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Multipartite entangled quantum states:
Transformation,Entanglement monotones and Application
by
Wei Cui
A thesis submitted in conformity with the requirementsfor the
degree of Doctor of Philosophy
Graduate Department of PhysicsUniversity of Toronto
c© Copyright 2013 by Wei Cui
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Abstract
Multipartite entangled quantum states: Transformation,
Entanglement monotones andApplication
Wei CuiDoctor of Philosophy
Graduate Department of PhysicsUniversity of Toronto
2013
Entanglement is one of the fundamental features of quantum
information science.
Though bipartite entanglement has been analyzed thoroughly in
theory and shown to
be an important resource in quantum computation and
communication protocols, the
theory of entanglement shared between more than two parties,
which is called multi-
partite entanglement, is still not complete. Specifically, the
classification of multipartite
entanglement and the transformation property between different
multipartite states by
local operators and classical communications (LOCC) are two
fundamental questions in
the theory of multipartite entanglement.
In this thesis, we present results related to the LOCC
transformation between multi-
partite entangled states. Firstly, we investigate the bounds on
the LOCC transformation
probability between multipartite states, especially the GHZ
class states. By analyzing
the involvement of 3-tangle and other entanglement measures
under weak two-outcome
measurement, we derive explicit upper and lower bound on the
transformation probabil-
ity between GHZ class states. After that, we also analyze the
transformation between
N-party W type states, which is a special class of multipartite
entangled states that
has an explicit unique expression and a set of analytical
entanglement monotones. We
present a necessary and sufficient condition for a known upper
bound of transformation
probability between two N-party W type states to be
achieved.
We also further investigate a novel entanglement transformation
protocol, the ran-
dom distillation, which transforms multipartite entanglement
into bipartite entanglement
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shared by a non-deterministic pair of parties. We find upper
bounds for the random dis-
tillation protocol for general N-party W type states and find
the condition for the upper
bounds to be achieved. What is surprising is that the upper
bounds correspond to en-
tanglement monotones that can be increased by Separable
Operators (SEP), which gives
the first set of analytical entanglement monotones that can be
increased by SEP.
Finally, we investigate the idea of a new class of multipartite
entangled states, the
Absolutely Maximal Entangled (AME) states, which is
characterized by the fact that
any bipartition of the states would give a maximal entangled
state between the two sets.
The relationship between AME states and Quantum secret sharing
(QSS) protocols is
exhibited and the application of AME states in novel quantum
communication protocols
is also explored.
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Acknowledgements
Firstly, I want to thank my supervisor, Prof Hoi-Kwong Lo.
During these five years, hisendless help and inspired discussion
guided me to explore the fantastic world of quantuminformation
theory, which was a really exciting and enjoyable journey because
of him.He taught me not only in science but also in other aspects
of life. He showed my howto do presentation, how to improve my
English, and more importantly, how to treat andwork with other
people. All the above and his kind help on my nonacademic life will
bevaluable and remembered for a lifelong time.
Secondly, I really appreciate the advices and suggestions from
my committee members,Daniel James and Aephraim Steinberg.
It has been my great pleasure to work with a group of pleasant
and brilliant col-leagues. I want to show my acknowledgement to
Eric Chitambar, Wolfram Helwig, BingQi, Christian Weedbrook,
Xiongfeng Ma, Benjamin Fortescue, Yi Zhao, Yuemeng Chi,Viacheslav
Burenkov, Feihu Xu, Kero Lau, Zhiyuan Tang, Felix Liao, and He Xu.
Specialthanks to Eric Chitambar for his brilliant discussions and
endless passion on the subject.And to Bing Qi for his support on
both of my academic and nonacademic life.
I have benefited a great deal from the discussion with many
excellent scientists. Specif-ically, I wish to thank Lin Chen,
Daniel Gottesman, Fred Fung, Debbie Leung, JonathanOppenheim, and
David Gosset.
I would like to thank Viacheslav Burenkov for his suggestions
and proof reading.Responsibility for any remaining mistakes rests
entirely with the author.
Also, I wish to thank Krystyna Biel and Diane Silva for their
great job in adminis-trative help.
The help from the Center of International Experience, family
care office and familyhousing of the University of Toronto is also
acknowledged. With their help, I had areally harmonious life with
my family while studying in the University of Toronto as
aninternational student.
Finally, the love and support from my family is greatly
appreciated. This thesis isdedicated to my parents, my wife Bilian,
and my lovely son Stephen.
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Contents
1 Introduction 11.1 Our results . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 2
1.1.1 List of papers and presentations . . . . . . . . . . . . .
. . . . . . 4
2 Background Information 62.1 Entanglement . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Entanglement in quantum physics . . . . . . . . . . . . .
. . . . . 62.1.2 Entanglement in Hilbert space . . . . . . . . . .
. . . . . . . . . . 72.1.3 Entanglement as a resource . . . . . . .
. . . . . . . . . . . . . . 8
2.2 Quantum operations and entanglement measures . . . . . . . .
. . . . . . 102.2.1 Quantum Operators . . . . . . . . . . . . . . .
. . . . . . . . . . 102.2.2 Local Operators and Classical
Communications . . . . . . . . . . 112.2.3 Separable operators . .
. . . . . . . . . . . . . . . . . . . . . . . . 132.2.4
Entanglement measures for pure bipartite states . . . . . . . . . .
142.2.5 Entanglement measures for mixed states . . . . . . . . . .
. . . . 17
2.3 Multipartite entangled pure states . . . . . . . . . . . . .
. . . . . . . . . 192.3.1 Tripartite entangled states . . . . . . .
. . . . . . . . . . . . . . . 192.3.2 W type entangled states . . .
. . . . . . . . . . . . . . . . . . . . 21
3 LOCC transformation bounds between multipartite pure states
233.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 243.2 Upper Bound for the Conversion from GHZ
state to a GHZ class state . 253.3 Failure Branch . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Conservation of interference term . . . . . . . . . . . .
. . . . . . 313.3.2 Conservation of normalization . . . . . . . . .
. . . . . . . . . . . 32
3.4 Upper Bound for a general case . . . . . . . . . . . . . . .
. . . . . . . . 353.4.1 interference term and the maximal value of
the 3-tangle of a GHZ-
class state . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 35
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3.4.2 "stop and reconstruct" procedure . . . . . . . . . . . . .
. . . . . 363.4.3 Example: |GHZ〉 → |φ〉 = γ(|000〉+ |aaa〉) . . . . .
. . . . . . . . 383.4.4 general case . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 45
3.5 Lower Bound for the Transformation . . . . . . . . . . . . .
. . . . . . . 483.6 Summary and Concluding Remarks . . . . . . . .
. . . . . . . . . . . . . 56
4 Optimal entanglement transformations among N-qubit W-type
states 574.1 Introduction . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 574.2 Upper bounds . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 594.3 Lower bounds . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4
General Features of symmetric transformations . . . . . . . . . . .
. . . . 664.5 Conclusion . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 68
5 Random distillation for W type states 695.1 Introduction . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2
Previous results and notation . . . . . . . . . . . . . . . . . . .
. . . . . 74
5.2.1 The generalized Fortescue-Lo protocol . . . . . . . . . .
. . . . . 745.2.2 Additional notation and the Kintas-Turgut
monotones . . . . . . 75
5.3 The least party out protocol . . . . . . . . . . . . . . . .
. . . . . . . . . 765.3.1 Phase I: Remove x0 component . . . . . .
. . . . . . . . . . . . . 765.3.2 Phase II: Equal or vanish (e/v)
subroutine . . . . . . . . . . . . . 775.3.3 Phase III: Obtaining
EPR pairs . . . . . . . . . . . . . . . . . . . 77
5.4 Main results: The LPO protocol on multipartite W type states
. . . . . . 815.4.1 Summary of results . . . . . . . . . . . . . .
. . . . . . . . . . . . 815.4.2 Three qubits . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 815.4.3 Four qubits . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 835.4.4 n qubits
and the entanglement monotones . . . . . . . . . . . . . 895.4.5
Interpretation of monotones . . . . . . . . . . . . . . . . . . . .
. 92
5.5 SEP VS LOCC . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 935.5.1 Random distillation by Separable
transformations . . . . . . . . . 935.5.2 Comparison between SEP
and LOCC . . . . . . . . . . . . . . . . 96
5.6 Applicaiton to the transformation |φ〉1,··· ,N → |WN〉 . . . .
. . . . . . . . 985.7 Conclusion . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 99
5.7.1 Open questions and concluding remarks . . . . . . . . . .
. . . . 99
6 Absolutely maximal entangled state and quantum secret sharing
1026.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 102
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6.2 Definition of AME states . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1046.3 Parallel Teleportation . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 1056.4 Quantum Secret
Sharing. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1076.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 110
7 Conclusion 1127.1 Future work . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 1137.2 Concluding words . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Appendix 1158.1 Appendix: Proof of Theorem 5 . . . . . . . . .
. . . . . . . . . . . . . . 1158.2 Appendix: proof of Theorem 14 .
. . . . . . . . . . . . . . . . . . . . . . 1178.3 Dual solution to
|WN〉 distillation by SEP . . . . . . . . . . . . . . . . . 119
Bibliography 122
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List of Figures
2.1 Structure of states that can be obtained from W3 state by
SLOCC. Thefirst level is the true W3 type state which is also the
genuine W classstate. The second level are bipartite entangled
states, such as (AB)-C(|ψ〉AB |φ〉C), (AC)-B (|ψ〉AC |φ〉B) and (BC)-A
(|ψ〉BC |φ〉A), and the thirdlevel is the product state |φ1〉A |φ2〉B
|φ3〉C . . . . . . . . . . . . . . . . . . 22
3.1 mapping type 1. c©2010 American Physical Society . . . . . .
. . . . . . 27
3.2 mapping type 2. c©2010 American Physical Society . . . . . .
. . . . . . 27
3.3 The value of pU as a function of a. In this figure, a = (
yy−1)
13 . So when a
goes from 0 to 1, y goes from 0 to ∞. Note that as y goes to
infinity, agoes to 1. We express the value as a function of a
because it will be easierfor us to combine different graphs into
one graph later. c©2010 AmericanPhysical Society. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 34
3.4 "stop and reconstruct" for a two-outcome measurement. c©2010
AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 36
3.5 The original protocol written in the many two-outcome
measurementsform. c©2010 American Physical Society. . . . . . . . .
