by a master equation of form

d�

dt��iH��i�H†�D�� . �2.3�

For D�� we choose the following ansatz �as in Ref. �37��:

D��PS�PL�PL�PS��

2 j�S ,L

†P j ,�P j ,��‡,�2.4�

where P j��K j��K j� ( j�S ,L) represent the projectors to theeigenstates of the Hamiltonian, and decoherence parameter� is positive, ��0.

Apart from its simplicity, ansatz �2.4� has the followingnice features.

�i� It generates a completely positive map since it is aspecial case of Lindblad’s general structure �38�

D��1

2 j

�A j†A j��A j

†A j�2A j�A j†� �2.5�

if we identify A j��P j , j�S ,L . Equivalently, it is a spe-cial form of the Gorini-Kossakowski-Sudarshan �39� expres-sion �see, e.g., Ref. �40��.

�ii� It conserves energy in case of a Hermitian Hamil-tonian since �P j ,H��0 �see, e.g., Ref. �41��.

�iii� The von Neumann entropy S(�)��Tr(� ln �) is notdecreasing as a function of time since P j

†�P j ; thus A j†

�A j in our case, which is a theorem due to Narnhofer andBenatti �42�.

With choice �2.4�, time evolution �2.3� decouples for thecomponents of � which are defined by

�� t �� i , j�S ,L

� i j� t ��Ki��K j�, �2.6�

and we obtain

�SS� t ��SS�0 �e�St,

�LL� t ��LL�0 �e�Lt,

�LS� t ��LS�0 �e�i�mt�t��t, �2.7�

with �m�mL�mS and � 12 (S�L). Only the off-

diagonal elements are affected by our model of decoherence.Before further discussing the model, we now proceed to thetwo-particle system.

B. The two-particle case

In the case of two entangled neutral kaons we make thefollowing identification:

�e1��KS� l � �KL�r and �e2��KL� l � �KS�r , �2.8�

and we consider, as common, the total Hamiltonian as a ten-sor product of the one-particle Hilbert spaces: H�Hl � 1r�1l � Hr , where l denotes the left-moving and r denotes theright-moving particle. Then the initial quasispin singlet state

��1

�2��e1��e2�� 2.9�

is equivalently given by density matrix

��0 ��

�1

2��e1��e1��e2��e2��e1��e2��e2��e1��.

�2.10�

Following the considerations of the one-particle case, wefind that the time evolution given by Eq. �2.3� with ourchoice �2.4�, where now operators P j��e j��e j� ( j�1,2)project to the eigenstates of the two-particle Hamiltonian H,also decouples:

�11� t ��11�0 �e�2t,

�22� t ��22�0 �e�2t,

�12� t ��12�0 �e�2t��t. �2.11�

Consequently, we obtain for the time-dependent density ma-trix

�� t ��1

2e�2t��e1��e1��e2��e2�

�e��t� �e1��e2��e2��e1��. �2.12�

The decoherence arises through factor e��t which only af-fects the off-diagonal elements. It means that for t�0 and��0 density matrix �(t) does not correspond to a pure stateanymore �for further discussion, see Sec. IV�.

Note that the assumption of CP invariance Eq. �2.2�,which is sufficient for our purpose, implying �e1�e2��0, iscrucial. Otherwise, we would have a time evolution into thefull four-dimensional Hilbert space of states.

C. Bounds from experimental data

In order to obtain information on possible values of � wecompare our model of decoherence with data of theCPLEAR experiment performed at CERN �35�. We have thefollowing point of view. The two-particle density matrix fol-lows the time evolution given by Eq. �2.3� with Lindbladgenerators A j��e j��e j� and undergoes thereby some de-coherence. We measure strangeness content S of the left-moving particle at time t l and of the right-moving particle attime tr . For sake of definiteness we choose tr�t l . For timestr�t�t l we have a one-particle state which evolves exactlyaccording to QM, i.e., no further decoherence is picked up.

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Mathematically, the measurement of the strangeness con-tent, i.e., the right-moving particle being a K0 or a K0 at timetr , is obtained by

Trr�1l � �S��S��r �� tr�� l� t�tr ;tr�, �2.13�

with strangeness S�� ,� and ��K0�, ��K0� .Consequently, � l(t;tr) for times t�tr is the one-particle den-sity matrix for the left-moving particle and evolves as a one-

particle state according to pure QM. At t�t l the strangenesscontent (S�� ,�) of the second particle is measured andwe finally have probability

P��S ,t l ;S�,tr��Trl��S��S� l � l� t l ;tr��. �2.14�

Explicitly, we find the following results for the like- andunlike-strangeness probabilities:

P��K0,t l ;K0,tr��P��K0,t l ;K0,tr��1

8�e�Stl�Ltr�e�Ltl�Str�e��tr2 cos��m�t �e�(t l�tr)�,

P��K0,t l ;K0,tr��P��K0,t l ;K0,tr��1

8�e�Stl�Ltr�e�Ltl�Str�e��tr2 cos��m�t �e�(t l�tr)�,

�2.15�

with �t�t l�tr .Note that at equal times t l�tr�t , like-strangeness probabilities

P��K0,t;K0,t ��P��K0,t;K0,t ��1

4e�2t�1�e��t� �2.16�

do not vanish, in contrast to the quantum-mechanical EPR correlation.The asymmetry of the probabilities is directly sensitive to the interference term and has been measured by the CPLEAR

Collaboration �35�. For pure QM we have

AQM��t ��P�K0,t l ;K0,tr��P�K0,t l ;K0,tr��P�K0,t l ;K0,tr��P�K0,t l ;K0,tr�

P�K0,t l ;K0,tr��P�K0,t l ;K0,tr��P�K0,t l ;K0,tr��P�K0,t l ;K0,tr��

cos��m�t �

cosh� 1

2��t � , �2.17�

with ��L�S , and for our decoherence model we find,by inserting probabilities �2.15�,

A�� t l ,tr��cos��m�t �

cosh� 1

2��t � e��min�t l ,tr�

�AQM��t �e��min�t l ,tr�. �2.18�

Thus the decoherence effect, simply given by factore��min�tl ,tr�, depends only, due to our philosophy, on the timeof the first measured kaon, in our case, min�tl ,tr��tr .

Comparing now our model with the results of theCPLEAR experiment �35� we recall that the experimentalsetup has two configurations: In the first configuration bothkaons propagate 2 cm. In the second, one kaon propagates 2cm and the other kaon 7 cm until they are measured by anabsorber.

Fitting decoherence parameter � by comparing asymme-try �2.18� with the experimental data �35� we find, when

averaging over both configurations,2 the following bounds on�:

��1.84�2.17�2.50��10�12 MeV and ��

�

S�0.25�0.32

�0.34 .

�2.19�

Results �2.19� are certainly compatible with QM (��0);nevertheless, the experimental data allow an upper bound

�up�4.34�10�12 MeV for possible decoherence in the en-

tangled K0K0 system.Results �2.19� can be compared with the analogous inves-

tigation of the entangled B0B0 system �37�, which gives �B�(�47�76)�10�12 MeV. Thus we find that bounds�2.19�of the K0K0 system are an order of magnitude morerestrictive.

2We have scaled �t in the QM asymmetry �2.17� in order toreproduce the QM curve in Fig. 9 of the CPLEAR Collaboration�35�.

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III. CONNECTION TO A PHENOMENOLOGICAL MODEL

There exists a one-to-one correspondence between modelof decoherence �2.3� and a phenomenological model �22,43–45� where the decoherence is introduced by multiplying the

interference term of the transition amplitude by factor(1��). Quantity � is called the effective decoherence pa-rameter. Our initial state is again spin singlet state ��which is given by mass eigenstate representation �2.9� Thenwe get for the like-strangeness probability

�3.1�

and the unlike-strangeness probability just changes the signof the interference term.

Value ��0 corresponds to pure QM and ��1 to totaldecoherence or spontaneous factorization of the wave func-tion, which is commonly known as Furry-Schrodinger hy-pothesis �36,46�. Effective decoherence parameter � , intro-duced in this way ‘‘by hand,’’ interpolates continuouslybetween these two limits and represents a measure for theamount of decoherence which results in a loss of entangle-ment of the total quantum state �we come back to this pointin Sec. IV�.

Calculating the asymmetry of the strangeness events withprobabilities �3.1� we obtain

A�� t l ,tr��AQM��t ��1�� t l ,tr�� . �3.2�

When we compare now the two approaches, i.e., Eq. �2.18�with Eq. �3.2�, we find formula

�� t l ,tr��1�e�� min�t l ,tr�. �3.3�

Of course, values �2.19� are in agreement with the corre-sponding � values �averaged over both experimental setups�:��0.13�0.15

�0.16 , as derived in Refs. �43,47�.We consider decoherence parameter � to be the funda-

mental constant, whereas the value of effective decoherenceparameter � depends on the time when a measurement isperformed. In the time evolution of state �� Eq. �2.9��,represented by density matrix �2.12�, we have relation

�� t ��1�e��t �3.4�

which, after the measurement of the left- and right-movingparticles at t l and tr , turns into formula �3.3� when decoher-ence is implemented at the two-particle level.

Our model can be compared with the case where decoher-ence in time evolution �2.3� happens at a one-particle level

and is transferred to the two-particle level by a tensor prod-uct of the one-particle Hilbert spaces �18�. Using the samestructure of decoherence term �2.4�, where now the operatorsproject to the one-particle states instead of states �2.8�, weobtain relation

�� t l ,tr��1�e��(t l�tr). �3.5�

Here parameter � depends explicitly on both times t l and tr ,the ‘‘eigentimes’’ of the left- and right-moving kaon, insteadof one time min�tl ,tr�, the time of the system when the firstkaon is measured.

Measuring the strangeness content of the entangled kaonsat definite times, we have the possibility to distinguish ex-perimentally these two models �3.3� and �3.5� on the basis oftime-dependent event rates. Indeed, it would be of high in-terest in future experiments to measure the asymmetry of thestrangeness events for several different times, in order toconfirm the time dependence of the decoherence effect. Infact, such a possibility is now offered in the B-meson system.Entangled B0B0 pairs are created with high density at theasymmetric B-factories and identified by detectors BELLE atKEK-B �see, e.g., Refs. �48,49�� and BABAR at PEP-II �see,e.g., Refs. �50,51�� with a high resolution at different dis-tances or times.

IV. DECOHERENCE AND ENTANGLEMENT LOSS

Term D�� in master equation �2.3� is usually called dis-sipative term or dissipator �see, e.g., Ref. �52��. In general,D�� describes two phenomena occurring in an open quan-tum system S, namely, decoherence and dissipation. Whensystem S interacts with environment E, the initial productstate evolves into entangled states of S�E in the course oftime �53,54�. It leads to mixed states in S—what means


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decoherence—and to an energy exchange between S andE—what is called dissipation.

