Measuring Quantum Entanglement

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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


E l f
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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


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


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


Measuring Entanglement
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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!


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!

http://find/

Measuring Quantum Entanglement

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