. . . . . . . . . . 37
3.6 "stop and reconstruct" for general protocol, I stands for
the interferenceterm. c©2010 American Physical Society. . . . . . .
. . . . . . . . . . . . 38
3.7 The new protocol, which can reconstruct the original one.
c©2010 AmericanPhysical Society. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 39
3.8 the relation between p̄s and p̄τABC . c©2010 American
Physical Society. . . 43
3.9 The upper bound for the transformation . . . . . . . . . . .
. . . . . . . 44
3.10 upper bound of transformation probability from |φ〉 to |ψ〉.
. . . . . . . . 47
3.11 Four-step method. c©2010 American Physical Society. . . . .
. . . . . . . 49
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4.1 An LOCC transformation tree from |x〉 to |y〉. For example,
the branchtraversing edges e(1,1) to e(n,1) is a success branch,
while the branch frome(1,1) to e(n,2) is a failure branch. Edge
e(n−1,1) is an intermediate edge.c©2010 American Physical Society.
. . . . . . . . . . . . . . . . . . . . . 60
4.2 The difference in maximum transformation probabilities when
only oneparty measures [pmax(s)] versus an identical filter by all
parties [qmax(s)].c©2010 American Physical Society. . . . . . . . .
. . . . . . . . . . . . . 66
5.1 A specified-pair versus random-pair distillation. For random
distillations,it is convenient to combine all the desired outcomes
into one configurationgraph G = (V,E) whose edge set encodes the
target pairs. Here, the targetpairs are AB and AC. The ” ≡ ”
indicates equivalent representations.c©2011 American Physical
Society. . . . . . . . . . . . . . . . . . . . . . 71
5.2 An N = 8 example of the "complete-type" distillations
considered byFortescue and Lo in [39]. Such a transformation is a
success if any two par-ties become EPR entangled, and this can be
achieved with a probabilityarbitrarily close to 1. Previous
research has not considered more generaltypes of configuration
graphs than this. c©2011 American Physical Society. 72
5.3 In Sec 5.4 we show that the optimal LOCC probability of
achieving thistransformation is 2/3, thus resolving an open problem
in Refs. [38]. Theinitial state is |W4〉 = 1/2(|1000〉 + |0100〉 +
|0010〉 + |0001〉). c©2011American Physical Society. . . . . . . . .
. . . . . . . . . . . . . . . . . . 72
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5.4 Equal or vanish subroutine (Phase II) for the normalized
state 11+3α
(α, α, α, 1)
and the configuration graph with edges {AB,AC,AD,BC}. 1.
David’scomponent is largest and Alice is a connected party to him
with a lessercomponent value. She performs an e/v measurement. 2.
For the outcome"vanish" (right branch) she is separated from the
system, and since Davidis not connected to either Bob or Charlie,
he immediately removes himselffrom the system leaving |ψ(BC)〉 with
some probability 1. For the outcome"equal" (left branch) the
components of all other parties receive a factorof α, and Alice’s
component is now maximum equaling David’s. Bob is aconnected party
to Alice with a lesser component value and he performsan e/v
measurement. 3. Again, either Bob vanishes (right branch) or
allother components except his receive a factor of α. In both
cases, Charlieis then a connected party to Bob with a lesser
component value and heperforms an e/v measurement. 4. The final
outcome states along thesebranches are |W4〉, |W (ABD)3 〉, |W (ACD)3
〉, and |ψAD〉. c©2011 AmericanPhysical Society. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 79
5.5 Phase III receives an input state W (S)|S| and a
configuration graph G. Partyk performs an e/v measurement. One
outcome is a standard W state withparty k removed, and the other is
the state 1|S|pα (α, · · · , α, 1, α, · · · , α).Phase II is
applied on this state outputting either W states or a
product(failure) state. Phase III will next be initiated on each of
the W states,and for any W state W (S
′)|S′| with |S ′| < |S|, the transformation success
probability from this point onward is given by PIII(W(S′)|S′| ,
G\S̄ ′); this value
is already known by recursion. However, for the state W (S)|S| ,
performingPhase III again will generate an indefinite loop, but one
whose overallsuccess probability converges to f(α)
1−α|S|−1 [see Eqs. 5.13 and 5.14]. c©2011American Physical
Society. . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.6 (Left) Configuration G∧. (Right) Configuration G∆. An upper
bound onthe success probability is given by Eqs. 5.17 and 5.19,
respectively, whichis effectively tight when x0 = 0. For |W3〉,
these probabilties are 2/3 and1, respectively. c©2011 American
Physical Society. . . . . . . . . . . . . . 82
5.7 Let GI , G′I , and G”I be the first, second, and third of
the above configura-tions, respectively. An upper bound on the
success probability is given byEq. 5.20 which is effectively tight
when x0 = 0. For |W4〉, this probabilityis 1/4 for each
configuration. c©2011 American Physical Society. . . . . . 83
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5.8 Let GII be the above configuration. An upper bound on the
success prob-ability is given by Eq. 5.21 which is effectively
tight when x0 = 0. For|W4〉, this probability is 3/4. c©2011
American Physical Society. . . . . . 84
5.9 Let GIII be any of the above configurations. In each of
these, (A,C) and(B,D) are unconnected pairs. An upper bound on the
success probabilityis given by Eq. 5.23 which is effectively tight
when x0 = 0. For |W4〉,this probability is 2/3 for each of these
configurations. c©2011 AmericanPhysical Society. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 84
5.10 Let GIV be the above configuration. We say two parties are
edge comple-mentary if their nodes have a different number of
connected edges. Forexample, A is edge complementary to both C and
D. An upper bound onthe success probability is given by Eq 5.30
which is effectively tight whenx0 = 0. For |W4〉, this probability
is 5/6. c©2011 American Physical Society. 85
5.11 Let GV be the above configuration. An upper bound on the
success proba-bility is given by Eq 5.32 which is effectively tight
when x0 = 0. For |W4〉,this probability is 1. c©2011 American
Physical Society. . . . . . . . . . . 87
5.12 Let GV I be the above configuration. For |W4〉, the LPO
protocol gives asuccess probability of 1
6(3 +√
3).We conjecture this to be optimal. c©2011American Physical
Society. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.13 Distillation configurations for η vs κ. Top: A
"combing-type" distillation:when x0 = 0, 2η(x) is the optimal
probability for a random distillation inwhich party n1 shares
one-half of each EPR pair. Bottom: A "complete-type" distillation:
when x0 = 0, κ(x) gives the optimal probability for arandom
distillation in which the target pairs are any two of the
parties.c©2011 American Physical Society. . . . . . . . . . . . . .
. . . . . . . . 92
5.14 LOCC vs SEP for the maximum probability of obtaining an EPR
pair be-tween any two parties as a function of s when the initial
state is
√s |100〉+√
1−s2
(|010〉+ |001〉). The LOCC probability is 2(1− s)− (1− s)2/4s.
Agap of 12.5% exists between SEP and LOCC. c©2012 American
PhysicalSociety. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 96
5.15 LOCC vs SEP for the maximum probability of party 1 becoming
EPRentangled as a function of N when the initial state is
√12|10 · · · 0〉 +√
12(1−N)(|010 · · · 0〉 + · · · + |0 · · · 01〉). The LOCC
probability is 1 − (1 −
1N−1)
N−1. A gap of 37% exists between SEP and LOCC. c©2012
AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 97
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5.16 The relative difference between the optimal separable
operation and theLPO protocol. The configuration graph consists of
N disjoint pairs. Sep-arable operations perform as PSEP =
√1N
whereas the LPO protocolobtains the rate of PLPO = 22N−1 . We
conjecture that the LPO protocolis LOCC optimal for this
configuration graph, as it is known to be whenN = 4. c©2012
American Physical Society. . . . . . . . . . . . . . . . . .
101
6.1 Parallel Teleportation scenarios of Theorem 20. Scenario (i)
is on the left,and (ii) on the right. Parties in A perform joint
quantum operations,parties in B only local quantum operations.
c©2012 American PhysicalSociety. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 105
6.2 (Color online) After D (blue) performs her teleportation
operation, anyset of m parties (red), A, A′, A′′ etc., can recover
the teleported state. Anyset of parties with m − 1 or less parties
(any set consisting only of greenparties) cannot gain any
information about the teleported state. c©2012American Physical
Society. . . . . . . . . . . . . . . . . . . . . . . . . . .
108
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Chapter 1
Introduction
Quantum information, as a field that employs entanglement, which
is the most impres-sive feature of quantum physics and a resource
to accomplish novel computation andcommunication protocols, has
been experiencing rapid development since the 1980’-s.The famous
EPR pair, which was expressed as a Bell state, |φ〉AB = 1√2(|00〉 +
|11〉)AB,has been shown to be the resource for various quantum
information protocols that areimpossible under classical physics.
EPR state shared between parties A and B cannot bewritten as a
direct product of two states of parties A and B respectively, which
is nowthe most important feature of the so called entangled
state.
Other than EPR pairs, there are infinitely many entangled
quantum states that canbe shared between several parties. It would
be rewarding to investigate the entangle-ment property of these
states and also their application in quantum information
proto-cols. Also, under real experimental conditions, one cannot
guarantee the generation ofa perfect EPR state, and it is necessary
to analyze how one can transform a quantumentangled state into an
EPR state that can be directly used in quantum information
pro-tocols. Because of that, the quantification of entanglement for
an entangled state and thetransformation property between two
entangled states are two main research directionsin entanglement
theory.
Based on the fact that protocols employing EPR pairs are
generally for nonlocalscenarios, the transformation between
entangled states is restricted under the scenario oflocal operators
and classical communications (LOCC). Under this scenario,
theentanglement theory for a bipartite pure state, which is an
entangled state shared betweentwo parties, is completely
constructed. The EPR state is shown to be the most powerfulresource
among pure states shared by two qubits since it could be
transformed into anyother pure state shared by two qubits with
probability 1. The optimal transformationprobability between any
two bipartite states is also derived.
1
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Chapter 1. Introduction 2
However, the corresponding results for higher dimensions and
more parties, whichis called multipartite entangled state, are
still lacking. The mathematical structureof multipartite entangled
states turns out to be much more complicated than the caseof
bipartite states. For example, tripartite states, which are the
quantum states sharedbetween three parties, can be classified into
two classes while states from different classescannot be
transformed into each other with nonzero probability.