The decoherence destroys the occurrence of long-rangequantum correlations by suppressing the off-diagonal ele-ments of the density matrix in a given basis and leads to aninformation transfer from S to E.

In general, both effects are present, however, decoherenceacts on a much shorter time scale �53–57� than dissipationand is the more important effect in quantum information pro-cesses.

Our model describes decoherence and not dissipation. Theincrease of decoherence of the initially totally entangledK0K0 system as time evolves means, on the other hand, adecrease of entanglement of the system. This loss of en-tanglement we can quantify and visualize explicitly.

In the field of quantum information the entanglement of astate is quantified by introducing certain measures. In thisconnection the entropy plays a fundamental role, which mea-sures ‘‘somehow’’ the degree of uncertainty of a quantumstate. A common measure is von Neumann entropy functionS, entanglement of formation E, and concurrence C.

A. von Neumann entropy

Since we are only interested in the effect of decoherence,we want to properly normalize state �2.12� in order to com-pensate the decay property3 of the non-Hermitian Hamil-tonian H:

�N� t �� t �

Tr�� t �. �4.1�

Then von Neumann’s entropy function for state �4.1� gives

S„�N� t �…��Tr��N� t �log2 �N� t ��

��1�e��t

2log2

1�e��t

2�

1�e��t

2log2

1�e��t

2.

�4.2�

At time t�0 the entropy is zero, there is no uncertainty inthe system, the quantum state is pure and maximally en-tangled. For t�0 the entropy becomes nonzero, increases,and approaches value 1 for t→� . Hence the state becomesmore and more mixed. Mixed states provide only partial in-formation about the system, and the entropy measures howmuch of the maximal information is missing �see, e.g., Ref.�58��. In Fig. 1 von Neumann entropy S„�N(t)… is plotted forthe mean value and upper bound of decoherence parameter� , given by Eq. �2.19�, as determined from the CPLEARexperiment �35�.

Let us consider next the reduced density matrices of thesubsystems, i.e., the propagating kaons on the left- �l� andright- �r� hand sides

�Nl � t ��Trr��N� t �� and �N

r � t ��Trl��N� t ��. �4.3�

Then, the uncertainty in subsystem l before subsystem r ismeasured is given by von Neumann entropy S„�N

l (t)… of thecorresponding reduced density matrix �N

l (t) �and alterna-tively we can replace l→r). In our case we find

S„�Nl � t �…�S„�N

r � t �…�1 �t�0. �4.4�

We see that the reduced entropies are independent of � . Thecorrelation stored in the composite system is, with increasingtime, lost into the environment—what is expectedintuitively—and not into the subsystems, i.e., the individualkaons.

Note that because of our chosen normalization �4.1� theeffects seen are only due to the introduced decoherence viaterm D�� 2.4� and not due to the decay of the system.

B. Lack of separability

For the subsequent considerations it is convenient to re-call the ‘‘quasispin’’ picture for the K0K0 system �see, e.g.,Ref. �8�� in order to express the formulas in terms of Paulispin matrices. Then the projection operators to the masseigenstates represent the spin projection operators ‘‘up’’ and‘‘down’’

PS��KS��KS��↑�1

2�1��z�� 1 0

0 0 � ,

PL��KL��KL��↓�1

2�1��z�� 0 0

0 1 � , �4.5�

and the transition operators are the ‘‘spin-ladder’’ operators

PSL��KS��KL��1

2��x�i�y�� 0 1

0 0 � ,

PLS��KL��KS��1

2��x�i�y�� 0 0

1 0 � . �4.6�

With the help of these spin matrices, density matrix �4.1� canbe expressed by

�N� t ��1

2��↑ � �↓��↓ � �↑�e��t�� ,

�4.7�

or by

�N� t ��1

4�1��z � �z�e��t��x � �x��y � �y��,

�4.8�

which at t�0 coincides with the well-known expression forthe pure spin singlet state �N(t�0)� 1

4 (1�� ); see, e.g.,Ref. �59�. Operators �4.7� and �4.8� can be nicely written as4�4 matrix:

3Note that for other physical situations, such as the verification ofBell inequalities, the decay property must not be neglected �8�.


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�N� t ��1

2 � 0 0 0 0

0 1 �e��t 0

0 �e��t 1 0

0 0 0 0

� . �4.9�

It is also illustrative to choose for density matrix �N(t)another basis, the so-called ‘‘Bell basis’’

�� and ��, �4.10�

with �� given by Eq. �2.9� and �� by

��1

�2��e1��e2��. �4.11�

States ��1/�2(�↑↑��↓↓�) �in spin notation� do notcontribute here. Then we are led to the following proposi-tion.

Proposition. The state represented by density matrix�N(t) �4.1� becomes mixed for 0t� but remains en-tangled. Separability is achieved asymptotically t→� withweight e��t. Explicitly, �N(t) is the following mixture ofBell states �� and ��:

�N� t ��1

2�1�e��t��

1

2�1�e��t��. �4.12�

Proof: �i� Diagonalizing matrix �4.9� we find � 12 (1

�e��t), 12 (1�e��t),0,0�, the eigenvalues together with

eigenvectors �� and ��, which proves Eq. �4.12�. For t�0 matrix �N(0) is a projector to a one-dimensional sub-space: �1,0,0,0�, Bell state ��, so the state is pure. For0t� matrix �N(t) is no longer a projector, thus the stateis mixed. The state becomes asymptotically t→� totallymixed, separable. Of course, �N(t) also satisfies the mixedstate criterion

�N2 � t ��

1

4 � 0 0 0 0

0 1�e�2�t �2e��t 0

0 �2e��t 1�e�2�t 0

0 0 0 0� ��N� t �

for t�0. �4.13�

�ii� The entanglement we prove via lack of separability.According to Peres �60� and the Horodeckis �61�, separabil-ity is determined by the positive partial transposition crite-rion: the partial transposition of a separable state with re-spect to any subsystem is positive �i.e., the operator haspositive eigenvalues�. Applying transposition operator T, de-fined by T(� i)kl�(� i) lk , to the left- or right-hand side, wefind a negative eigenvalue:

�1l � Tr��N� t ��1

2��↑ � �↓��↓ � �↑

�e��t��

�1

2 � 0 0 0 �e��t

0 1 0 0

0 0 1 0

�e��t 0 0 0

� �” 0,

�4.14�

with eigenvalues � 12 , 1

2 , 12 e��t,� 1

2 e��t�.Alternatively, we could use Horodecki’s �62� reduction

criterion: �N(t) is separable iff �Nl (t) � 1r��N(t)�0 or

1l � �Nr (t)��N(t)�0. In our case we have

1l � �Nr � t ��N� t ��

1

2 � 1 0 0 0

0 0 e��t 0

0 e��t 0 0

0 0 0 1� �” 0,

�4.15�

where the eigenvalues are the same as for the previous cri-terion. For a general separability criterion, i.e., a generalizedBell inequality, see Ref. �59�. �

C. Entanglement of formation and concurrence

For pure quantum states entanglement can be measuredby the entropy of the reduced density matrices. For mixedstates, however, the von Neumann entropy is not generally agood measure for entanglement, whereas another measure,entanglement of formation �63�, is very suitable. It has beenconstructed by Bennett et al. �63� to quantify the resourcesneeded to create a given entangled state.

1. Definitions

Every density matrix � can be decomposed in an en-semble of pure states � i�� i�� i� with probabilities pi , i.e.,�� ipi� i . The entanglement of formation for a pure state isgiven by the entropy of either of the two subsystems. For amixed state the entanglement of formation for a bipartitesystem is then defined as the average entanglement of thepure states of the decomposition, minimized over all decom-positions of �:

E��min i

piS�� il�. �4.16�

Bennett et al. �63� found a remarkably simple formula forentanglement of formation:

E��E„f ��…, �4.17�

where function E„f (�)… is defined by


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E„f ��…�H� 1

2��f �1� f � � for f �

1

2, �4.18�

and E„f (�)…�0 for f 12 . Function H represents the familiar

binary entropy function H(x)��x log2 x�(1�x)log2(1�x).Quantity f is called the fully entangled fraction of �

f �max�e��e�, �4.19�

being the maximum over all completely entangled states �e�.For general mixed states � , function E„f (�)… is only a lowerbound to entropy E(�). For pure states and mixtures of Bellstates �the case of our model� the bound is saturated, E�E,and we have formula �4.18� for calculating the entanglementof formation.

Wootters and Hill �64–66� found that entanglement offormation for a general mixed state � of two qubits can beexpressed by another quantity, concurrence C,

E��E„C��…�H� 1

2�

1

2�1�C2� with 0�C�1.

�4.20�

Explicitly, function E(C) looks like

E�C ��1��1�C2

2log2

1��1�C2

2

�1��1�C2

2log2

1��1�C2

2�4.21�

and is monotonically increasing from 0 to 1 as C runs from0 to 1. Thus C itself is a kind of entanglement measure in itsown right.

Defining spin flipped state � of � by

��y � �y��*��y � �y�, �4.22�

where �* is the complex conjugate and is taken in the stan-dard basis, i.e., basis ��↑↑�,�↓↓�,�↑↓� ,�↓↑��, concurrence Cis given by formula

C��max�0,�1��2��3��4�. �4.23�

The � i’s are the square roots of the eigenvalues, in decreas-ing order, of matrix �� .

2. Applications to our model

For density matrix �N(t), Eq. �4.1� of our model, which isinvariant under spin flip �see, e.g., Eq. �4.7��, i.e., �N��N

and thus �N�N��N2 , we get for concurrence,

C„�N� t �…�max�0,e��t��e��t, �4.24�

and for the fully entangled fraction of �N(t),

f „�N� t �…� 1

2�1�e��t�, �4.25�

which is simply the largest eigenvalue of �N(t). Clearly, inour case functions C and f are related by

C„�N� t �…�2 f „�N� t �…�1. �4.26�

Finally, for the entanglement of formation of the K0K0 sys-tem we have

E„�N� t �…��1��1�e�2�t

2log2

1��1�e�2�t

2

�1��1�e�2�t

2log2

1��1�e�2�t

2.

