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Measuring Quantum Entanglement
John Cardy
University of Oxford
Max Born Lecture
University of Gttingen, December 2012
A B
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Quantum Entanglement
Quantum Entanglement is one of the most fascinating andcounter-intuitive aspects of Quantum Mechanics
its existence was first recognised in early work of the
pioneers of quantum mechanics1
it is the basis of the celebrated Einstein-Podolsky-Rosenpaper2 which argued that its predictions are incompatible
with locality
1Schrdinger E (1935). "Discussion of probability relations between
separated systems". Mathematical Proceedings of the Cambridge
Philosophical Society31 (4): 555-563.Communicated by M Born2Einstein A, Podolsky B, Rosen N (1935). "Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?". Phys. Rev. 47
(10): 777-780.Measuring Quantum Entanglement
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this was part of Einsteins programme to refute the
probabilistic interpretation of quantum mechanics, due to
Born and others
Die Quantenmechanik ist sehr achtunggebietend. Aber eineinnere Stimme sagt mir, da das noch nicht der wahre Jakob
ist. Die Theorie liefert viel, aber dem Geheimnis des Alten
bringt sie uns kaum nher. Jedenfalls bin ich berzeugt, da
der Alte nicht wrfelt.3
Quantum mechanics is certainly impressive. But an inner
voice tells me that it is not yet the real thing. The theory tells us
a lot, but it does not bring us any closer to the secrets of the
ancients. I, at any rate, am convinced that the old fellow does
not play dice."
Bell4 showed that EPRs explanation, involving hidden
variables, is inconsistent with the predictions of quantum
mechanics this was subsequently tested experimentally3
Letter to Max Born, 4 December 19264Bell JS (1964). "On the EPR paradox". Physics1, 3, 195-200.
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this was part of Einsteins programme to refute the
probabilistic interpretation of quantum mechanics, due to
Born and others
Die Quantenmechanik ist sehr achtunggebietend. Aber eineinnere Stimme sagt mir, da das noch nicht der wahre Jakob
ist. Die Theorie liefert viel, aber dem Geheimnis des Alten
bringt sie uns kaum nher. Jedenfalls bin ich berzeugt, da
der Alte nicht wrfelt.3
Quantum mechanics is certainly impressive. But an inner
voice tells me that it is not yet the real thing. The theory tells us
a lot, but it does not bring us any closer to the secrets of the
ancients. I, at any rate, am convinced that the old fellow does
not play dice."
Bell4 showed that EPRs explanation, involving hidden
variables, is inconsistent with the predictions of quantum
mechanics this was subsequently tested experimentally3
Letter to Max Born, 4 December 19264Bell JS (1964). "On the EPR paradox". Physics1, 3, 195-200.Measuring Quantum Entanglement
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Outline
in this talk I am going to assume that conventionalquantum mechanics (and the Copenhagen interpretation)holds and will address the questions:
what is quantum entanglement and is there a universalmeasure of the amount of entanglement?
how does this behave for systems with many degrees offreedom?
how might it be measured experimentally?
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A simple example
a system consisting of two qubits (spin- 12
particles) with a
basis of states
(| A, | A) (| B, | B)
Aliceobserves qubit A,Bobobserves qubitB
the state
| = 12
| A| B + | A| B
isentangled: before Alice measureszA, Bob can obtain
either resultz
B= 1, but after she makes themeasurement the state in Bcollapses and Bob can only
get one result
moreover this still holds if the subsystemsAandBare far
apart (the EPR paradox)
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anentangledstate is different from aclassically correlated
state, eg with density matrix
= 1
2| A| BA |B |+ 1
2| A| BA |B |
in both cases AzBz = 1but for the entangled state
A
xBx
= 1
while it vanishes for the classically correlated state
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Entanglement of pure states
is there a good way of characterising the degree of
entanglement (of pure states)?