Another question is the application of multipartite entangled
states in novel quantumprotocols. Here the interest is on the
protocols that could reveal the advantage of mul-tipartite
entanglement over bipartite entanglement. That is to say, for some
protocols,employing the multipartite entangled states directly
would achieve a better result thanconverting the multipartite
entangled state into EPR pairs and then using the EPR pairsfor the
protocol.
1.1 Our results
During my Ph. D study, I mainly worked on the LOCC
transformation probability be-tween pure multipartite entangled
states. In the following I will provide a quick overviewof my work,
which is also an outline of this thesis.
In chapter 2, I will provide the background information on
entanglement theory thatis related to my research.
In chapter 3, upper and lower bounds on the transformation
probability betweenmultipartite entangled states will be derived.
The result is mainly related to the trans-formation between
tripartite states while some of our results can be generalized into
moreparties. There is still a gap between the upper bound and lower
bound we found, whichfuture work will investigate. The result of
this chapter was published in [33]. As thefirst author, I proposed
the four-step-method and discovered the upper bound and lowerbound
for the transformation probability.
In chapter 4, another type of multipartite entangled state, the
W-type state, is ana-lyzed. Based on the explicit form of a general
W-type state given in [52], we derive thelower bound on
transformation probability between any two W-type states under
LOCC.Also, we find the condition under which the lower bound is
actually optimal. The resultof this chapter was published in [31].
As the first author, I proposed the LOCC trans-formation protocol
and discovered the necessary and sufficient condition for the
boundto be achieved.
In chapter 5, the question regarding the conversion from
multipartite entangled stateinto bipartite state is explored.
Specifically, we consider the following question: given a
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Chapter 1. Introduction 3
multipartite state, how can it be converted into EPR pairs
shared between any two par-ties? This protocol, called random
distillation, was first proposed in [39]. Here we findupper bound
for this type of transformation by discovering a new type of
entanglementmonotones. One surprising result is that the
entanglement monotones discovered can beincreased by separable
operators (to be defined in chapter 2), which gives the first setof
analytic entanglement monotones that can be increased by separable
operators. Theresult of this chapter was published in [32][23][19].
As the first author of [32] and thesecond author of [23] and [19],
I proposed the least party out protocol and discovered
theentanglement monotones for three and four qubit systems.
In chapter 6, a new type of multipartite entangled states, which
is called the ab-solutely maximal entangled states (AME states) (to
be defined in chapter 6), isstudied. This type of state is
characterized by the property that any bipartition of theparties
could lead to a maximal entangled state between the two sets of
parties. Theclose relationship between AME states and quantum
secret sharing protocols will be ex-hibited. Also, the possible
application of AME states is proposed. The result of thischapter
was published in [48]. As the second author, I collaborated on the
analysis of themultipartite teleportation protocol and on the proof
of the one-to-one correspondencerelationship between AME state and
quantum secret sharing protocol (to be defined inchapter 6).
In chapter 7, a summary of my Ph.D research is provided, and the
future work thatcan be developed from this thesis is also
discussed.
In chapter 8, the appendix, we provide the detailed proofs for
some important theo-rems in this thesis.
The significance of our work can be summarized as the following
three points. Firstly,the transformation probability between
multipartite pure states is a complex problem onwhich little work
has been done before ours. Our work sheds some light on the
investi-gation of this problem by finding various upper and lower
bounds. Secondly, instead oftrying to classify all types of
multipartite entangled states, we focus on some specific typesof
multipartite entangled states and analyze their properties and
potential applications,which is shown to be very helpful. Finally,
our work on random distillation demonstratesthat the mathematical
structure of local operators and classical communication (LOCC,to
be defined in chapter 2) is more complex than expected.
-
Chapter 1. Introduction 4
1.1.1 List of papers and presentations
Papers
1. Wei Cui, Wolfram Helwig, Hoi-Kwong Lo, Bounds on the
probability of transfor-mation between multipartite pure states,
Physics Review A, 81, 012111, 2010
2. Wei Cui, Eric Chitambar, Hoi-Kwong Lo, Optimal Entanglement
TransformationsAmong N-qubit W-Class States, Physics Review A, 82,
062314, 2010
3. Wei Cui, Eric Chitambar, Hoi-Kwong Lo, Randomly distilling
W-class states intogeneral configurations of two-party
entanglement, Phys. Rev. A 84, 052301, 2011
4. Eric Chitambar, Wei Cui, Hoi-Kwong Lo, Increasing
Entanglement by SeparableOperations and New Monotones for W-type
Entanglement, Phys. Rev. Lett. 108,240504, 2012. (This work was
selected as a plenary talk at QIP, one of my colleagues(Eric
Chitambar) did the presentation. A plenary talk is the most
prestigious talkat the QIP conference, which is the most
prestigious theory conference in the field.)
5. Eric Chitambar, Wei Cui, Hoi-Kwong Lo, Entanglement monotones
for W-typestates, Phys. Rev. A 85, 062316, 2012
6. Wolfram Helwig, Wei Cui, Arnau Riera, Jose I. Latorre,
Hoi-Kwong Lo, AbsoluteMaximal Entanglement and Quantum Secret
Sharing, Phys. Rev. A 86, 052335,2012
Presentations
1. August 2011, AQIS 11, Pusan National University, Pusan,
Korea, "Randomly dis-tilling W-class states into general
configurations of two-party entanglement" by W.Cui, E. Chitambar,
H. -K. Lo
2. March 2011, APS March Meeting, Dallas Convention Center,
Dallas, Texas, UnitedStates of American, "Optimal Entanglement
Transformations Among N-qubit W-Class States" by W. Cui, E.
Chitambar, H. -K. Lo
3. July 2010, University of Calgary, presentation, "Bounds on
the probability of trans-formation between Multipartite quantum
states", by W. Cui, W. Helwig, H.-K. Lo
4. June 2010, CAP 2010, University of Toronto, presentation,
"Bounds on the prob-ability of transformation between Multipartite
quantum states" by W. Cui, W.Helwig, H.-K. Lo
-
Chapter 1. Introduction 5
5. January 2010, QIP 2010, ETH, Switzerland, Rump session,
"Bounds on the prob-ability of transformation between Multipartite
quantum states" by W. Cui, W.Helwig, H.-K. Lo
-
Chapter 2
Background Information
In this chapter we provide a brief summary of some important
results from the field ofquantum entanglement related to our work.
People who are familiar with the theory ofentanglement can skip
this chapter on a first reading.
2.1 Entanglement
In this section we provide a description of entanglement, in
both the physical and math-ematical aspects.
2.1.1 Entanglement in quantum physics
The distinction between quantum physics and classical physics is
the description of aphysical system as a state with uncertain
parameters. Specifically, from the uncertaintyprinciple, it is
impossible to measure the exact position and momentum of a
particlesimultaneously. However, in 1935, Einstein, Podolsky, and
Rosen proposed a quantumstate shared between two parties A and B in
the following form, which is called an EPRstate [35].
Φ(x1, x2) =
∫ +∞−∞
e(2πi/h)(x1−x2+x0)pdp. (2.1)
For this state, the two particles have a fixed midpoint at x0
while their momentashould be opposite to each other. In this sense,
the center of mass of the system ABwould be at the original
location while the total momentum for them remains zero. So,by
measuring the position of A, one can determine the position of B.
Or we can say thecollapse of the wave function of A leads to the
collapse of the wave function of B, which
6
-
Chapter 2. Background Information 7
appears to indicate an interaction between A and B that is
faster than light.As an explanation of this feature of an EPR
state, it was claimed that quantum
physics is not a complete theory and that there is a set of
hidden variables that actuallydetermine the property of a physical
system. The local hidden variable conjecture basedon this
assumption was shown to be incorrect by the experimental violation
of Bell’sinequality [5]. To this day, the interpretation of quantum
mechanics remainsl an openproblem. However, the EPR state turns out
to be a stronger resource than a classicalbit from an information
theory perspective. In the following subsection we will
introducethe mathematical structure of general entangled
states.
2.1.2 Entanglement in Hilbert space
The mathematical framework for the analysis of quantum states is
the Hilbert space.Given a quantum system that could evolve as a
superposition of n possible eigenstates ofthe Hamiltonian, one can
express the state in an n-dimensional Hilbert space. A quantumstate
can be expressed as a density matrix in the corresponding Hilbert
space. For apure state, the density matrix can be expressed as |φ〉
〈φ| while for a mixed state, whichis considered an ensemble of
several quantum states, the density matrix is
∑i pi |φi〉 〈φi|.
Here we have Trρ =∑
i pi = 1.
Example 1. Let us take spin as an example. Suppose there is an
electron whose spincould be up or down. We can treat the Hilbert
space of spin as {|↑〉 , |↓〉}. A pure state inthis space could be
written as |φ〉 = a |↑〉+b |↓〉 (density matrix (a |↑〉+b |↓〉)(a∗
〈↑|+b∗ 〈↓|)where |a|2 + |b|2 = 1. And a mixed state could be
written as ρ = |a|2 |↑〉 〈↑| + |b|2 |↓〉 〈↓|where |a|2 + |b|2 = 1
also.
Mathematically, one can also define |0〉 = |↑〉 and |1〉 = |↓〉.
Thus the pure state andmixed state could be expressed as |φ〉 = a
|0〉+ b |1〉 (density matrix (a |0〉+ b |1〉)(a∗ 〈0|+b∗ 〈1|)) and ρ =
|a|2 |0〉 〈0|+ |b|2 |1〉 〈1|.
Remark 1. One question here is how to distinguish pure states
and mixed states. Math-ematically, given a hermitian matrix ρ, if
the eigenvalues {λi} are all nonnegative and∑
i λi = 1, then this matrix can represent the density matrix of a
quantum state. Further-more, if one of the eigenvalues is 1 while
all the other eigenvalues are 0, the correspondingquantum state is
a pure state. Otherwise, it is a mixed state.
For two particles A and B, the whole system would lie in a
Hilbert space given by thetensor product of the two individual
Hilbert spaces.