�4.27�

Recalling our relation �3.4� between decoherence parameters� and � , we find a direct connection between the entangle-ment measure and the amount of decoherence of the quan-tum system. Defining the loss of entanglement as one minusentanglement and expanding expression �4.27� for small val-ues of � or � �which is denoted by �� we obtain

1�C„�N� t �…�� t �, �4.28�

1�E„�N� t �…� 1

ln 2�� t ��

�

ln 2t . �4.29�

The loss of entanglement of the propagating K0K0 systemin terms of concurrence C �Eq. �4.28�� equals precisely theamount of decoherence parameter � which describes the fac-torization of the initial spin singlet state into product states�KS� l � �KL�r or �KL� l � �KS�r �Furry-Schrodinger hypoth-esis�. In terms of entanglement of formation, Eq. �4.29�, thedecoherence parameter is weighted by a factor 1/ln 2�1.44

FIG. 1. The time dependence of the von Neumann entropy�dashed lines�, Eq. �4.2�, and the loss of entanglement of formation1�E �solid lines�, given by Eq. �4.27�, are plotted for the experi-

mental mean value ��1.84�10�12 MeV �lower curve� and upper

bound �up�4.34�10�12 MeV �upper curve�, given by Eq. �2.19�,of decoherence parameter � . Time t is scaled vs lifetime �s of theshort-lived kaon KS : t→t/�s . The vertical lines represent propaga-tion time t0 /�s�0.55 of one kaon, including the experimental errorbars, until it is measured by the absorber in the CPLEAR experi-ment.


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�factor ln 2 reflects the dimension 2 of the K-meson qua-sispin space� and the entanglement loss increases linearlywith time.

D. Discussion of the results

In Fig. 1 we have plotted loss of entanglement 1�E ,given by Eq. �4.27�, as compared to the loss of information,the von Neumann entropy function S given by Eq. �4.2�, withrespect to time t/�s of the propagating K0K0 system. Thecurves are shown for the mean value and upper bound ofdecoherence parameter � , given by Eq. �2.19�. The von Neu-mann entropy function visualizes the loss of the informationabout the correlation stored in the composite system. Re-member that the information flux is not flowing into the sub-systems but into the environment; see Eq. �4.4�. The loss ofentanglement of formation increases more slowly with timeand shows the resources needed to create a given entangledstate. At t�0 pure Bell state �� is created and becomesmixed for t�0 by the other Bell state ��. In the total statethe amount of entanglement decreases until separability isachieved �exponentially fast� for t→� .

For example, in the case of the CPLEAR experiment,where one kaon propagates about 2 cm, which correspondsto a propagation time t0 /�s�0.55, until it is measured by anabsorber, the loss of entanglement is about 18% for the meanvalue and maximal 38% for the upper bound of decoherenceparameter � . These values, however, could diminish consid-erably in future experiments.

V. SUMMARY AND CONCLUSIONS

We have considered a simple model of decoherence of theentangled K0K0 state due to some environment, i.e., a masterequation of Liouville–von Neumann type with an additionalterm D�� . As generators of D�� causing the decoherenceeffect with strength � we choose the projectors to the eigen-states of the ‘‘effective mass’’ Hamiltonian and, for simplic-ity, CP invariance is assumed. For this choice the time evo-lution for the components of � decouples and only the off-diagonal elements are effected by our modification.

We apply the model to the data of the CPLEAR experi-ment, where we follow the philosophy that only the two-particle state is affected by decoherence, whereas the one-particle state evolves according to pure QM. We estimate in

this way strength � of the occurring decoherence, Eq. �2.19�.Moreover, we can relate the model to the case of a phe-

nomenologically introduced decoherence parameter � andfind a one-to-one correspondence �Eq. �3.4��. However, theexisting data are not yet sufficient to measure the time de-pendence of � as predicted by our model �Eq. �3.3��. So,further measurements of the time-dependent asymmetryterm, Eqs. �2.18� and �3.2�, would be of high interest infuture experiments and will sharpen considerably the boundsof parameters � and � .

We can directly relate the decoherence of the K0K0 stateto its loss of entanglement. In this connection we considerentanglement measures frequently discussed in the field ofquantum information. We demonstrate that the initially puresinglet state of the entangled K0K0 becomes mixed for0t� but remains entangled and achieves separability fort→� �see Proposition�.

We find that entanglement loss in terms of the concur-rence equals precisely decoherence parameter � �Eq. �4.28��,and in terms of entanglement of formation the loss is verywell approximated by �/ln 2 or t�/ln 2 �Eq. �4.29��, which isone of our main results. We can propose in this way how tomeasure experimentally the entanglement of the K0K0 sys-tem.

In Fig. 1 we visualize both the loss of information givenby the von Neumann entropy, which flows totally into theenvironment and not into the subsystems of the two-particlesystem, and the loss of entanglement of formation. Insertingthe mean value and upper bound of parameter � , which wehave determined from the CPLEAR experiment, we obtaindefinite bounds for both the information loss and the en-tanglement loss of the propagating K0K0 system with respectto the time. These values, however, could be improved con-siderably in future experiments.

ACKNOWLEDGMENTS

The authors want to thank Caslav Brukner, Franz Em-bacher, Walter Grimus, Heide Narnhofer, and Walter Thirringfor helpful discussions. This research was supported by theFWF, Project No. P14143-PHY, of the Austrian ScienceFoundation. The aid of the Austrian-Czech Republic Scien-tific Collaboration, Project No. KONTAKT 2001-11, and ofthe EU project EURIDICE EEC-TMR program HPRN-CT-2002-00311 is acknowledged.

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Berry phase in entangled systems: A proposed experiment with single neutrons

Reinhold A. Bertlmann,1 Katharina Durstberger,1,* Yuji Hasegawa,2 and Beatrix C. Hiesmayr3,†

1Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria2Atominstitut de Österreichischen Universitäten, Stadionällee 2, 1020 Vienna, Austria

3Grup de Física Teòrica, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain(Received 15 September 2003; published 25 March 2004)

The influence of the geometric phase, in particular the Berry phase, on an entangled spin-12 system is studied.

We discuss in detail the case, where the geometric phase is generated only by one part of the Hilbert space. Weare able to cancel the effects of the dynamical phase by using the “spin-echo” method. We analyze how theBerry phase affects the Bell angles and the maximal violation of a Bell inequality. Furthermore we suggest anexperimental realization of our setup within neutron interferometry.

DOI: 10.1103/PhysRevA.69.032112 PACS number(s): 03.65.Vf, 03.65.Ud, 03.75.Dg, 42.50.�p

I. INTRODUCTION

Entanglement is one of the most profound aspects ofquantum mechanics (QM). It occurs in quantum systems thatconsist of two (or more) parts which can be separated, ortypically, in systems whose observables belong to disjointHilbert spaces. Its deep meaning has been already realized bySchrödinger in 1935 in his famous papers on “The presentsituation of quantum mechanics” [1]. In Schrödinger’s viewthe whole system is in a definite state whereas the individualparts are not.

In the same year Einstein, Podolsky, and Rosen (EPR) [2]recognized—what nowadays is called “EPR paradox”—thatQM exhibits very peculiar correlations between two physi-cally distant parts of the total system. It is possible to predictthe outcome of the measurement of one part by looking atthe distant part.

About three decades later Bell [3] reanalyzed the EPRparadox. He discovered inequalities, commonly known asBell inequalities (BI’s), that can be violated by QM but haveto be satisfied by (all) local realistic theories [4]. A violationof a BI demonstrates the presence of entanglement and thusaccording to Bell’s theorem the occurrence of nonlocal fea-tures in the quantum systems. Generalizations of such in-equalities serve as a criterion for entanglement or separabil-ity [5].

Experiments with photons (see, e.g., Ref. [6] for a review)confirm QM with its nonlocality in an impressive way. In thelast years there have also been considerable activities to testentangled massive systems in particle physics [7–13]. En-tanglement is the basis for quantum communication andquantum information (see, e.g., Ref. [6]) and it became animportant issue of investigation nowadays.

Geometric phases such as the Berry phase [14] play aconsiderable role in physics and arise in a quantum systemwhen its time evolution is cyclic and adiabatic. Its deep geo-metric origin is given by a holonomy of the line bundle ofthe states where the phase emerges from the integral of the

connection (or curvature) of the bundle over the parameterspace [15]. The quantum holonomy also appears for mixedstates [16,17]. Generalizations to nonadiabatic evolutions[18] and noncyclic and nonunitary settings [19,20] do existas well as extensions to off-diagonal geometric phases[21,22]. The application of geometric phases in quantumcomputation has been suggested by several authors [23–25].Experimentally, geometric phases have been tested in variouscases, e.g., with photons [26–28], with neutrons [29,30] andwith atoms [31].

Whereas the geometric phase in a single-particle system isalready studied very well, both theoretically and experimen-tally, its effect on entangled quantum systems is less known.However, there is increasing interest to combine both quan-tum phenomena, the geometric phase and the entanglementof a system [32–35].

In this paper we are studying the influence of the Berryphase on the entanglement of a spin-1

2 system by consideringa BI. The Berry phase is generated by implementing an adia-batically rotating magnetic field into one of the paths of theparticles. Our goal is to propose an explicit experimentalsetup which eliminates the dynamical phase, which wouldspoil the geometric effect, so that we are sensitive just to thegeometric phase. We can achieve this within neutron inter-ferometry [36,37] which is an almost ideal tool to investigatethe evolution of a spin-1

2 system. In particular, when using apolarized beam [38] we have entanglement between differentdegrees of freedom, i.e., the spin and the path of the neutron.In this case it is physically rather noncontextuality than lo-cality which is tested experimentally [39,40].

Noncontextuality means that the value of an observabledoes not depend on the experimental context; the measure-ment of the observable must yield the value independent ofother simultaneous measurements. The question is whetherthe properties of individual parts of a quantum system (orensemble) do have definite or predetermined values beforethe act of measurement—a main hypothesis in hidden vari-able theories. The “no-go theorem” of Bell-Kochen-Specker[41,42] states that noncontextual theories are incompatiblewith QM. More precisely, it is, in general, impossible toassign to an individual quantum system a definite value foreach set of observables (see, e.g., Refs. [43,44]).

*Electronic address: [email protected]†Electronic address: [email protected]

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In our case the observables, which belong to mutuallydisjoint Hilbert spaces, are the spin and the path of the neu-tron in the interferometer and we use a BI containing theseobservables to test noncontextual hidden variable theoriesversus QM [39].

II. THE BERRY PHASE FOR SPIN-12 PARTICLES

We concentrate on the spin-12 system where it is rather

simple to implement a geometrical phase and we use Berry’s[14] construction for the system evolving adiabatically andcyclically in time.