Schmidt decomposition theorem: any state in HA HB canbe written
| =
j
cj |jA |jB (S)
where the states are orthonormal, cj >0, and
jc2j = 1
thec2j are the eigenvalues of the reduced density matrix
A = TrB ||
if there is only one term in (S), | is unentangledif all the cj are equal,
|
is maximally entangled
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a suitable measure is the entanglemententropy
SA =
j
c2j log c2j = TrAAlog A = SB
it is zero for unentangled states and maximal when all the cjare equal
it is convex: SA1A2 SA1+ SA2it is basis-independent
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it increases under Local Operations and Classical
Communication
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even for many-body systems it is often computable(analytically, or numerically by density matrix
renormalization group methods or matrix product states)
but it is not the only such measure: eg the Rnyi entropies
S(n)A log TrAAn
are equally useful, and for different ngive information
about the whole entanglement spectrum ofA
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Entanglement Entropy in Extended Systems
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Entanglement Entropy in Extended Systems
consider a system whose degrees of freedom are
extended in space, e.g. a quantum magnet described by
the Heisenberg model with hamiltonian
H=r,r
J(r r)(r) (r)
the temperature is low enough so the system is in the
ground state |0 ofH
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A B
supposeAis a large but finite region of space: what is the
degree of entanglement of the spins within Awith the
reminder inB?sinceSA = SB it cant be the volume of AorBin fact in almost all cases we have thearea law:
SA
C
Area of boundary
whereDis the dimensionality of space. The constant Cis
1/(lattice spacing)D1 and is non-universal in general.entanglement occurs only near the boundary
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One dimension
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One dimension
whenD = 1something interesting happens: the constantis proportional to a logarithm
SA C log(correlation length)
now the constant Cis dimensionless anduniversal
at a quantum critical point diverges and so does the
entanglement entropy
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
S
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Measuring Entanglement
measuring entropy of many-body systems is conceptually
difficult: even at finite temperature we do it by integrating
the specific heat
however the situation is better for the Rnyi entropies
SA(n) log Tr An = log
j
c2nj
to simplify the discussion assume n = 2and consider twoindependent identical copies of the whole system, so the
composite system is in the state
|1|2=
j1
j2
cj1cj2 |j1A1|j1B1|j2A2|j2B2
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let S be a swap operator which interchanges the states in
A1 with those inA2 but leaves states inB1 andB2 the
same:
S |j1A1|j1B1|j2A2|j2B2= |j2A1|j1B1|j1A2|j2B2
then 1|2|
S|1|2
=
j
c4j = TrA2
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on a system with local interactions, S can be implemented
locally as a quantum switch: eg a 1D quantum spin chain:
A B
initially the two decoupled chains have a hamiltonian
H= H1 +H2 with a ground state |0 = |1|2after the switch, the hamiltonian isH = SHS1, with aground state
|0 = S
|0
with thesameenergy
we need to measure the overlap
M= 0|0 = Tr2A
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two proposals for how to do this:
5
prepare the system in ground state |0 ofHflip the switch so the new hamiltonian is H
the system finds itself in a higher energy state than |0 anddecays to this eg by emission of quasiparticlesdecay rate
|M
|2
6
introduce tunnelling between |0 and |0, equivalent toadding a term S to the hamiltonianthe tunnelling amplitude is Mthis can be detected by preparing in one state and
observing Rabi oscillations
5JC, Phys. Rev. Lett. 106, 150404, 20116
Abanin DA and Demler E, Phys. Rev. Lett. 109,020504,2012Measuring Quantum Entanglement
Summary
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Summary
entanglement is a fascinating and useful property of
quantum mechanics
entropy is a useful measure of entanglement for
characterising many-body ground states(and also in
quantum information theory)
in principle it can be measured in condensed matter or
cold atom experiments
although the theory tells us a lotdie Altenstill have many
secrets!
Measuring Quantum Entanglement
Summary
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Summary
entanglement is a fascinating and useful property of
quantum mechanics
entropy is a useful measure of entanglement for
characterising many-body ground states(and also in
quantum information theory)
in principle it can be measured in condensed matter or
cold atom experiments
although the theory tells us a lotdie Altenstill have many
secrets!
Measuring Quantum Entanglement
http://find/
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