H = HA ⊗HB (2.2)
-
Chapter 2. Background Information 8
The density matrix of their joint state is denoted by ρAB. If it
can be written in the formof
ρAB =∑i
piρAi ⊗ ρBi , (2.3)
where pi ≥ 0 and∑
i pi = 1, it is called a separable state. Otherwise, it is
calledentangled. The most important entangled state to consider for
a bipartite system, sayparty A and B, is the EPR state
|Φ〉AB =1√2
(|00〉+ |11〉)AB. (2.4)
The above definition easily be generalized into higher
dimensions and more parties.In general, for an n-party state
ρ1···n, if it can be written as
ρ1···n =∑i
piρ1i ⊗ · · · ρni , (2.5)
where pi ≥ 0 and∑
i pi = 1, it is called a separable state. Otherwise, it is
entangled.
One thing to note is that, for a multipartite system, one can
divide the parties intodifferent groups and talk about the
entanglement property between these groups. Forexample, the
state
|Φ〉ABC =1√2
(|00〉+ |11〉)AB |0〉C , (2.6)
is an entangled state between A, B, and C. However, it is a
separable state between thesystems (AB) and C.
2.1.3 Entanglement as a resource
Now let us explain the application of a qubit from a quantum
information theory per-spective. We denote a classical bit as a
cbit, which stands for either 0 or 1. Also, wedefine a two-level
quantum mechanical system (a |0〉+b |1〉) as 1 qubit. Another
resourceto consider is one in which there is an entangled state
shared between the parties initially.Specifically, if Alice and Bob
share an EPR pair, we denote it as 1 ebit.
Quantum superdense coding
Suppose that Alice wants to communicate two bits of information
{00, 01, 10, 11} to Bob.One choice is to send 2 cbits via a
classical channel. However, if they share an EPR
-
Chapter 2. Background Information 9
pair (1 ebit), they have another choice that allows them to send
1 qubit instead. Thisprotocol is called superdense coding [11].
Let us explain how the protocol works and its implication. In
the beginning, Aliceand Bob share an EPR state, so that we have
|φ〉AB =1√2
(|00〉+ |11〉)AB. (2.7)
To send the information, Alice could use an encoding scheme by
implementing a localunitary operator on her subsystem, which has
the following rules:
00 : IA |φ〉AB =1√2
(|00〉+ |11〉)AB
01 : XA |φ〉AB =1√2
(|10〉+ |01〉)AB
10 : ZA |φ〉AB =1√2
(|00〉 − |11〉)AB
11 : XAZA |φ〉AB =1√2
(|10〉 − |01〉)AB
(2.8)
where X, Z are Pauli matrices defined as
X =
(0 1
1 0
), Y =
(0 −ii 0
), Z =
(0 1
1 0
). (2.9)
After that, Alice can send her qubit to Bob. Notice that the
four resulting states areorthogonal to each other so that if Bob
has the full copy of the state, he can identifywhich state it is
via a Bell measurement. Thus Bob could recover the 2 classical
bitsAlice wants to send to him.
In the above protocol, Alice and Bob have 1 ebit in the
beginning, and they transmit1 qubit of information. In all, they
communicate 2 cbits of information. We thus have
1 qubit+ 1 ebit ≥ 2 cbits. (2.10)
Quantum teleportation
In quantum teleportation, Alice wants to teleport a qubit, or an
unknown quantum state(|φ〉 = a |0〉 + b |1〉) to Bob [7]. Supposing
they share an EPR state in the beginning,Alice could attach the
unknown state to her subsystem so we have
-
Chapter 2. Background Information 10
(a |0〉+ b |1〉)A′1√2
(|00〉+ |11〉)AB
=1
2(|00〉+ |11〉)A′A(a |0〉+ b |1〉)B +
1
2(|00〉 − |11〉)A′A(a |0〉 − b |1〉)B
+1
2(|01〉+ |10〉)A′A(a |1〉+ b |0〉)B +
1
2(|01〉 − |10〉)A′A(a |1〉 − b |0〉)B
(2.11)
Alice could make a Bell measurement on her system and
communicate the measure-ment result (2 cbits) to Bob. With that
information, Bob can recover the unknownquantum state by applying
the corresponding Pauli matrices on his quantum system.
During the above process, Alice and Bob possess 1 ebit in the
beginning and they use2 cbits to transmit 1 qubit of information.
We thus have
1 ebit+ 2 cbits ≥ 1 qubit (2.12)
2.2 Quantum operations and entanglement measures
In this section we will describe the general quantum operators
and especially Local Op-erators and Classical Communications
(LOCC). After that we will show that LOCCprovides the framework to
quantify how much entanglement a system contains.
2.2.1 Quantum Operators
Given a quantum state, how can we transform it into another
state? The transformationa state will undergo during a physical
process is described by quantum operators.Mathematically, a quantum
operator can be described by a linear and completely positivemap,
ψ, from the set of density operators onto itself.
Remark 2. A linear map ψ is positive if ψ(ρ) is positive for any
positive ρ on the Hilbertspace. And it is completely positive if ψ⊗
1p(ρ⊗ 1p) is positive for any positive integer p.
Mathematically, a quantum operator can always be expressed in
the form [26]
φ(ρ) =∑j
VjρV+j (2.13)
where ∑j
V +j Vj = 1 (2.14)
-
Chapter 2. Background Information 11
Each term VjρV +j can also be treated as a branch, with the
resulting stateVjρV
+j
tr(VjρV+j )
and
the probability tr(VjρV +j ).
Example 2. Measurement: For example, suppose we have a quantum
state |φ〉 = α |0〉+√1− |α|2 |1〉 in the {|0〉 , |1〉} Hilbert space, a
measurement in {|0〉 , |1〉} can be performed
on the state. With probability |α|2 the state will collapse onto
|0〉 while with probability1− |α|2 the state will collapse onto |1〉.
This operator can be described as
M(ρ) = |0〉 〈0| ρ |0〉 〈0|+ |1〉 〈1| ρ |1〉 〈1| (2.15)
The resulting state is given by
ρ = 〈0|ρ|0〉 |0〉 〈0|+ 〈1|ρ|1〉 |1〉 〈1| (2.16)
and the possible resulting states are |0〉 (density matrix |0〉
〈0|) and |1〉 (density matrix|1〉 〈1|), with
p(|0〉) = 〈0|ρ|0〉 = |α|2 and p(|1〉) = 〈1|ρ|1〉 = 1− |α|2
(2.17)
Since entanglement is the resource for quantum information, an
important questionwould be how entanglement evolves under quantum
operations, especially measurements.Here we want to emphasize that
measurement could induce the collapse of a quantumwave function,
which means the resulting state might have significantly different
entan-glement properties from the original state. For example,
given an EPR state shared byAlice and Bob, |φ〉AB = 1√2(|00〉+ |11〉),
if Alice or Bob measures in {|0〉 , |1〉} basis, thestate would
collapse into 1
2|00〉 〈00|+ 1
2|11〉 〈11|, which is a separable state.
2.2.2 Local Operators and Classical Communications
Since entanglement is used for the transmission of information
between parties far apartfrom each other, we restrict the quantum
operations to be locally implemented. Also,we only allow classical
information to be transmitted between the distant parties.
Thisscheme is called LOCC (Local Operators and Classical
Communications), a stan-dard scheme in which we could quantify the
amount of quantum resource we have.
In general, given three parties A, B, and C, one could describe
an LOCC protocol inthe following way: Party A makes a local
operator and passes the information regardingthis operator and the
measurement result (classical information) to Bob and Charlie.Based
on the information from Alice, Bob or Charlie could implement
another local
-
Chapter 2. Background Information 12
operator, and so on. Finally, their joint state would end up
being some density matrix,which is the resulting state of this LOCC
protocol.
Suppose that we use LOCC to transform a state from |φ〉 into |ψ〉.
If there is apositive probability less than 1 with which the
transformation can be successful, then thecorresponding protocol is
called SLOCC (Stochastic Local Operators and
ClassicalCommunications) protocol.
Two-outcome weak measurement decomposition of LOCC
If one party performs a local measurement that has several
possible outcomes, the statecould collapse into a new state far
from the original state. This phenomenon is an obstaclefor the
investigation of the behavior of some quantitative parameters under
LOCC sinceit is not continuous and hard to formulate
mathematically. To overcome this problem,one could first decompose
a local measurement into several two-outcome measurements.This is
shown in [3].
Another important result is that any two-outcome measurement
could be decomposedinto many steps of two-outcome weak
measurements, while during each step the resultingquantum states
are almost identical with the original state. The idea is similar
to arandom walk, where one needs to stop when the resulting state
becomes one of the tworesulting states of the original two-outcome
measurement [63].
By using the above two techniques, one can discuss a general
LOCC protocol underthe restriction of two-outcome weak measurement,
during which the state can be seen aschanging continuously. This
method will be explored further in chapters 3, 4, and 5.
Entanglement Monotone
Given a quantum state, how much entanglement does it contain? To
answer this question,one needs a quantitative measure for the
amount of entanglement a state possesses.
In general, entanglement is used for nonlocal missions. That is
to say, many partiesshare a quantum system, while they can only
perform local operations on their subsys-tems. Entanglement, as a
resource, can only be consumed, rather than created, duringthis
process. In this sense, entanglement monotone, as a quantification
of how muchentanglement one quantum system has, is defined in the
following way:
Definition 1. For a quantum system ρ, any magnitude µ(ρ) that
does not increase onaverage under local transformations and
classical communications is called entanglementmonotone (EM)
[77].
-
Chapter 2. Background Information 13
An important application of entanglement monotone is that it can
be used to boundthe optimal transformation probability between two
quantum states ρ and ρ′ underLOCC. More precisely, the optimal
successful probability for the conversion from ρ to ρ′
under LOCC, denoted by P (ρ LOCC−−−→ ρ′), is given by
P (ρLOCC−−−→ ρ′) = minµ
µ(ρ)
µ(ρ′)(2.18)
where the minimization is to be performed over the set of all
EMs [77]. It is straight-forward to see that the optimal
transformation probability should be upper bounded byany µ(ρ)
µ(ρ′). At the same time, the transformation probability itself
is also an entanglement
monotone, and this upper bound is hence tight.In general, it is
not easy to find the optimal probability for an LOCC
transformation.