The scenario is as follows. The particle, without loss ofgenerality moving in y direction, couples to a time-

dependent magnetic field B� �t� with unit vector n�� ; t� and

constant norm B= �B� �t��. Field B� �t� rotates adiabatically withan angular velocity �0 around the z axis under an angle �(for proper adiabaticity conditions, see Ref. [45]). The inter-action is described by the Hamiltonian

H�t� =

2B� �t�� , �1�

where the coupling constant is given by =g B, the Landéfactor g times the Bohr magneton B= 1

2 �e /m��.Consequently, the instantaneous eigenstates of the

spin operator in direction n�� ; t�—thus of Hamiltonian(1)—expanded in the �z basis are given by

�⇑n;t� = cos�

2�⇑z� + sin

�

2ei�0t�⇓z� ,

�2�

�⇓n;t� = − sin�

2�⇑z� + cos

�

2ei�0t�⇓z� .

The corresponding time-independent energy levels are

E± = ± B

2= ± ��1, �3�

where �1 : = �E+−E−� /2�= B /2� denotes the energy differ-ence of spin ⇑n and spin ⇓n and represents the characteristicfrequency of the system. Let us consider an adiabatic �whichmeans �0 /�1�1� and cyclic time evolution for the period�=2� /�0 of these eigenstates. Then each eigenstate picks upa phase factor that can be split into a geometrical and adynamical part of the following form:

�⇑n;t = 0� → �⇑n;t = �� = ei�+��ei�+�⇑n;t = 0� ,

�4��⇓n;t = 0� → �⇓n;t = �� = ei�−��ei�−�⇓n;t = 0�

with

�+�� = − ��1 − cos �� ,�5�

�−�� = − ��1 + cos �� = − �+�� − 2� ,

�+ = −1

�E+� = − 2�

�1

�0, �− = +

1

�E−� = + 2�

�1

�0= − �+.

�6�

Symbol �± denotes the Berry phase which is precisely half ofthe solid angle 1

2� swept out by the magnetic field during therotation and �± is the familiar dynamical phase.

Now we are going to eliminate the dynamical effectwhich would dominate the geometrical one, by using the socalled “spin-echo” method. First the propagating particle issubjected to the rotating magnetic field in the direction n��for one period and therefore picks up the phases given by Eq.(4). Afterwards the particle passes another rotating fieldwhich points in direction −n��−�� again for one period.Then the states change according to

�⇑n� � �⇓−n� → ei�−��−��ei�−�⇓−n� � ei�+��ei�−�⇑n� ,

�7��⇓n� � �⇑−n� → ei�+��−��ei�+�⇑−n� � ei�−��ei�+�⇓n� .

Therefore we get the following net effect after two rotationperiods:

�⇑n� → e2i�+��⇑n�, �⇓n� → e2i�−��⇓n� , �8�

or for two half periods of rotation we have

�⇑n� → ei�+��⇑n�, �⇓n� → ei�−��⇓n� , �9�

where the dynamical effects totally disappear.

III. THE BERRY PHASE AND THE ENTANGLED STATE

Let us consider an entangled state of two spin-12 particles,

e.g., the antisymmetric Bell singlet state ��−�. One of theparticles (e.g., the left side moving particle) interacts twicewith the adiabatically rotating magnetic fields as described inSec. II. Thus only one subspace of the Hilbert space is influ-enced by the phases.

To locate the Berry phase we decompose the initial Bellsinglet state into the eigenstates of the interaction Hamil-tonian

��t = 0�� = ��−�� =12

�⇑n�l � �⇓n�r − �⇓n�l � �⇑n�r� .

�10�

According to our “spin-echo” construction, after one cycle,the state �10� picks up precisely the geometric phase �9�,

��t = �� =12

ei�+�⇑n�l � �⇓n�r − ei�−�⇓n�l � �⇑n�r� ,

�11�

which can be rewritten by neglecting an overall phase factor�from now on �+��

��t = �� =12

�⇑n�l � �⇓n�r − e−2i��⇓n�l � �⇑n�r� . �12�

Increasing the Berry phase �� :0→� /2→�, which weachieve by varying the magnetic-field angle � :0→60°

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→90°, we move continuously from the antisymmetric Bellsinglet state ��−� to the symmetric Bell state ��+� and backto ��−�.

As in common Bell experiments, we want to measuresimultaneously the spin components of the particles on theleft and right side, see Fig. 1. First we define the projectionoperator onto an up �+� and a down �−� spin state along anarbitrary direction �� ,

P±�� = �±�� ±�� 13�

with

�+ �� = cos�1

2�⇑n� + sin

�1

2ei�2�⇓n� ,

�14�

�− �� = − sin�1

2�⇑n� + cos

�1

2ei�2�⇓n� ,

where �1 denotes the polar angle measured from the n� direc-tion and �2 the azimuthal angle. Then we calculate the jointprobability for finding spin up on the left side under an angle�� from the quantization axis n� and spin up (or spin down) on

the right side under an angle �� :

P�� ⇑n,�� ⇑n� = ��t = ��P+l �� P+

r �� t = ��

=1

4 1 − cos �1 cos �1 − cos��2 − �2

+ 2��sin �1 sin �1�

→ 1

4 1 − cos��1 − �1�� for ��2 − �2� → − 2� ,

�15�

P�� ⇑n,�� ⇓n� = ��t = ��P+l �� P−

r �� t = ��

=1

4 1 + cos �1 cos �1 + cos��2 − �2

+ 2��sin �1 sin �1�

→ 1

4 1 + cos��1 − �1�� for ��2 − �2� → − 2� .

�16�

Introducing the observable

Al�� = P+l �� − P−

l �� , �17�

and similarly Br�� , we calculate the expectation value of thejoint measurement

E�� ,�� = ��t = ��Al�� Br�� t = ��

= − cos �1 cos �1 − cos��2 − �2 + 2��sin �1 sin �1

→ − cos��1 − �1� for ��2 − �2� → − 2� . �18�

We observe that we always can compensate the effect of theBerry phase by simply changing the difference of the azi-muthal angles ��2−�2�→−2� of the two measuring direc-

tions �� and �� and regain the familiar expressions withoutBerry phase.

To test experimentally the influence of a pure geometricphase in the entangled state we have to vary only the openingangle � of the magnetic field, i.e., the geometry of the setup,which is related to the Berry phase by formula (5). Then wemeasure expectation value (18) with respect to � at certain

fixed directions �� and �� . By rotating the measurement planesby the angle difference ��2−�2�=−2� the geometric effect isbalanced. Experimentally we propose to test this featurewithin neutron interferometry, see Sec. IV.

When considering a BI to test the local features of thestates we find the following behavior, which we want to il-

FIG. 1. Schematic view of the setup. The vector n� denotes the quantization direction; �� and �� are the measurement directions whichdetermine the measurement planes.

BERRY PHASE IN ENTANGLED SYSTEMS: A… PHYSICAL REVIEW A 69, 032112 (2004)

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lustrate by considering the CHSH (Clauser-Horne-Shimony-Holt) inequality [46], an experimentally testable type of a BI,

S � 2, �19�

where the S function is defined by

S�� ,�� ,�� ,�� ;�� = �E�� ,�� − E�� ,��

+ �E�� ,�� + E�� ,��

= �− sin �1 cos��2 − �2 + 2��sin �1

− cos��2 − �2� + 2��sin �1��

− cos �1�cos �1 − cos �1��

+ �− sin �1� cos��2� − �2 + 2��sin �1

+ cos��2� − �2� + 2��sin �1��

− cos �1��cos �1 + cos �1�� . �20�

Without loss of generality we can eliminate one angle bysetting, e.g., �� =0 ��1=�2=0�, which gives

S�� ,�� ,�� ;��= �− sin �1� cos��2� − �2 + 2��sin �1 + cos��2�

− �2� + 2��sin �1�� − cos �1��cos �1 + cos �1��

+ � − cos �1 + cos �1�� . �21�

We always can reach as maximal value of S the standardvalue 22. We keep the polar angles �1�, �1, and �1� con-stant at the Bell angles �1�=� /2, �1=� /4, �1�=3� /4 andadjust the azimuthal parts

S��2�,�2,�2�;��

= 2 + �−2

2 cos��2� − �2 + 2�� + cos��2� − �2� + 2�� .

�22�

The maximum 22 is reached for �2=�2� and �2�−�2�=−2� �mod ��. For convenience we canchoose �2�=0. The measurement planes on both sides en-close an angle of 2�, see Fig. 1.

On the other hand, keeping the azimuthal angles fixed,e.g., �2=�2�=�2=�2�=0, the polar Bell angles determined by

the maximum of the S function change with respect to theBerry phase �. By calculating the derivatives (the extremumcondition)

� S

� �1= − sin �1 cos �1� sin �1 ± cos�2��sin �1� cos �1 = 0,

� S

� �1�= sin �1� cos �1� sin �1� ± cos�2��sin �1� cos �1� = 0,

�23�

� S

� �1�= sin �1��cos �1 + cos �1�� ± cos�2��cos �1��sin �1

+ sin �1�� = 0,

the solutions are given by (± corresponds to the case eitherf10 and f20 or to f10 and f2�0, when we denote S= �f1 � + �f2�)

�1 = ± arctan cos�2�� ,

�1� = � − �1, �24�

�1� =�

2,

and are plotted in Fig. 2 and Fig. 3 (of course we may inter-change �1↔�1�).

With these angles the S function shows the behavior plot-ted in Fig. 4. We see that the maximal S decreases for � :0→� /4 and touches at �=� /4 even the limit of the CHSHinequality S=2, where we are unable to distinguish betweenQM and local realistic theories. It increases again to the fa-miliar value S=22 at �=� /2, which corresponds to theBell state ��+�.

IV. PROPOSED NEUTRON EXPERIMENT

Let us now consider how the predicted behavior of S canbe measured in practice. In our polarized neutron interferom-eter experiment the wave function of each neutron is definedover a tensor product of Hilbert spaces which describe thespatial and spin components of the wave function and isentangled analogously to the two spin-1

2 system (10)

FIG. 2. The Bell angles �1 and �1� with respect to the Berry phase � for the cases f10 and f20.


032112-4

166

�� =12

�I� � �sI� − �II� � �sII�� . �25�

The states �sI� and �sII� denote the spin states of beams I andII, and �I� and �II� denote the states in the two beam pathsin the interferometer.

A schematic view of the experimental setup is shown inFig. 5. In addition to an auxiliary phase shifter, two radiofrequency (RF) spin flippers [47] are inserted into one beampath and a direct current � spin flipper into the other path ofthe interferometer. The former two flippers enable the neu-tron spinors to evolve along a particular curve inducing onlya geometric phase �B without any dynamical component[48,49], see Fig. 6. The latter flipper produces the entangledstate, like ��t=�� in Eq. (12). Thus, after the spinor evo-lution the total wave function is represented by

��B�� =12

�I� � �⇑n� − ei�B�II� � �⇓n�� , �26�

with the geometrical phase

�B = 12� = �1 − �2. �27�

Our observables can be decomposed in the following form:

Ap�� = P+p�� − P−

p�� and Bs�� = P+s �� − P−

s �� ,

�28�

where P±p�� and P±

s �� denote the projection operators ontopath and spin states, respectively

P±p�� = �±p��±p� and P±

s �� = �±n��±n�� 29�

with

�+ p� = cos�

2�I� + sin

�

2�II� ,

�30�

�+ n�� = cos�1

2�⇑n� + sin

�1

2ei�2�⇓n� ,

�− p� = − sin�

2�I� + cos

�

2�II� ,

�− n�� = − sin�1

2�⇑n� + cos

�1

2ei�2�⇓n� .

Once the state ��B�� is produced, joint measurementson the path and spin are performed by choosing the phase

shift � and the angle �� of the spinor analysis appropriately.

Note that � and �� play the role of the angles �� and �� de-scribed before in Sec. III.

FIG. 4. The maximum of the S function (21) with respect to theBerry phase � with the choice of zero azimuthal angles �2�=�2

=�2�=0. FIG. 5. Schematic view of the experimental setup.

FIG. 3. The Bell angles �1 and �1� with respect to the Berry phase � for the cases f10 and f2�0.


032112-5

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Experimentally, the joint probabilities are given by thenumber of counts

N++��,�� = ��B��P+p�� P+

s ��B�� ,

N+−��,�� = ��B��P+p�� P+

s ��1 + �,�2��B��

= N++„�,��1 + �,�2�… ,

N−+��,�� = ��B��P+p�� + �� P+

s ��B��

= N++�� + �,�� , �31�

N−−��,�� = ��B��P+p�� + �� P+

s ��1 + �,�2��B��

= N++„� + �,��1 + �,�2�… .

Then the expectation value

E��,�� = ��B��Ap�� Bs��B�� 32�

is experimentally represented by

E��,�� =N++��,�� − N+−��,�� − N−+��,�� + N−−��,��

N++��,�� + N+−��,�� + N−+��,�� + N−−��,��.

�33�

The outcome of Eq. �33� coincides with formula �18�. Vary-ing the opening angle of the magnetic field—here it is therelative phase of the RF between the two spin rotators—onecan demonstrate experimentally the influence of the puregeometric phase of the entangled state on the expectationvalue �33�. Considering the CHSH inequality Smax isachieved for ��1=0 ��2=0�, which corresponds to thechoice of one path, and ��1�=� /2 ��2�=0�, whichmeans an equal superposition of the states �I� and �II�,whereas the angles �� and �� of the spinor analysishave to be chosen accordingly.

V. SUMMARY AND CONCLUSION

We have studied the influence of the Berry phase on anentangled spin-1

2 system, specifically the case where theBerry phase is generated by one subspace of the system. Dueto our special setup with external opposite rotating magnets,the “spin-echo” method, we are able to eliminate the dy-

namical phase such that only the geometrical part remains.To analyze the effects of the Berry phase we consider theexpectation value of spin measurements and test the localfeature via a CHSH inequality.

A phase—like our geometrical one—in a pure entangledsystem does not change the amount of entanglement andtherefore not the extent of nonlocality of the system, which isdetermined by the maximal violation of a BI. Therefore sucha phase just affects the Bell angles in a definite way.

We always can achieve the familiar maximum value 22of the S function by rotating the Bell angles with respect tothe Berry phase � by the azimuthal amount ��2−�2�=−2� asdemonstrated in Fig. 1. This occurrence of the maximalvalue is in accordance with theorems of Horodecki [50] andGisin [51].

On the other hand, keeping the measurement planes fixedthe polar Bell angles vary according to formula (24) and themaximum of the S function varies with respect to the Berryphase � as shown in Fig. 4. It even touches the boundary ofthe CHSH inequality at �=� /4, making any distinction be-tween QM and local realistic theories obsolete for this setup.Other Bell inequalities like Bell’s original one [3] show asimilar behavior.

It is our free choice after all whether we rotate the mea-surement planes accordingly in order to achieve the maxi-mum value Smax=22 or keep them fixed at some azimuthalangle and cope with smaller values of Smax. In the secondcase we show explicitly the dependence of Smax on the geo-metric phase.

Entanglement is a quantum property of states defined overa tensor product of Hilbert spaces, no matter what kind ofspaces they are. In this sense we certainly can entangle theinternal (spin) with the external (space) degrees of freedomof one and the same particle. Then the physical interpretationof a BI, however, differs from the usual nonlocality case andit is the more general concept of contextuality of the states,which is tested, similarly to the Bell-Kochen-Specker theo-rem [41–43]. Noncontextuality here means that the value forthe observable spin of the neutron does not depend on theexperimental context, i.e., on the other comeasured observ-able, the path.

Noncontextuality is a rather restrictive demand for atheory, which is incompatible with QM. We propose the ex-perimental test with single particles—the test of noncontex-tuality versus QM—to be performed within neutron interfer-ometry which is an excellent tool to study the properties ofspin-1

2 systems. In our case we study both the influence ofthe Berry phase on entanglement and on contextuality of thestates. The neutron experiment which can be easily carriedout is in progress.

ACKNOWLEDGMENTS

The authors want to thank Gerhard Ecker, Stefan Filipp,Heide Narnhofer, and Helmut Rauch for helpful discussions.This research has been supported by the EU projectEURIDICE EEC-TMR program HPRN-CT-2002-00311 andthe FWF, Project Nos. F1513 and SFB 015 P06 of the Aus-trian Science Foundation.

FIG. 6. Schematic representation of the spinor evolution withthe use of the Poincaré sphere.


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168

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Decoherence within neutron interferometry and testing of the Krausoperator decomposition

Reinhold A. Bertlmann,1 Katharina Durstberger,1, ∗ and Yuji Hasegawa2,3

1Institute for Theoretical Physics, University of ViennaBoltzmanngasse 5, 1090 Vienna, Austria

2Atominstitut der Osterreichischen UniversitatenStadionallee 2, 1020 Vienna, Austria

3PRESTO, Japan Science and Technology Agency4-1-8 Honcho Kawaguchi, Saitama, Japan

We consider two different decoherence modes for bipartite two-level systems where the decoherenceenters via special Lindblad-terms in the master equation. The solutions for general and for Belldiagonal states are given. The case of Bell diagonal states is investigated in detail via considerationsof the degrees of mixedness and the amount of entanglement present in the time-evolved state.

We propose an implementation of the decoherence modes within neutron interferometry by ap-plying random magnetic fields. An experimental test of the Kraus operator decomposition for eachmode is presented.

PACS numbers: 03.65.Yz, 03.75.Dg, 42.50.-pKeywords: Decoherence, Lindblad generators, neutron interferometry, Kraus operator decomposition, Belldiagonal states

I. INTRODUCTION

Closed quantum systems are idealizations which do not exist in real physical world. One has to dealwith open quantum systems which arise due to an interaction of the system under consideration withan external environment (e.g. reservoir, heat bath) [1, 2]. The interaction between the system andthe environment causes a phenomenon known as decoherence: quantum correlations and interferencesare destroyed as time goes on; the system shows more and more classical behavior. The theory ofdecoherence is one candidate to solve the question why our world looks so classical [3, 4].The total Hamiltonian of system and environment generates a unitary evolution U and is of the formHSE = H ⊗ �E + � ⊗HE + HI , where H , HE and HI are, respectively, the system, environment andinteraction Hamiltonians. The evolution of the system or the reduced dynamics is obtained by tracingover the environmental degrees of freedom ρ(t) = TrE(ρS+E(t)) = TrE(UρS+E(0)U †) and thus gaining anonunitary evolution of the system in contrast to closed systems.In most of the cases we do not have access or information about the dynamics of the environment.Therefore we have to describe the evolution of the system by an effective dynamics: the Liouville-vonNeumann master equation. Thereby it is not so important to know the exact Hamiltonian and thenature of the environment but only its effects on the system. Our tactic in this paper is to proposeseveral effective models which do not care about the exact nature of decoherence but provide scenarioshow decoherence can affect a system.Under several assumptions [1], such as Markovian semigroup approach, complete positivity, initial decou-pling of system and environment, and weak coupling, the dynamics of the system can be described by aLiouville-von Neumann master equation

∂

∂tρ(t) = −i[H(t), ρ(t)]−D[ρ(t)] . (1)

Lindblad, Gorini, Kossakowski and Sudarshan [5, 6] derived the most general structure of the dissipator

D[ρ(t)] =12

∑k

(A†

kAkρ(t) + ρ(t)A†kAk − 2Akρ(t)A†

k

), (2)

∗Electronic address: [email protected]

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2

where Ak represent the so-called Lindblad generators. The sum is taken over an arbitrary number ofcomponents but maximally up to n2 − 1, where n denotes the dimension of the system. For simplicitywe choose the generators to be projectors such that Ak =

√λkPk with P 2

k = Pk (see Ref.[7]) which givesfor the dissipator

D[ρ] =12

∑k

λk

(Pkρ + ρPk − 2PkρPk

). (3)

The starting point for our analysis is a dissipator with projective generators.The paper is organized as follows. In the next section we introduce and discuss two possible decoher-ence scenarios for a two qubit system by choosing different projection operators Pk. In Sect.III the twodecoherence modes are discussed for the special case of Bell diagonal states. Sect.IV proposes an imple-mentation of the decoherence modes within neutron interferometry via random magnetic fields. In Sect.Vwe present the Kraus operator decomposition. The action of this decomposition is mathematically equalto the Lindblad form of the Liouville-von Neumann equation. We can test this equivalence by a simpleexperiment with single neutrons.

II. DECOHERENCE MODES IN A TWO QUBIT SYSTEM

Let us consider a two qubit system with Hilbertspace H = H(1) ⊗ H(2) = C2 ⊗ C2 where {|ek〉}k=1,...,4

denotes an eigenbasis defined by H |ek〉 = Ek|ek〉, with H = H(1) ⊗ � + � ⊗ H(2) the Hamiltonianof the undisturbed system. A general state ρ of the system can be expressed in the eigenbasis ρ =∑

k,j ρkj |ek〉〈ej |, where (ρkj) denotes the 4× 4 coefficient matrix.We consider Lindblad generators Pk that project onto one-dimensional subspaces and fulfil

∑k Pk = �,

furthermore we assume only one dissipation parameter λ that parameterizes the strength of the interactionand therefore of the decoherence. Then the dissipator, Eq.(3), can be written as

D[ρ] = λ(ρ−

4∑k=1

PkρPk

). (4)

In the following sections we solve the Liouville-von Neumann equation (1) with the dissipator (4) byassuming different projection operators Pk which we call decoherence modes.