However, the known entanglement monotones could be used to find
an upper bound onthe transformation probability. Also, if any
transformation probability coincides with aknown bound obtained
from some entanglement monotone, we can say for certain thatit is
optimal.
2.2.3 Separable operators
One drawback of LOCC is that it is hard to be analyzed
mathematically, because itdoes not have an explicit analytical
definition. Also, its mathematical structure is verycomplex. For a
good summary of the mathematical structure of LOCC, we refer to
[22].
To overcome this problem, separable operators (SEP) were
introduced. Mathe-matically, SEP on an n-party state is defined
as
Ω(ρ) =k∑i=1
AiρA+i (2.19)
wherek∑i=1
A+i Ai = 1 (2.20)
andAj = A1j ⊗ A2j ⊗ · · · ⊗ Anj (2.21)
where Akj is a local operator implemented by party k, which
means AkjρA+kj
should bea positive matrix defined on the Hilbert space of one
party.
The motivation for the introduction of SEP for quantum
information is the factthat separable operators have an explicit
mathematical structure and can be analyzed
-
Chapter 2. Background Information 14
numerically by programs like semi-definite programming [75].
Also, since every LOCCprotocol can also be implemented by SEP, SEP
can be used to identify entanglementmonotones for a quantum system.
More concretely, if a physical quantity could not beincreased by
SEP, then it is impossible for it to be increased by any LOCC
protocol and itis an entanglement monotone. However, the converse
is not true: there are entanglementmonotones that can be increased
by SEP [24] [23].
However, given an SEP, it is unclear how to check whether it can
or cannot beimplemented by LOCC. What we can do is that, if an SEP
can accomplish a missionthat is impossible by LOCC, one can be sure
that this SEP cannot be implemented by anyLOCC protocol. For
example, separable operations that can increase some
entanglementmonotone [23], or distinguished states that could not
be perfectly distinguished by LOCC[9], cannot be implemented by
LOCC. A general method to check whether a protocolwith separable
operations can be implemented by LOCC within a given number of
roundswas presented in [30]. In chapter 5, we will show the gaps
between LOCC and SEP forsome quantum information protocols.
2.2.4 Entanglement measures for pure bipartite states
In this section we review the known results for the entanglement
properties of purebipartite states.
Schmidt decomposition and Schmidt number
Bipartite pure states have been analyzed thoroughly in quantum
information theory. Thispartly comes from the existence of Schmidt
decomposition for bipartite states. Given anybipartite pure state,
we can always write it as
|φ〉AB =k∑l=1
√al |il〉A |il〉B (2.22)
where∑k
l=1 al = 1 and {i1, · · · , in} is a set of orthogonal state
vectors. This form is calledSchmidt decomposition. For each
bipartite pure state, one can uniquely determinethe values of its
Schmidt coefficients (together with their degeneracies) [62]. Based
onthis decomposition, the LOCC transformation rules are well
developed.
Remark 3. Notice that uniqueness of Schmidt decomposition means
that the set of co-efficients are unique, the actual states may not
be unique if some of the coefficients aredegenerate.
-
Chapter 2. Background Information 15
Given a bipartite pure state |φ〉AB, to find its Schmidt
decomposition? One firstneeds to compute the reduced density matrix
for one party, say ρA. The eigenvalues andeigenvectors of ρA are
als and |il〉As in Eq 2.22 respectively.
Example 3. For example, given a pure state |φ〉AB = 1√2 |00〉AB
+12(|10〉 + |11〉)AB, let
us find its Schmidt decomposition.Firstly, by tracing out party
B, one can find that
ρA = 〈0B|ρAB|0B〉+ 〈1B|ρAB|1B〉 =1
2|0〉 〈0|+
√2
4(|0〉 〈1|+ |1〉 |0〉) + 1
2|1〉 〈1| . (2.23)
By diagonalizing it, we can find the eigenvalues and
corresponding eigenvectors of ρAare
a1 =1
2−√
2
4: |i1〉A =
1√2
(|0〉 − |1〉); a2 =1
2+
√2
4: |i2〉A =
1√2
(|0〉+ |1〉) (2.24)
Similarly, for ρB, we have the same eigenvalues and the
corresponding eigenvectorsare
|i1〉B =1√
4 + 2√
2(|0〉 − (1 +
√2) |1〉); |i2〉B =
1√4− 2
√2
(|0〉+ (√
2− 1) |1〉). (2.25)
Finally, the Schmidt decomposition is given by
|φ〉AB =√a1 |i1〉A |i1〉B +
√a2 |i2〉A |i2〉B (2.26)
Remark 4. Note that for a quantum system with more parties,
Schmidt decompositiondoes not always exist.
Also, given a bipartite pure state
|φ〉AB =k∑l=1
√al |il〉A |il〉B , (2.27)
the Schmidt number is defined as k, the number of non-zero terms
in the Schmidtdecomposition. A generalization of Schmidt number is
called Schmidt rank, whichis the minimum number of non-zero terms
needed to write a multipartite state as thesuperposition of product
states.
In the following subsections, we will review some entanglement
monotones for bipar-tite states and the transformation rules
between any two bipartite states.
-
Chapter 2. Background Information 16
Entropy of entanglement
Entropy of entanglement is one of the most important
entanglement measures of bipartitepure states since it is defined
from the information theoretic perspective. For a pure state|φ〉AB,
the entropy of entanglement �(|φ〉) is defined as the von Neumann
entropy of thereduced density matrix for either party, or we
have
�(|φ〉) = S(ρA) = S(ρB) (2.28)
where S(ρ) = −Trρ log2 ρ, ρA = TrB(|φ〉AB 〈φ|), ρB = TrA(|φ〉AB
〈φ|).Entropy of entanglement is closely related to entanglement
concentration and
entanglement dilution [6]. More concretely, given N copies of a
bipartite pure state |φ〉,asymptotically (in the limit of large N)
one can use LOCC transformation to concentratethem into N�(|φ〉)
copies of EPR pairs. Also, given N copies of EPR pairs,
asymptoticallythey can be distilled into N
�(|φ〉) copies of |φ〉.
Transformation probability between the states
Based on the Schmidt decomposition of bipartite pure states, the
optimal LOCC trans-formation probability between any two bipartite
pure states is discovered in [77]. Giventwo pure bipartite
states
|φ〉 = √a1 |i1i1〉+√a2 |i2i2〉+ · · ·
√an |inin〉 , ai ≥ ai+1 ≥ 0, (2.29)
and
|ψ〉 =√b1 |j1j1〉+
√b2 |j2j2〉+ · · ·
√bn |jnjn〉 , bi ≥ bi+1 ≥ 0, (2.30)
where {i0, i1, · · · , in} and {j0, j1, · · · , jn} are two sets
of orthogonal vectors, the optimaltransformation probability from
|φ〉 to |ψ〉 under LOCC is given by
P (φLOCC−−−→ ψ) = min
l∈[1,n]
∑ni=l ai∑ni=l bi
. (2.31)
Notice that if we have∑n
i=l ai ≥∑n
i=l bi for any given l, the transformation canbe done by LOCC
with probability 1. This is called the majorization relationshipand
was discovered in [61]. Based on this result, the EPR state serves
as the maximalentangled state for bipartite quantum system since it
can be transformed deterministicallyinto any other pure bipartite
state of two qubits under LOCC.
-
Chapter 2. Background Information 17
2.2.5 Entanglement measures for mixed states
The situation becomes more complex for mixed states. In general,
a mixed state couldbe written as
ρ =∑i
pi |φi〉 〈φi| (2.32)
where∑
i pi = 1.To quantify the entanglement of ρ, one natural choice
is to consider the corresponding
entanglement measures for the pure states in this ensemble, and
define the correspondingentanglement measure for mixed state as
∑i piE(|φi〉).
However, one needs to note that the decomposition of a mixed
state into an en-semble of pure states is not unique, which means
that we need to consider all possibledecompositions and find the
one that yields the minimum value. Or we have
E(ρ) = min{pi,|φi〉}
∑i
piE(|φi〉) (2.33)
Entanglement of formation and entanglement cost
If we choose entropy of entanglement as the corresponding
entanglement measure forpure state, we can define entanglement of
formation as [10]
Ef (ρ) = min{pi,|φi〉}
∑i
pi�(|φi〉) (2.34)
Note that in large N limit, one can prepare N copies of state
|φi〉 with N�(|φ〉) copiesof EPR pairs. In general, one can prepare
all states in this ensemble with EPR pairsand combine them together
to form ρ. Thus we achieve an operational interpretation
ofentanglement of formation: with NEf (ρ) copies of EPR pairs, one
can prepare N copiesof ρ using LOCC in the limit of large N.
This definition leads to another concept called entanglement
cost, which is the min-imum value of the average number of EPR
pairs needed to prepare one copy of a stateusing LOCC [46]. If n
EPR pairs are needed to produce m copies of a state ρ, thenEC(ρ
⊗m) = min nm. Here the minimum value is chosen from all possible
LOCC protocols.
Entanglement of formation is not always equal to entanglement
cost, but in the large mlimit, they are equal. Or we have
Ec(ρ) = limm→∞
1
mEf (ρ
⊗m). (2.35)
-
Chapter 2. Background Information 18
Distillable entanglement and Bound entanglement
Conversely, distillable entanglement is defined as the number of
EPR pairs one can distillfrom a given state [8]. In particular,
suppose one can distill n copies of EPR pairsfrom m copies of state
ρ, one has ED(ρ) = max limm→∞ nm . Here we need to considerall
possible LOCC protocols to maximize the number of EPR pairs
produced. Dueto the complexity of LOCC protocols, there is no
analytical expression for distillableentanglement. However, PPT
criterion can be used to check if distillable entanglementis zero
for a bipartite mixed state [65][50].
In general, entanglement cost is larger than distillable
entanglement. The extremecondition is that for some states, the
distillable entanglement is zero while the entan-glement cost is
positive, which means EPR pairs needs to be consumed to prepare
thestate while no EPR pairs can be distilled back from that state.
This is called boundentanglement [50].
Concurrence
Concurrence is a widely used entanglement measure defined for
mixed states of a two-qubit system [80]. We firstly introduce the
definition of the "spin-off" density matrixbetween AB as
ρ̃AB = (σy ⊗ σy)ρ∗AB(σy ⊗ σy), (2.36)
based on which concurrence is defined as
Definition 2. Concurrence between A and B for a density matrix
ρAB, is
CAB = max{λ1 − λ2 − λ3 − λ4, 0} (2.37)
where λis are the square roots of the eigenvalues of ρABρ̃AB in
decreasing order (λ1 ≥λ2 ≥ λ3 ≥ λ4).