A. Mode A

The first mode depicts the simplest possible case. The Lindblad generators are chosen to be projectorsPk = |ek〉〈ek| onto the eigenbasis of the Hamiltonian (see Refs.[8, 9]). In this mode of decoherence thetime evolution (1) for the coefficient matrix is given by

˙ρkj =(−i(Ek − Ej)− λA

)ρkj for k �= j

˙ρkk = 0 ,(5)

which can be easily solved

ρkj(t) = e−i(Ek−Ej)te−λAtρkj(0) for k �= j

ρkk(t) = ρkk(0) .(6)

The decoherence affects only the off-diagonal elements and leaves the diagonal elements untouched.

B. Mode B

For the second mode the Lindblad generators are chosen to be projectors Pk = |ek〉〈ek| onto the followingstates

|e1,3〉 =1√2(|e1〉 ± |e3〉) , |e2,4〉 =

1√2(|e2〉 ± |e4〉) , (7)

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3

where the upper (lower) sign corresponds to the fist (second) index.The time evolution (1) of the coefficient matrix gives 3 types of differential equations. Type I holds forthe components ρ12, ρ14, ρ23, ρ34 and has the structure

˙ρ12 =(−i(E1 − E2)− λB

)ρ12 , (8)

in analogy to mode A. Type II holds for the diagonal components and reveals pairwise coupled differentialequations of the form

˙ρ11 = −λB

2ρ11 +

λB

2ρ33 , ˙ρ33 =

λB

2ρ11 −

λB

2ρ33 . (9)

Type III also gives pairwise coupled differential equations

˙ρ13 =(−i(E1 − E3)−

λB

2)ρ13 +

λB

2ρ31 , ˙ρ31 =

λB

2ρ13 +

(i(E1 − E3)−

λB

2)ρ31 , (10)

and holds for the components ρ13, ρ31, ρ24, ρ42. The solution for type I, Eq.(8), is given by

ρ12(t) = e−i(E1−E2)te−λBtρ12(0) , (11)

type II, Eq.(9), is solved by

ρ11(t) =12(1 + e−λBt)ρ11(0) +

12(1− e−λBt)ρ33(0) ,

ρ33(t) =12(1− e−λBt)ρ11(0) +

12(1 + e−λBt)ρ33(0) ,

(12)

and type III, Eq.(10), leads to

ρ13(t) = e−λBt

2

((cosh

μt

2− 2i(E1 − E3)

μsinh

μt

2

)ρ13(0) +

λB

μsinh

μt

2ρ31(0)

),

ρ31(t) = e−λBt

2

((cosh

μt

2+

2i(E1 − E3)μ

sinhμt

2

)ρ31(0) +

λB

μsinh

μt

2ρ13(0)

),

(13)

where μ =√

λ2B − 4(E1 − E3)2.

The decoherence mode affects not only the off-diagonal elements of the density matrix but also thediagonal ones. Eq.(7) indicates the basis where the density matrix gets diagonal for t→∞.

It is worth noting here that the choice of projection states, Eq.(7), corresponds to a rotation of statesin one subspace. Suppose we split the eigenstates of the undisturbed Hamiltonian in eigenstates of thesubspace Hamiltonians {|a1〉, |a2〉} and {|b1〉, |b2〉} in the following way

|e1,3〉 = |a1,2〉|b1〉 , |e2,4〉 = |a1,2〉|b2〉 . (14)

Now consider a rotation of the first subbasis, {|+〉 = 1√2(|a1〉+ |a2〉), |−〉 = 1√

2(|a1〉 − |a2〉)}, the second

subbasis is left untouched. The basis of the total Hilbertspace changes

|e1〉 = |+〉|b1〉 =1√2(|e1〉+ |e3〉) , |e2〉 = |+〉|b2〉 =

1√2(|e2〉+ |e4〉) ,

|e3〉 = |−〉|b1〉 =1√2(|e1〉 − |e3〉) , |e4〉 = |−〉|b2〉 =

1√2(|e2〉 − |e4〉) ,

(15)

which corresponds exactly to the states used in Eq.(7). Therefore decoherence mode B can be called“R ⊗ E” to indicate the rotation of the first subspace and the untouched eigenbasis in the secondsubspace whereas mode A can be called “E ⊗ E”.

In the case of photons (see e.g. [10]) the eigenbasis E corresponds to horizontal |H〉 and vertical |V 〉polarization whereas the rotated basis R represents polarization states |+45◦〉 and |−45◦〉. In the case ofneutral kaons (for an overview see e.g. [11]) we can identify the eigenbasis E with the states |KS〉 and|KL〉 and the rotated basis R with |K0〉 and |K0〉.

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4

III. INITIAL CONDITIONS – BELL DIAGONAL STATES

We want to illustrate the above discussed decoherence modes by choosing a certain class of states asinitial conditions – the so called Bell diagonal states.Bell diagonal states ρ =

∑i νi|Ψi〉〈Ψi| with

∑i νi = 1 are diagonal in the so called Bell basis

|Ψ1,2〉 =1√2(|e1〉 ± |e4〉) , |Ψ3,4〉 =

1√2(|e2〉 ± |e3〉) . (16)

In matrix form in the standard basis they are given by

ρ =12

⎛⎜⎝ν1 + ν2 0 0 ν1 − ν2

0 ν3 + ν4 ν3 − ν4 00 ν3 − ν4 ν3 + ν4 0

ν1 − ν2 0 0 ν1 + ν2

⎞⎟⎠ =:12

⎛⎜⎝Σ1 0 0 Δ1

0 Σ2 Δ2 00 Δ2 Σ2 0

Δ1 0 0 Σ1

⎞⎟⎠ . (17)

States can be characterized for instance by two quantities: mixing and entanglement.The mixedness, defined as δ = Trρ2, ranges between 1 (pure states) and 1

4 (maximally mixed states) andis given by δ = ν2

1 + ν22 + ν2

3 + ν24 for Bell diagonal states.

The concurrence C [12–14] is a quantity that measures the entanglement contained in a state ρ. It isdefined by C(ρ) = max{0, μ1−μ2−μ3−μ4}, where μi are the square roots of the eigenvalues in decreasingorder of the matrix R = ρ (σy ⊗ σy) ρ∗ (σy ⊗ σy) and ρ∗ denotes complex conjugation in the standardbasis. The concurrence can take values between 1 (maximally entangled states) and 0 (separable states)and is given by C = max

{0, 2 max{νi}− 1

}for Bell diagonal states, depending on which weight νi is the

largest. A Bell diagonal state can only be entangled (C > 0) if the largest eigenvalue fulfills μi > 12 .

The special case of a pure and maximally entangled Bell state, e.g. the Bell singlet state |Ψ4〉 whereν4 = 1 and ν1 = ν2 = ν3 = 0 or Σ1 = Δ1 = 0 and Σ2 = −Δ2 = 1, gives δ = 1, C = 1.The quantities mixedness δ and entanglement C are unchanged by dynamical evolution induced purelyby the Hamiltonian of the system H . Therefore it is justified to ignore the dynamical aspects in theevolution, that means we consider solutions for mode A and B of the Liouville-von Neumann masterequation which are of the form

∂

∂tρ(t) = −D[ρ(t)] . (18)

A. Mode A

The initial state is a Bell diagonal state, Eq.(17), and after a time evolution according to mode A, Eq.(6),we end up with the state

ρ(t) =12

⎛⎜⎜⎝Σ1 0 0 e−λAtΔ1

0 Σ2 e−λAtΔ2 00 e−λAtΔ2 Σ2 0

e−λAtΔ1 0 0 Σ1

⎞⎟⎟⎠ . (19)

We get for the mixedness δ = 12

(Σ2

1 + Σ22 + e−2λAt(Δ2

1 + Δ22)) and the concurrence can be calculated by

C(ρ) = max{0, 2 max{μi} − 1

}where μ1,2 = 1

2 (Σ1 ± e−λAtΔ1) and μ3,4 = 12 (Σ2 ± e−λAtΔ2).

For our special case of the pure Bell singlet state |Ψ4〉 we get that the mixedness δ = 12 (1 + e−2λAt)

ranges from a pure state (δ = 1) to a mixed but not maximally mixed state (δ t→∞−−−→ 12 ). The concurrence

C(ρ) = e−λAt goes from a maximally entangled state (C = 1) to an separable state asymptotically(C t→∞−−−→ 0).The behavior of δ and C is plotted in Fig.1.

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5

B. Mode B

The second mode, Eqs.(11)-(13), gives a slightly different density matrix

ρ(t) =14

⎛⎜⎜⎝1 + e−λBt(−Σ1 + Σ2) 0 0 2e−λBtΔ1

0 1 + e−λBt(Σ1 − Σ2) 2e−λBtΔ2 00 2e−λBtΔ2 1 + e−λBt(Σ1 − Σ2) 0

2e−λBtΔ1 0 0 1 + e−λBt(−Σ1 + Σ2)

⎞⎟⎟⎠ .

(20)We receive for the mixedness δ = 1

4

(1 + e−2λBt

(2Δ2

1 + 2Δ22 + (Σ1 + Σ2)2

))and for the amount of

entanglement C(ρ) = max{0, 2 max{μi} − 1

}where μ1,2 = 1

4 (1 + e−λBt(Σ1 − Σ2 ± 2Δ2)) and μ3,4 =14 (1 + e−λBt(−Σ1 + Σ2 ± 2Δ1)).The special case of |Ψ4〉 gives the following results. The mixedness δ = 1

4 (1 + 3e−2λBt) goes from a pure

state (δ = 1) to a maximally mixed state (δ t→∞−−−→ 14 ). The concurrence C(ρ) = max

{0, 1

2 (3e−λBt − 1)}

starts at the maximally entangled state (C = 1) decreases and reaches the border of separability (C = 0)at finite time t = ln 3

λBwhere the mixing has the value of δ = 1

3 .In Fig.1 the dependence of δ and C with respect to λt is shown.

1

δ

λt

ln 3

12

14

1

C

λt

ln 3

FIG. 1: Graphical comparison of the mixedness δ and the concurrence C in dependence of λt for mode A and Bfor the Bell singlet state |Ψ4〉. The upper curves correspond to mode A and the lower to mode B.

IV. IMPLEMENTATION OF DECOHERENCE MODES FOR NEUTRON STATES

The different decoherence modes presented in Sect.II and discussed for Bell diagonal states in Sect.III canbe implemented within neutron interferometry [15]. Single neutrons can show entanglement between aninternal degrees of freedom (spin) and an external degree of freedom (path)[16–18] which can be describedformally with a bipartite Hilbertspace H = Hspin ⊗ Hpath. For instance the neutrons can be producedin the antisymmetric Bell state

|Ψ4〉 =1√2(|⇑〉 ⊗ |II〉 − |⇓〉 ⊗ |I〉) ≡ 1√

2(|e2〉 − |e3〉) . (21)

where |⇑〉 and |⇓〉 represent ±z polarized spin states whereas |I〉 and |II〉 denote the paths in theinterferometer.In the following treatment we consider a separable state of the form

|Ψexp〉 =1√2(|⇑〉 ⊗ |I〉+ |⇑〉 ⊗ |II〉) ≡ 1√

2(|e1〉+ |e2〉) . (22)

Quantum state tomography [22] confirmed that this state has four main poles in the real-part of thedensity matrix [23], which is one of the characteristics of the Bell basis. In addition, this state can bemanipulated more simply and accurately in the experiment. Thus, the state described by Eq.(22) is verysuitable to implement and investigate the decoherence modes with neutrons.

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6

The final states of both modes are then given by

ρA(t) =12

⎛⎜⎜⎝1 e−λAt 0 0

e−λAt 1 0 00 0 0 00 0 0 0

⎞⎟⎟⎠ , ρB(t) =14

⎛⎜⎜⎝1 + e−λBt 2e−λBt 0 02e−λBt 1 + e−λBt 0 0

0 0 1− e−λBt 00 0 0 1− e−λBt

⎞⎟⎟⎠ . (23)

Here one sees the fading of the off-diagonal elements for both modes and homogenization of the diagonalelement for mode B.

A. Decoherence via random magnetic fields

For the implementation of decoherence we use magnetic fields which act randomly on an ensemble ofneutrons produced in a specific state ρ.The action of a magnetic field B = Bn in the direction n on a neutron state is described by the unitaryoperator U(α) = ei α

2 �n·�σ where α = 2μBt = ωLt denotes the rotation angle and ωL the Larmor frequency(μ magnetic moment, B magnetic field strength, σ Pauli matrices).Suppose the neutron beam passes a fluctuating magnetic field B(t) in such a way that each neutronwhich is part of the quantum mechanical ensemble described by ρ experiences a time independent fieldB|t = const. This corresponds to a unitary operator U(α) with constant rotation angle α. For the wholeensemble we have to take the integral over all possible rotation angles α

ρ −→ ρ′ =∫

U(α) ρ U †(α)︸︷︷︸ρ(α)

P (α)dα , (24)

where P (α) denotes a distribution function. Due to the integration the evolution of the whole ensemblestate is described by a nonunitary operation.

We are using Gaussian distribution functions P (α) = 1√2πσ

e−α2

2σ2 with standard deviation σ.

B. Mode A

The state |Ψexp〉 is prepared after going through the beam-splitter. This initial state suffers from thedissipative magnetic fields oriented along the z-axis in each path of the interferometer, see Fig.2. Therotations U(α) and U(β) caused by the fields are independent but their distributions have the samedeviation σ.The action of the two magnetic fields can be described by a conditioned operation. Depending on thestate of the spatial degree of freedom either operation U(α) or U(β) is applied to the spin state

|ψspin〉 ⊗ |I〉 −→ U(α)|ψspin〉 ⊗ |I〉|ψspin〉 ⊗ |II〉 −→ U(β)|ψspin〉 ⊗ |II〉 . (25)

For a single neutron the application of the conditioned operation on the initial state |Ψexp〉 (22) gives

ρ(α, β) =12

⎛⎜⎜⎝1 ei α+β

2 0 0e−i α+β

2 1 0 00 0 0 00 0 0 0

⎞⎟⎟⎠ , (26)

which after integration over α and β turns into

ρ′ =∫

ρ(α, β)P (α)P (β)dα dβ =12

⎛⎜⎜⎜⎝1 e−

σ24 0 0

e−σ24 1 0 0

0 0 0 00 0 0 0

⎞⎟⎟⎟⎠ . (27)

175

7

|I〉

|II〉

�B(1)z

�B(2)z

Phase shifter χ

Spin rotator ξ

FIG. 2: Schematic experimental setup for the implementation of mode A. The magnetic field �B(1)z and �B

(2)z

produce independent rotations U(α) and U(β) respectively. With the phase shifter χ and the spin rotator ξ thefinal state can be analyzed.

By comparison of Eq.(23) and Eq.(27) we immediately see that

λAt =σ2

4(28)

the decoherence parameter λA is directly related to the deviation σ of the fluctuating magnetic fields.

Note, that for only one magnetic field located in one of the paths fluctuating with deviation σ the aboverelation is given by λAt = σ2

8 , and for one field acting on both paths in the same kind we get λAt = σ2

2 .

C. Mode B

For mode B, although the same preparation procedure is applied, slightly different fluctuating magneticfields are applied, as shown in Fig.3 shows. The different unitary operations caused by the magneticfields are assumed to be independently but the distributions have the same deviation σ.

|I〉

|II〉

�B(1)x

�B(1)z

�B(2)x

�B(2)z

Phase shifter χ

Spin rotator ξ

FIG. 3: Schematic experimental setup for the implementation of mode B. The magnetic fields �B(1)x , �B

(2)x , �B

(1)z

and �B(2)z generate independent unitary rotations. The order of the magnetic fields in each path does not matter

in this context. With the phase shifter χ and the spin rotator ξ the final state can be analyzed.

176

8

We get for the ensemble state after the integrations

ρ′ =14

⎛⎜⎜⎜⎜⎝1 + e−

σ22 2e−

σ22 0 0

2e−σ22 1 + e−

σ22 0 0

0 0 1− e−σ22 0

0 0 0 1− e−σ22

⎞⎟⎟⎟⎟⎠ , (29)

where we can establish via Eq.(23) and Eq.(29) the relation

λBt =σ2

2, (30)

between decoherence parameter λB and the deviation σ of the Gaussian distribution function.

V. CONNECTION TO THE KRAUS OPERATOR DECOMPOSITION

In the following section we give a connection to the Kraus operator decomposition. Theory tells us thatnon-unitary evolutions can be described by Kraus operators. We want to demonstrate that by a simpleexperiment within neutron interferometry [15] we can check if the theoretically predicted Kraus operatorscorrespond to the implemented decoherence models discussed in Sect.IV.

A. Kraus operator decomposition

The completely positive time evolution generated by the Liouville-von Neumann master equation (1)together with the Lindblad form of the generator (2) can also be represented by the so-called Krausoperator decomposition [2, 19]

ρ(0) �→ ρ(t) =∑

k

Mkρ(0)M †k , (31)

where the Kraus operators Mi fulfil∑

k M †kMk = �. The first approach gives a continuous time-

dependence whereas the second approach models decoherence via discrete state changes. Both viewsare equivalent and a correspondence between Lindblad generators Ak and Kraus operators Mk exists forsmall δt(see e.g. Ref.[20])

M0 = �− (iH +12

∑A†

kAk)δt

Mk =√

δt Ak .

(32)

Kraus operators are not unique and can be transformed by unitary operations.

B. Mode A

We use the concept of a bipartite neutron Hilbert space Hspin ⊗ Hpath where entanglement can occurbetween spinor and spatial degrees of freedom [16–18].

Mode A represents a kind of phase damping channel [2, 20] which destroys the coherence in the system.The Kraus operators are given by

M0 =

√1− 3p

4�

s ⊗ �p M1 =√

p

4�

s ⊗ σpz

M2 =√

p

4σs

z ⊗ �p M3 =√

p

4σs

z ⊗ σpz ,

(33)

177

9

�Bz(π)

�Bz(π) Phase shifter (π)

Phase shifter (χ)

Spin rotator (ξ)

|I〉

|II〉

FIG. 4: Schematic experimental setup for the realization of the Kraus operator σsz ⊗ σp

z for mode A.

where p = λt is the probability for the decoherence taking place which leads in first order of t to the stateρ(t), Eq.(6).In the experiments, neutrons are prepared in the initial state after going through the beam splitter andsuffer from the effect represented by the Kraus operators. The identity-operators �s and �

p do nothingto the spinor and spatial degree of freedom. The operator σs

z on the spinor part operates only on the spinstate |⇓〉 where it induces a phase shift of π. This phase shift difference of spin up and spin down canbe implemented by a magnetic field Bz in z-direction (modulo an overall phase shift). The operator σp

z

on the spatial subspace can be realized by a phase shifter in the path |II〉 which acts with a fixed phaseshift of π.The states produced by the four Kraus operators are measured and the weighted sum according to (33)will show the expected behavior.

C. Mode B

Mode B is a combination of a bit flip channel and a phase flip channel [2, 20]. The corresponding Krausoperators are

M0 =

√1− 3p

4�

s ⊗ �p M1 =√

p

4�

s ⊗ σpz

M2 =√

p

4σs

x ⊗ �p M3 =√

p

4σs

x ⊗ σpz ,

(34)

and create in first order of t the state given by Eqs. (11), (12) and (13) (p = λt).The Kraus operator for the spatial part σp

z is the same as for mode A, by a phase shift of π. The onlydifference is the σs

x operator for the spin part. It can be realized by two magnetic fields Bx in both armspointing in the x-direction which cause a spin flip. The experimental setup can be seen in Fig.5.The weighted sum of the measured states produced by the Kraus operators according to (34) will showthe expected behavior.

VI. SUMMARY AND CONCLUSION

We have considered the Liouville-von Neumann equation where decoherence is implemented by the dis-sipator in Lindblad form. We study two kinds of decoherence modes where the Lindblad generators aregiven by different projection operators: decoherence in the eigenbasis of the Hamiltonian, mode A, anddecoherence in a rotated basis, mode B. The two modes are analyzed in detail for Bell diagonal states,where it turns out that in mode B the state gets more mixed and the decrease of the entanglement is

178

10

�Bx

�Bx Phase shifter (π)

Phase shifter (χ)

Spin rotator (ξ)

|I〉

|II〉

FIG. 5: Schematic experimental setup for the realization of the Kraus operator σsx ⊗ σp

z for mode B.

higher than in mode A. For the Bell singlet state |Ψ4〉 there exists a point λt = ln 3 where in mode B wereach the border of separability whereas in mode A the state is still entangled by an amount of 33%, seeFig.1 and Ref.[21].The implementation of the proposed decoherence modes uses the bipartite Hilbert space construction ofneutron interferometry where entanglement in single neutrons occurs between an internal (spin) and anexternal (path) degree of freedom.We create decoherence via random magnetic fields in the interferometer (Sect.IV). There we can establishsimple relations between the decoherence parameter λ and the deviation of the random distribution ofthe fields σ, Eqs. (28) and (30). This allows an experimental control of the implemented decoherencein each mode. The strength of decoherence does not depend on the actual rotation parameter α of themagnetic field but only on the width of the distribution. We are able to construct any state predicted bythe two decoherence modes just by varying σ. The states (27) and (29) are measured by state tomography[22, 23].In the second type of proposed neutron experiment we want to test experimentally the validity of theKraus operator decomposition which describe completely positive time evolutions (Sect.V). Via relation(32) we can determine the Kraus operators for each mode and implement them in the neutron interfer-ometer. We show that by 4 different Kraus operators for each decoherence mode the final states canbe modelled where each Kraus operator must be implemented separately. Mode A represents a kind ofphase damping channel and mode B describes a bit flip channel together with a phase damping channel.The state determination is done by state tomography measurements.