Up to now, concurrence is the only known analytical entanglement
monotone formixed states. Under a general noisy environment, a pure
state will be transformed intoa mixed state because of decoherence.
To quantify the involvement of entanglement forthis quantum system,
the change in concurrence serves as an important criterion.
Forexample, entanglement sudden death was based on the calculation
of concurrence underthe effect of classical noise [85].
-
Chapter 2. Background Information 19
2.3 Multipartite entangled pure states
While bipartite pure entangled states have been analyzed
thoroughly in theory, the gen-eralization of these results into
multipartite pure states remainsl an open problem.
2.3.1 Tripartite entangled states
For a general tripartite pure entangled state, the Schmidt
decomposition might not alwaysexist. Instead, a unique form called
generalized Schmidt decomposition of tripartite stateswas
discovered in [2] as
|φ〉 = λ0 |000〉+ λ1 |100〉+ λ2 |101〉+ λ3 |110〉+ λ4 |111〉
(2.38)
where λi ≥ 0, 0 ≤ φ ≤ π, µi ≡ λ2i ,∑
i µi = 1.
Based on this form, truly entangled three-qubit states (that is
to say, other thanproduct states or bipartite entangled states that
are separable under the bipartition A-BC, AB-C and C-AB) can be
divided into two classes [34]. When λ4 6= 0, it is calleda GHZ
class state; otherwise, it is called a W class state. States that
belong todifferent classes cannot be transformed into each other by
LOCC even with some non-zero probability. However, the optimal
transformation probability between states in thesame class is still
a open problem.
Further generalization of this result turns out to be a very
complex problem. Actually,for four-qubit entangled states, no
unique form has been proposed yet and it has beenproved that
four-qubit entangled states can be classified into nine families
[76], betweenwhich the transformation rules are not yet known.
Entanglement monotones for tripartite pure entangled states
Given a tripartite state shared between A, B, and C, the
entanglement can be shared byA and B, A and C, A and BC, etc. Is
there any entanglement that is shared by ABCaltogether? If so, how
can we quantify it? The answer lies in concurrence and a
quantitycalled 3-tangle [29].
Firstly, it should be clarified that the concurrence between A
and B, CAB, is not anentanglement monotone. In fact, it can be
increased significantly by LOCC. For example,for a GHZ state
|GHZ〉ABC = 1√2(|000〉+ |111〉)ABC , if Charlie performs a
measurementon the basis {|+〉 , |−〉} and communicates the
measurement result to Alice and Bob, ABwill end up sharing an EPR
pair. Notice that for the original state, the density matrix
-
Chapter 2. Background Information 20
for AB isρAB =
1
2(|00〉 〈00|+ |11〉 〈11|) (2.39)
which is a separable state with CAB = 0. But for the final EPR
state, we clearly haveCAB = 1. During this transformation, CAB has
been increased from 0 to 1 with Charlie’sassistance.
However, the concurrence between A and BC, CA(BC) is an
entanglement monotone,similarly for CB(AC) and CC(AB). Also, based
on concurrence, one can define 3-tangle -the entanglement shared by
ABC altogether - as the difference between C2A,BC and thesummation
of C2AB and C2AC , or we have
τABC = C2A(BC) − C2AB − C2AC . (2.40)
Remark 5. Also, we can define τABC using different orders of the
parties, which willgive the same result. Or we have
τABC = C2B(AC) − C2BA − C2BC = C2C(AB) − C2CA − C2CB. (2.41)
τABC is shown to be an entanglement monotone [29]. From the
definition of τABC wecan see that it quantifies the entanglement
not attributed to any two-party entanglement,or we can say that it
is the genuine entanglement shared by three parties altogether.
τABCserves as a clear distinction between the GHZ class state and W
class state because τABCis zero for any W class state. Or we can
say, the entanglement of a W class state canall be attributed to
two-party entanglement [34]. Because of that, the
transformationprobability from a W class state to any GHZ class
state is always zero. In chapter 3,we will show the result we
obtain for upper and lower bounds of the transformationprobability
between GHZ class states.
Schmidt rank for tripartite pure entangled states
Another concept that can be used to distinguish GHZ class and W
class states is theSchmidt rank. As we defined earlier, it is the
minimum number of product statesneeded in the superposition form of
a pure state. For a GHZ class state, the Schmidtrank is 2 while any
W class state has Schmidt rank 3. Notice that Schmidt rank cannot
even be increased by SLOCC, which means it can not increase even
with a nonzeroprobability. Since W class states have a higher
Schmidt rank than any GHZ class state,the transformation
probability from a GHZ class state to a W class state is also zero
[34].
-
Chapter 2. Background Information 21
Remark 6. For tripartite pure states with higher dimensions, it
is proved that the cal-culation of Schmidt rank is an NP hard
problem [21].
2.3.2 W type entangled states
Multipartite entangled states have a very complex mathematical
structure when thenumber of parties is higher than three. However,
for a specific class of multipartitestates, the W-type states, a
unique form was discovered and the transformation rulesbetween
these states turn out to be very explicit [52]. In the following we
will introducethe definition of W type state as a generalization of
W state. We will also describe theentanglement monotones and
transformation rules associated with W type states.
From the definition of the W state
|W 〉 = 1√3
(|100〉+ |010〉+ |001〉) (2.42)
one can easily generalize it into more parties as
|Wn〉 =1√n
(|10 · · · 0〉+ |010 · · · 0〉+ · · ·+ |0 · · · 01〉) (2.43)
which is called standard n-party W state. Then we can give the
following definitionof W-type state:
Definition 3. W type states are the states that can be obtained
by SLOCC from a stan-dard n-party W state, where n is any integer
no less than three.
Remark 7. Notice that based on this definition, product states
also belong to W typestates since one can easily obtain a product
state from an n-party W state by making ameasurement on some
parties. An illustration of W type state for three qubits is
shownin 2.1.
It has been shown that W-type states have a very explicit unique
form as
|φ(−→x )〉 = √x0 |0 · · · 0〉+√x1 |10 · · · 0〉+
√x2 |010 · · · 0〉+ · · ·+ |xn〉 |0 · · · 01〉 . (2.44)
where xi ≥ 0 and∑n
i=0 xi = 1. It was proved that the above form is unique when
atleast three xis are nonzero [52]. Also, we can use a vector
notation for the above state
−→x = (x1, · · · , xn). (2.45)
-
Chapter 2. Background Information 22
Figure 2.1: Structure of states that can be obtained from W3
state by SLOCC. The firstlevel is the true W3 type state which is
also the genuine W class state. The second levelare bipartite
entangled states, such as (AB)-C (|ψ〉AB |φ〉C), (AC)-B (|ψ〉AC |φ〉B)
and(BC)-A (|ψ〉BC |φ〉A), and the third level is the product state
|φ1〉A |φ2〉B |φ3〉C .
LOCC transformation rule for W-type states
In the following we will show the behavior of W-type state under
LOCC transformation.Since any LOCC transformation will turn a
W-type state into another W-type state, wecan simply discuss the
change to xis. Given an initial W-type state −→x = (x1, · · · ,
xn),consider a local operator being applied on party j. This leads
to several possible finalstates −→xk with corresponding probability
pk, and we have [52]
(i) xk,i = skxi where i 6= 0, j;
(ii)∑
k pksk = 1;
(iii)∑
k pk√skxk,0 ≥
√x0.
It is not hard to see that all xis where i > 0 can never
increase on average underLOCC, which means they are all
entanglement monotones. Further investigation for thetransformation
probability between W type states will be shown in chapter 4.
Also,by analyzing random distillation protocol for W type states,
we find new entanglementmonotones for W type states that can be
increased by SEP [20]. The detail of this resultwill be discussed
in chapter 5 of this thesis.
-
Chapter 3
LOCC transformation bounds betweenmultipartite pure states
In this chapter, the upper and lower bounds on the
transformation probability betweenmultipartite pure states will be
derived. For tripartite pure states, it is well knownthat there are
two inequivalent classes of genuine tripartite entangled states,
namelythe Greenberger-Horne-Zeilinger (GHZ) class and the W class.
Any two states withinthe same class can be transformed into each
other via stochastic local operations andclassical communication
with a nonzero probability. The optimal conversion
probability,however, is only known for special cases. Here, lower
and upper bounds are derivedfor the optimal probability of
transformation from a GHZ state to other states of theGHZ class. A
key idea in the derivation of the upper bounds is to consider the
actionof the local operations and classical communications (LOCC)
protocol on a differentinput state, namely 1√
2|000〉− |111〉), and to demand that the probability of an
outcome
remains bounded by 1. We also find an upper bound for more
general cases by usingthe constraints of the so-called interference
term and 3-tangle. Moreover, some of theresults are generalized to
the case in which each party holds a higher dimensional system.In
particular, the GHZ state generalized to three qutrits, that is,
|GHZ3〉 = 1√3(|000〉 +|111〉+|222〉) shared among three parties can be
transformed to any tripartite three-qubitpure state with
probability 1 via LOCC. Some of our results can also be generalized
tothe case of a multipartite state shared by more than three
parties. The content of thischapter is mainly based on [33].
23
-
Chapter 3. LOCC transformation bounds between multipartite pure
states24
3.1 Introduction
Entanglement is the most peculiar feature that distinguishes
quantum physics from clas-sical physics and lies at the heart of
quantum information theory. Thus it is importantto get a good
understanding of entanglement properties of quantum states. These
prop-erties are well understood for bipartite pure states. In the
standard distant laboratoryparadigm, suppose two distant parties,
Alice and Bob, shared a bipartite entangled state.They may apply
local operations and classical communications (LOCC) to convert it
intoanother partite state. Bennett et al [6] has answered the
question for the rate of LOCCtransformation between bipartite pure
states. It is quantified by the von Neummanentropy of a reduced
density matrix. For the single-copy case, the optimal
conversionprobabilities are known for any pure state transformation
[56, 61, 77]. For an LOCCtransformation protocol, if it can succeed
with probability 1, we call it deterministic, ifit can only succeed
with a nonzero probability smaller than 1, we call it stochastic,
orSLOCC (Stochastic Local Operators and Classical Communications).