Acknowledgments

The authors want to thank Stefan Filipp, Jurgen Klepp and Helmut Rauch for helpful discussions.This research has been supported by the EU project EURIDICE EEC-TMR program HPRN-CT-2002-00311 (R.A.B., K.D.), the University of Vienna (Forderungsstipendium of K.D.) and the FWF-projectP17803-N02 of the Austrian Science Foundation (Y.H.).

[1] H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, New York,2002).

[2] M. Nielsen and I. Chuang, Quantum computation and quantum information (Cambridge University Press,Cambridge, 2000).

[3] D. Giulini et al., Decoherence and the Appearance of a Classical World in Quantum Theory (Springer Verlag,Berlin, 1996).

[4] W. Zurek, Physics Today 36 (1991).[5] G. Lindblad, Comm. Math. Phys. 48, 119 (1976).

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[6] V. Gorini, A. Kossakowski, and E. Sudarshan, J. Math. Phys. 17, 821 (1976).[7] R. A. Bertlmann and W. Grimus, (2002).[8] R. A. Bertlmann, K. Durstberger, and B. C. Hiesmayr, Phys. Rev. A 68, 012111 (2003).[9] R. A. Bertlmann and W. Grimus, Phys. Rev. D 64, 056004 (2001).

[10] D. Bouwmeester, A. Ekert, and A. Zeilinger, The physics of quantum information: Quantum Cryptography,Quantum Teleportation, Quantum Computation (Springer-Verlag, Berlin, 2000).

[11] B. Hiesmayr, Ph.D. thesis, University of Vienna, 2002.[12] S. Hill and W. Wootters, Phys. Rev. Lett. 78, 5022 (1997).[13] W. Wootters, Phys. Rev. Lett. 80, 2245 (1998).[14] W. Wootters, Quantum Information and Computation 1, 27 (2001).[15] H. Rauch and S. Werner, Neutron Interferometry: Lessons in Experimental Quantum Mechanics (Oxford

University Press, Oxford, 2000).[16] Y. Hasegawa et al., Nature 425, 45 (2003).[17] Y. Hasegawa et al., J. Opt. B: Quantum and Semiclass. Opt. 6, 7 (2004).[18] R. A. Bertlmann, K. Durstberger, Y. Hasegawa, and B. Hiesmayr, Phys. Rev. A 69, 032112 (2004).[19] K. Kraus, States, Effects and Operations: Fundamental Notations of Quantum Theory, Vol. 190 of Lecture

notes in physics (Springer, Berlin, 1983).[20] J. Preskill, http://theory.caltech.edu/people/preskill/ph229/ .[21] T. Yu and J. Eberly, quant-ph/0503089 (2005).[22] D. James, P. Kwiat, W. Munro, and A. White, Phys. Rev. A 64, 052312 (2001).[23] Y. Hasegawa et al., to be published .

180

Curriculum Vitae CURRICULUM VITAE

Curriculum Vitae von Katharina Durstberger

Personliche Daten:

Ich wurde am 16.06.1978 in Kirchdorf an der Krems (OO) als erstes Kind von Heinzund Christa Durstberger geboren.

Schulische Laufbahn:

1984 - 1988 Besuch der offentlichen Volksschule in Roitham (OO)1988 - 1996 Besuch des Realgymnasiums im Stift Lambach (OO)

Schwerpunkt: Darstellende GeometrieWahlpflichtfacher: Mathematik und Informatik

Februar 1996 Verfassen einer Fachbereichsarbeit in Physik im Rahmen derMatura bei Mag. Christian Kitzberger zum Thema:“Die Theorie der Quarks unter besonderer Berucksichtigungder QuantenChromoDynamik”

Juni 1996 Matura mit Ausgezeichnetem ErfolgOktober 1996 Beginn des Diplomstudiums der Physik und Astronomie an

der Universitat WienJuli 1998 erste Diplomprufung in Physik bestanden, erste

Diplomprufung in Astronomie mit Auszeichnung bestandenOkt. 2000 – Dez. 2001 Diplomarbeit in Physik bei Univ.Prof.Dr. R.A. Bertlmann

zum Thema:“Geometrical Phases in Quantum Theory”

Janner 2002 zweite Diplomprufung in Physik mit AuszeichnungMarz 2002 Beginn des Doktoratsstudiums der Physik16. Mai 2003 Verleihung des Alfred-Wehrl-Preises fur Mathematische

Physik fur die DiplomarbeitJuli – Sept. 2003 Mitorganisation der Triangle Graduate School 2003 in Raach

am Semmering(siehe http://www.ap.univie.ac.at/users/raach)

WS 2003/04 Tutorium zur LVA “Theoretische Physik T1 - Mechanik”Sept. 2003 – April 2004 Erstellung und Betreuung der Homepage EU3CM fur das 3.

Meeting der EURIDICE-Collaboration(siehe http://univie.ac.at/euridice/EU3CM)

12. Nov. 2004 Verleihung der Talentforderungspramie des LandesOberosterreich in der Sparte Wissenschaft

Marz 2002 – Okt. 2005 Dissertation in Physik bei Univ.Prof.Dr. R.A. Bertlmannzum Thema:“Geometry of Entanglement and Decoherence in QuantumSystmes”

WS 2005/06 Ubungen zur LVA “Theoretische Physik T1 - Mechanik”

181

CURRICULUM VITAE Curriculum Vitae

Konferenzen und Graduiertenschulen:

September 2000 Triangle Summer School in Praha (Tschechien)09. – 15.09. 2001 Triangle Summer School in Modra-Harmonia (Slowakei)17. – 18.09. 2001 OPG-FAKT (Osterreichische Physikalische Gesellschaft,

Fachausschuss Kern- und Teilchenphysik) Tagung in Hol-labrunn

09. – 14.11. 2001 Konferenz “Quantum[Un]speakables, conference in commemo-ration of John S. Bell” in Wien

23. – 28.03. 2002 EURESCO Konferenz “Quantum Information” in San Feliu deGuixols (Spanien)

13. – 18.07. 2002 Konferenz uber “Quantum Information: Conceptual Founda-tions, Developments and Perspectives” in Oviedo (Spanien)

02. – 07.09. 2002 Internationale Schule uber “Quantum Computation and Infor-mation” in Lissabon (Portugal)

23. – 28.09. 2002 OPG-FAKT Tagung in Seggauberg08. – 10.10. 2002 Small Triangle Meeting in Snina (Ostslowakei)01. – 06.06. 2003 Konferenz “Quantum Theory: Reconsideration of Foundations

– 2” in Vaxjo (Schweden)22. – 26.09. 2003 Triangle Graduate School in Raach02. – 04.10. 2003 OPG-FAKT Tagung in Strobl am Wolfgangsee19. – 20.12. 2003 SFB-Meeting in Innsbruck28.02 – 06.03. 2004 Winterschule fur Theoretische Physik in Schladming15. – 17.09. 2004 Workshop “Gedankenexperimente Traunkirchen”26. – 27.09. 2004 OPG-FAKT Tagung in Weyer28. – 30.09. 2004 OPG Tagung in Linz23. – 25.05. 2005 IUPAP Konferenz “Women in Physics” in Rio de Janeiro12. – 14.09. 2005 Workshop “Gedankenexperimente Traunkirchen II”27. – 29.05. 2005 OPG Tagung in Wien

Wissenschaftliche Vortrage:

28.09. 2002 OPG-FAKT Tagung in Seggauberg09.10. 2002 Small Triangle meeting in Snina (Slowakei)03.06. 2003 Konferenz “Quantum Theory: Reconsideration of Foundations -

2” in Vaxjo (Schweden)03.10. 2003 OPG-FAKT Tagung in Strobl am Wolfangsee24.10. 2003 Gastvortrag, Seminar fur Neutronen- und Festkorperphysik, Prof.

Helmut Rauch, Atominstitut der Osterreichischen Universitaten10.11. 2003 Gastvortrag, Seminar des Instituts fur Experimentalphysik, Prof.

Anton Zeilinger, Universitat Wien19.12. 2003 Gastvortrag, SFB-Meeting Innsbruck29.09. 2005 OPG Tagung in Wien

182

Curriculum Vitae CURRICULUM VITAE

Preise:

• Verleihung des Alfred-Wehrl Preises fur Mathematische Physik fur die Diplo-marbeit “Geometrische Phasen in der Quanten Theorie” am 16. Mai 2003

• Verleihung der Talentforderungspramie des Landes Oberosterreich in der Sparte“Wissenschaften” am 12. November 2004

Projekte:

• EU- Projekt EURIDICE (European Investigation on Daphne and other Inter-national Collider Experiments), Netzwerk aus 12 Institutionen, Mitarbeit inder Quantenmechanikgruppe (http://www.univie.ac.at/euridice), 2002 – 2006

• Forschungsprojekt des Magistrats der Stadt Wien, Magistratsabteilung 7 –Kultur, Wissenschafts- und Forschungsforderung, September 2002 – Septem-ber 2003

• Projekt “Gender Mainstreaming and Gender Sensibility” im Rahmen des Pro-jekts IMST2 “Innovations in Mathematics, Science and Technology Teaching”,Fuhren von Interviews und Erstellen von Lebenslaufen von PhysikerInnen furdie homepage LISE (http://lise.univie.ac.at), seit Marz 2003

• Forschungsprojekt der Universitat Wien, Fakultat fur Naturwissenschaftenund Mathematik, Janner 2004 – Dezember 2005

• Forschungskooperation Wien-Bratislava “nonlocal seminar” (mehr Informatio-nen http://mailbox.univie.ac.at/reinhold.berlmann oder http://www.quniverse.sk),seit Oktober 2004

183

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