For mixed states,the question of what the optimal rate of
transformations is between them is still largelyopen.
For multipartite states, however, the problem is much more
complicated. There existdifferent types of entanglement and
therefore the transformations are rather involved.For the case of
tripartite pure three qubit states, a characterization into six
differententanglement classes, of which two contain true tripartite
entanglement, exists [34]. Oneis the GHZ class state, which is
defined as
|φGHZ〉 =√K(cδ |0〉 |0〉 |0〉+ sδeiϕ |ϕA〉 |ϕB〉 |ϕC〉) (3.1)
where
|ϕA〉 = cα |0〉+ sα |1〉 (3.2)|ϕB〉 = cβ |0〉+ sβ |1〉 (3.3)|ϕC〉 = cγ
|0〉+ sγ |1〉 (3.4)
and K=(1 + 2cδsδcαcβcγcφ)−1 ∈ [12 ,∞), cδ = cos δ, sδ = sin δ,
the same for α, β, γ, φ.The range for the parameters are δ ∈ (0,
π
4], α, β, γ ∈ (0, π
2] and ϕ ∈ [0, 2π).
Another is W class state, which is defined as a state that is
unitarily equivalent to
|φ〉 = (√c |0〉+√d |1〉) |00〉+ |0〉 (√a |01〉+
√b |10〉) (3.5)
-
Chapter 3. LOCC transformation bounds between multipartite pure
states25
with c+ d+ a+ b = 1.
A transformation between any two states of the same class is
always possible with non-zero probability. However, here comes the
key point. The optimal conversion betweenthe states within the same
class of genuine tripartite entangled states is not
known.Incidentally, a similar characterization into nine different
classes exists for four qubits[76]. In 2000, the optimal rate of
distillation of a GHZ state from any GHZ-class statewas found [2].
Very recently, a necessary and sufficient condition for
deterministically(i.e., with probability 1) transforming
multipartite qubit states with Schmidt rank 2 [36]have been given
[74].
In this chapter, we present new upper and lower bounds for
multipartite entanglementtransformations. In particular, we focus
on transformations among states with the sameSchmidt rank [36]. We
put an emphasis on the transformation from a GHZ state to
aGHZ-class state.
But our upper bound can also be generalized to general
transformations from oneGHZ class state to another. And some of the
results are derived for the more generalcase of higher dimensions
and more parties. Especially, we find that all tripartite purethree
qubit states can be transformed from 3-term GHZ state 1√
3(|000〉 + |111〉 + |222〉)
with probability 1. This is a new result. Moreover, some of the
general theorems fordeterministic transformation are also
derived.
This Chapter is structured as follows. In Section 3.2, we derive
upper bounds for thetransformation of the GHZ-state to any other
state in the GHZ-class. The upper boundsare only non-trivial for a
subclass of the GHZ-class. Thus Section 3.3 and 3.4 use adifferent
approach that results in upper bounds for a wider class of states.
More specific,for any GHZ class state which does not have a known
way to be transformed from GHZstate with probability 1, we can find
a nontrivial upper bound for the probability ofthis transformation.
And our upper bound can also be effective for the
transformationfrom a GHZ class state to a large class of other GHZ
class states. Lower bounds forthe transformation of higher
dimensional GHZ-states distributed among three or moreparties to
states with the same Schmidt rank are given in Section 3.5.
3.2 Upper Bound for the Conversion from GHZ state
to a GHZ class state
In this section, we derive an upper bound for the conversion of
the GHZ-state to anyother state of the GHZ-class via LOCC. This
upper bound will be nontrivial (i.e., smaller
-
Chapter 3. LOCC transformation bounds between multipartite pure
states26
than 1) for ϕ ∈ (12π, 3
2π). The transformation under consideration is given by
|GHZ〉 = 1√2(|000〉+ |111〉)
LOCC−→ |Ψ〉 =√K(cδ |0〉 |0〉 |0〉+ sδeiϕ |ϕA〉 |ϕB〉 |ϕC〉), (3.6)
with the parameters defined in introduction.The LOCC operation
is represented by Kraus operators {Oi = Ai ⊗ Bi ⊗ Ci}. In
the following we will refer to different Kraus operators of the
LOCC protocol as differentbranches. Furthermore, a branch Oi |GHZ〉
= |Φ〉 is called a success branch if |Φ〉 ∝ |Ψ〉,and a failure branch
if there exists no LOCC-operation that can transform |Φ〉 into|Ψ〉
with a non-zero probability, if a branch is neither success nor
failure, we call it anundecided branch. An optimal protocol only
consists of success and failure branches.
For the following analysis we first recall two known results
[34, 2]
Lemma 1. For a GHZ-class state |Ψ〉 we have:
a) The Schmidt rank of |Ψ〉 is 2 [34]. This means that the
minimum number of productstates necessary to write |Ψ〉 as a
superposition of them is 2:
|Ψ〉 =2∑i=1
αi |aibici〉 , (3.7)
with αi ∈ (0, 1) and 〈aibici|aibici〉 = 1.
b) This product state decomposition, i.e., the set {(α1,
|a1b1c1〉), (α2, |a2b2c2〉)} is unique[1].
This result leads to
Lemma 2. For a successful LOCC operation within the
GHZ-class,
|Ψ〉 = α1 |a1b1c1〉+ α2 |a2b2c2〉LOCC−→ |Ψ′〉 = α′1 |a′1b′1c′1〉+ α′2
|a′2b′2c′2〉 , (3.8)
described by the operator O1, we must either have the
mapping
O1 |a1b1c1〉 = o1α′1α1|a′1b′1c′1〉 (3.9)
O1 |a2b2c2〉 = o1α′2α2|a′2b′2c′2〉 (3.10)
-
Chapter 3. LOCC transformation bounds between multipartite pure
states27
or
O1 |a1b1c1〉 = o1α′2α1|a′2b′2c′2〉 (3.11)
O1 |a2b2c2〉 = o1α′1α2|a′1b′1c′1〉 (3.12)
with some proportionality constant o1, which can be chosen to be
real. See Figure 3.1,Figure 3.2.
Figure 3.1: mapping type 1. c©2010 American Physical Society
Figure 3.2: mapping type 2. c©2010 American Physical Society
Proof: Since a LOCC Kraus operator is always of the form O1 = A1
⊗ B1 ⊗ C1, aproduct state is always transformed into a product
state. With that observation and the
-
Chapter 3. LOCC transformation bounds between multipartite pure
states28
fact that the two-term product decomposition of a tripartite
GHZ-class state is unique(Lemma 1), Lemma 2 follows. �
Theorem 1. An upper bound for the conversion probability for
|GHZ〉 = 1√2(|000〉+ |111〉)
→ |Ψ〉 =√K(cδ |000〉+ sδeiϕ |ϕAϕBϕC〉), (3.13)
where the parameters are defined in Equation (3.6), is given
by
p ≤ min{
1,1 + 2cδsδcαcβcγcϕ1− 2cδsδcαcβcγcϕ
}(3.14)
Idea of the Proof: From Lemma 2 we know that, for a success
branch, each productstate (in the Schmidt term) of the input states
have to be mapped to a product stateof the output state. This
allows us to infer how the same LOCC protocol acts on thephase
flipped GHZ state, i.e., 1√
2(|000〉 − |111〉). From the requirement that the sum of
the probabilities for the output states have to sum to 1 for
this transformation, we canderive a bound for the parameters
arising in the original transformation. This gives abound on the
successful transformation probability.
Proof: Consider the optimal LOCC strategy, given by the Kraus
operators Oi =Ai ⊗ Bi ⊗ Ci. According to Lemma 2, there are two
possibilities to have a successfulbranch. They are
Oi |000〉 = oicδ |000〉 (3.15)Oi |111〉 = oieiϕsδ |ϕAϕBϕC〉
(3.16)
for i = 1, . . . , n1, and
Oi |000〉 = oieiϕsδ |ϕAϕBϕC〉 (3.17)Oi |111〉 = oicδ |000〉
(3.18)
for i = n1 + 1, . . . , n1 + n2. Both cases give the desired
transformation
Oi |GHZ〉 =1√2oi(cδ |000〉+ eiϕsδ |ϕAϕBϕC〉) =
oi√2K|Ψ〉 (3.19)
-
Chapter 3. LOCC transformation bounds between multipartite pure
states29
for i = 1, . . . , n1 + n2. The successful conversion
probability is then given by
p =1
2K
n1+n2∑i=1
o2i . (3.20)
To get an upper bound for∑n1+n2
i=1 o2i , we consider how
1√2
(|000〉 − |111〉) (3.21)
behaves when put through the Kraus Operator Oi. We see that
Oi1√2(|000〉 − |111〉)
= 1√2oi(cδ |000〉 − eiϕsδ |ϕAϕBϕC〉) = oi√2K′ |Ψ
′〉 (3.22)
with|Ψ′〉 =
√K ′(cδ |000〉 − eiϕsδ |ϕAϕBϕC〉), (3.23)
where K ′ = 11−2cδsδcαcβcγcϕ
, for i = 1, . . . , n1 +n2 (up to an overall minus sign for i =
n1 +1, . . . n1 +n2). Thus the conversion probability for this
process is given by 12K′
∑n1+n2i=1 o
2i .
Being a probability, this has to be bounded by 1,
giving∑n1+n2
i=1 o2i ≤ 2K ′. This together
with Equation (3.20) gives the upper bound
p ≤ K′
K=
1 + 2cδsδcαcβcγcϕ1− 2cδsδcαcβcγcϕ
(3.24)
for the process described by Equation (3.13). �
Special Case: Regarding the special case, where we have |ϕA〉 =
|ϕB〉 = |ϕC〉, cα =cβ = cγ = λa, ϕ = 0, and cδ = sδ = 1√2 , i.e.,
|Ψ〉 = 1√2√
1− λ3a(|000〉 − |aaa〉), (3.25)
we get
p ≤ 1− λ3a
1 + λ3a. (3.26)
Theorem 1 gives a non-trivial upper bound for the transformation
from the GHZ-state to a GHZ-class state for all values of ϕ with
cosϕ < 0, i.e., φ ∈ (π
2, 3π
2). This nicely
shows, that unlike in the bipartite case, where the maximally
entangled EPR-state can
-
Chapter 3. LOCC transformation bounds between multipartite pure
states30
be tranformed into any other pure two qubit state with
probability one, the GHZ-state,which exhibits maximal genuine
tripartite entanglement as it maximizes the 3-tangle [29]and
tracing out one qubit results in a totally mixed state, cannot be
transformed to allother states in the same class with probability
one.
3.3 Failure Branch
Recall in the last section that Eq. 3.14 gives a trivial bound
for the case φ ∈ (π2, 3π
2).
Here, we will derive a useful bound for a larger class of
states: we find a upper boundnontrivial for all the cases except φ
= π
2or 3π
2and 〈000|ϕAϕBϕC〉 = 0. In fact, it was
shown that these two kinds of transformations can succeed with
probability 1 [74]. Ourproof has two important ingredients, namely,
the conservation of a new quantity definedas "interference term"
under positive operator valued measures (POVMs) and that thethree
tangle is an entanglement monotone, which we will discuss in detail
in the following.
The idea of our discussion is that, firstly, recall our
definition of "failure branch"as one can not be successful with any
nonzero probability, we will prove the weightsummation of the
so-called interference terms and normalizations of all the branches
inan LOCC protocol should be constant during the transformation,
which is included insection 3.3. After that, we find that three
tangle is bounded for a fixed interferenceterm which will be
defined in this section. Then, we try to see the whole process
fromthe weak measurement aspect, which divides the whole process
into many infinitesimalsteps and each step changes the state very
little. That is to say, the state is changingcontinuously. Then we
stop in the middle and investigate whether there will be a newupper
bound. Surprisingly we find there are some new upper bounds and
these upperbounds will still be effective in the following steps,
even when we reach the end. So itcan be used to derive a new upper
bound for the supremum success probability of thewhole LOCC
protocol. Detailed discussion will be showed in section 3.4.
Theorem 2. For the transformation from GHZ to GHZ-class state
|φ〉, failure branchesshould end with a state with at least one
parties’ reduced matrix with rank 1.
Proof: Suppose we would like to get a GHZ-class state |φ〉 =√K(cδ
|0〉 |0〉 |0〉 +
sδeiϕ |ϕA〉 |ϕB〉 |ϕC〉), where |0A〉 is linearly independent of
|ϕA〉, the same for B and C. If
there is a state whose reduced density matrices are all of full
rank, |φ〉 =√K ′(c′δ |0〉 |0〉 |0〉+
s′δeiϕ′ |ϕ′A〉 |ϕ′B〉 |ϕ′C〉), where |0A〉 is linearly independent
of |ϕ′A〉, the same for B and C.
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Chapter 3. LOCC transformation bounds between multipartite pure
states31
Then it is easy to see, the equation
OA |0〉 = |0〉 (3.27)OA |ϕ′A〉 = |ϕA〉 (3.28)
always has a non-trivial solution, the same for B and C. That
means we can alwaystransform this state into |φ〉 with nonzero
probability. �
3.3.1 Conservation of interference term
To go further, we want to use the following property of the LOCC
Kraus operators. Fora complete set of Kraus operators Oi = Ai ⊗Bi ⊗
Ci, we have
∑O+i Oi = 1.
Suppose that a Kraus operator O satisfies
O |000〉 = α |a1b1c1〉 (3.29)
O |111〉 = β |a2b2c2〉 (3.30)
with 〈a1b1c1|a1b1c1〉 = 〈a2b2c2|a2b2c2〉 = 1.Then it can transform
|GHZ〉 = 1√
2(|000〉 + |111〉) into |ψ〉 = 1√
2p(α |a1b1c1〉 +
β |a2b2c2〉, where 1√2p is the normalization factor and one can
check p is exactly theprobability of getting |ψ〉. From here we
define interference term and normalization inthe following:
Definition 4. For a normalized GHZ-class state |γ〉 where 〈γ|γ〉 =
1, written in theform |γ〉 = 1√
2(α |a1b1c1〉 + β |a2b2c2〉), suppose 〈a1b1c1|a2b2c2〉 = k, then we
call the real
part of α∗βk the interference term I of |γ〉.
It is easy to see if an operator O transforms |GHZ〉 to a state
|ψ〉, the interferenceterm of |ψ〉 is in fact the real part of
p, where p is the probability of the
branch corresponding to operator O.
Remark 8. In fact, one can find I = 1− 12(|α|2 + |β|2).
Remark 9. Note also that −∞ < I ≤ 1. In other words, it can
be unbounded below.This fact will become important in our
discussion in Section 3.4.
Remark 10. Notice that a failure branch gives a state that is
outside the GHZ class.For such a state, the actual value of
interference term depends not only on the state itself,
-
Chapter 3. LOCC transformation bounds between multipartite pure
states32
but also on the particular Kraus operator, Oi, and the initial
state, φi, used to reach thestate. So, when we talk about the
interference term of failure branches of an SLOCCtransformation, we
need to be careful: We are not talking about the interference term
ofthe state given by the failure branches, but the interference
term determined by the wholetransformation protocol.
Theorem 3. For a complete set of LOCCs which transforms GHZ
state to other states,in which the operators are {Oi}, the weighted
sum of the interference terms in all thebranches should be
zero.
0 =∑
p(Oi |GHZ〉)I(Oi |GHZ〉) (3.31)
where p(Oi |GHZ〉) is the probability of branch corresponding to
the Kraus operator Oi,and I(Oi |GHZ〉) denotes the interference term
I for a state Oi |GHZ〉.
Proof: Suppose the corresponding complete set of Kraus operators
consists ofOi = Ai ⊗Bi ⊗ Ci.Then we have
∑O+i Oi = Id. So, we should have
0 = < 000|111 >=< 000|I|111 >= < 000|∑O+i Oi|111
>=
∑< 000|O+i Oi|111 >
=∑p(Oi |GHZ〉)
p(Oi|GHZ〉) (3.32)
From the definition of interference term I we know the real part
of the right side ofEquation (3.32) is exactly the weighted sum of
I of each branch. As the right side ofEquation (3.32) is equal to
zero, its real part should also be zero, which means for
atransformation from |GHZ〉 to other states, the average value of
the interference termsof all the states we get in each branch
should be zero. We call this the conservation ofinterference term.
�
3.3.2 Conservation of normalization
Definition 5. For a two-term tripartite state |γ〉, written in
the form |γ〉 = 1√2(α |a1b1c1〉+
β |a2b2c2〉, then we call 12(|α|2 + |β|2) the normalization of
|γ〉.
Easy to see if an operator O transforms |GHZ〉 to the state |ψ〉,
the normalization of|ψ〉 is in fact +
2p, where p is the probability. And because OρO+
is positive, normalization should be always no less than
zero.
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Chapter 3. LOCC transformation bounds between multipartite pure
states33
Suppose the corresponding complete set of Kraus operators
consists of {Oi = Ai ⊗Bi ⊗ Ci}. Then we have
∑O+i Oi = I. So we should have
1 = < GHZ|GHZ >= 1
2(< 000|+ < 111|)(|000 > +|111 >)
= 12(〈000|000〉+ 〈111|111〉)
= 12(∑ 〈000|O+i Oi|000〉+∑ 〈111|O+i Oi|111〉)
=∑p(Oi |GHZ〉)+
2p(Oi|GHZ〉) (3.33)
From the definition of normalization we know it is exactly the
weighed sum of thenormalization of each branch. That is to say, for
a transformation from |GHZ〉 to otherstates, the average value of
the normalization of all the states we get in each branchshould be
1. And recall that normalization can be no less than zero. So each
term in thesummation should be no larger than 1, which means for
each branch, the product of itsprobability and the normalization of
the state it gets should be no larger than 1.
In fact, the conservation of normalization can be derived from
conservation of inter-ference term. However, conservation of the
normalization also gives the following. Foreach branch, the product
of its probability and the normalization of the state it getsshould
be no larger than 1. The fact is also useful in determining the
upper bound oftransformation probability.
The basic idea is that, if we know the state we want and the
state failure branch gives,equations (3.32) and (3.33) combined
with the fact that the summation of probabilityshould be one can
give us some implication about the supremum success probability.
Forexample, we can have the following theorem:
Theorem 4. For a transformation protocol from GHZ state to a
GHZ-class state |φ〉whose interference term is x, which is positive
(negative), if there exists a y (y > 0), suchthat, the
interference term of all the failure branches are larger than -y
(smaller than y),we have an upper bound for its successful
probability ps in the following:
if x > 0:
ps ≤ pU(−y) =y
x+ y. (3.34)
if x < 0:ps ≤ pU(y) = −
y
x− y . (3.35)
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Chapter 3. LOCC transformation bounds between multipartite pure
states34
See
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
a
pu
Figure 3.3: The value of pU as a function of a. In this figure,
a = ( yy−1)
13 . So when a goes
from 0 to 1, y goes from 0 to ∞. Note that as y goes to
infinity, a goes to 1. We expressthe value as a function of a
because it will be easier for us to combine different graphsinto
one graph later. c©2010 American Physical Society.
Proof: Take x > 0, suppose there are n failure branches,
whose probabilities arepf1 , pf2 , ·, pfn , and the corresponding
interference terms are −y1,−y2, ·−yn, then we have
psx−∑pfiyi = 0 (3.36)
ps +∑pfi = 1 (3.37)
Rewrite it in the following form,
psx− pfty′ = 0 (3.38)ps + pft = 1 (3.39)
where pft =∑pfi and y′ =
∑pfiyipft
. The solution of it is
ps =y′
x+ y′(3.40)
As the interference term of all the failure branches are larger
than -y, we have y′ < y,then we can get ps < pU(−y) = yx+y .
The discussion for the case when x < 0 is similar.�
Remark 11. Recall the range of the I can be −∞ < I ≤ 1, which
means I can be
-
Chapter 3. LOCC transformation bounds between multipartite pure
states35
unbounded below. Then in the x > 0 case, if the I of the
failure branch goes to −∞, or wecan say y goes to∞, we will have
pU(−y) arbitrary close to 1. Therefore, theorem 4 aloneis not
enough for establishing a non-trivial upper bound. To derive a
non-trivial upperbound, we need to find some additional constraints
which are related to the i