Quantifying quantum discord

and Entanglement of Formation

via Unified Purifications

岑理相lixiangcen@scu.edu.cn

四川大学 物理科学与技术学院

Outline

Background

• A brief introduction to quantum discord

• Recent studies on quantum discord and related topics

Quantifying Qd and Eof via unified purifications

• Trilateral relation between Qd and Eof in purifications

• Qd and Eof in pure states of three qubits

• Qd and Eof in rank-2 mixed states of 4 X 2 systems

Summary and acknowledgement

Different measures of quantum correlations

Quantum correlations: the key resource to realize QIP

Entanglement:nonlocality

Quantum discord:quantumness

Introduction to quantum discord

Conception of mutual information:

--conditional entropy: the residual entropy (unknown information) of S given the state of A

The two definitions are equivalent for the classical world

Definition I:

Definition II:

Classical correlations

Introduction to quantum discord (cont.)

For a given quantum state of bipartite systems:

kk

kBkAB

BBAB

ASpS

ASSJ

k)|(min)(

)|()(

}{

Conditional entropy

Quantum discord:

Total correlations:

Classical correlations:

Generally not identical to entanglement! (except for pure states)

Quantum discord for particular examples

1. Classical correlated states:

k

kBAAkAB kkp

)|()()( ASSS BAAB

2. Werner states:

zIz

AB

44

1

0&)( ABABAB QJI

Separable when

Separable states could have nonzero discord!

Ollivier & Zurek, PRL(2002)

Quantum discord and Maxwell’s demon

Szilard’s engine （ 1929 ）

Work produced in the isothermal expansion:

Erasure of 1-bit informationhas an energy cost (Landauer’s principle)

2ln1 kTbit

)](2[ln)( SkTWex

Informationvs

energy

Fig. from Maruyama etc, RMP 2009

Quantum discord and Maxwell’s demon(cont.)

)](2ln2[ ABqex SkTW

)]}|()([2ln2{ ASSkTW BAcex

Quantum demon (nonlocal):

Classical demon (local):

Difference between the efficiencydefines the quantum discord

ABcex

qex kTQWW

Dynamics of quantum correlations under decoherence

separable

Dynamics of quantum discord?

Only investigated for some particular cases owing to the difficulty to quantify Qd

Werner states / Bell-diagonal states

See: Maziero etc., PRA 2009 Werland etc., PRA 2009

Uf

H(n)

Quantum discord or entanglement?

Resources for quantum computation:quantum discord or entanglement?

Discord in deterministic quantum computation

arXiv:10062460

DQC: See E. Knill and R. Laflamme, (1998)

Both quantum discord and entanglement are responsible for the QC speedup.

Eof: arbitrary two-qubit states (Wootters, PRL 2008)

Qd: Bell diagonal states (Luo etc., PRA 2008)

Two-qubit X-states (Ali etc., PRA 2010)

Qd: Arbitrary two-qubit mixed states with rank two

Arbitrary rank-2 mixed state of 4 X 2 systems

Eof: A sort of rank-2 mixed states of 4 X 2 systems

Studies on quantification of Qd and Eof

Previous results:

Our results:Intrinsic relation between Qd and Eof

Qd versus Eof in Unified Purifications

A B

CPurification

Set: )(dim ABrankC

ABCABC '~ Locally equivalent

Properties:

Qd versus Eof in Unified Purifications (cont.)

A B

CConditional entropy:

--Two different definitions:

1. von-Neumann projective measurement

2. Positive operator-valued measurement

Trilateral relationship of Qd and Eof in Unified purifications

A B

C Eof of

IIAC

IIAB QQ &

Consequence: quantify quantum discord via Eof & vice versa

Quantifying quantum correlations: quantum discord versus entanglement of formation

Quantifying Eof via Qd

dimA=n dimB=2

dimC=2

4 2

2

Quantifying Qd via Eof

Eof of two-qubit systems:Wootters’ formula

Discord of systems with rank two2n

1. Bell-diagonal States

2. Two-qubit X-states

Entanglement for corresponding mixed

states (rank-2) of systems24

Qd and Eof in pure states of three qubits

Concurrence: Entanglement of formation

Qd and Eof in pure states of three qubits(cont.)

222)( ACABBCAABC CCC 3-tangle:

2)(

22)( )( CABBCBCA

rAB CCCQ

)(2XY

rXY CQ

Deriving quantum discord via entanglement of formation

States of a systems with no more than two nonzero eigenvalues:24

2

121 pp

)( 224 ABC

( : Bell-diagonal state)

Deriving quantum discord via entanglement of formation (cont.)

)1(log)1(log)(

1)(

log)(

22

2

xxxxE

S

S

BC

AB

iiiA

)cos1(2

1

)cos1(2

1

2204,3

1212,1

k

k

}0,2max{

)11(2

1 2

iimC

Cx

)1,();,();,0(

2

1cos,1cos

21

21

31

312

1

21

k

Deriving entanglement of formation via quantum discord

)1,();,0(

2

1cos,1cos

51

512

1

21

k

Analytical expression for Eofother than two-qubit systems!

Comparison: Amount of Qd and Eof

ACCACBCB

CAACABAB

QQEQ

QQEQ

)(

)(

A B

C

)()(

)()(

CBCBABAB

CBCBABAB

EQEQ

EQEQ

)()( CBABCBAB EEQQ

Result 1: Result 2:

Fanchini etc., arXiv:10062460

Trilateral relation

or

Applications: dynamics under decoherence

Initial state:

Evolution under a phase-damping process

J.-S. Xu, etc., Nat. Commun. 1:7 doi:10.1038/ncomms1005 (2010).

Experimentally realizable via optical

systems

Eof

Entanglement of formaiton: Quantum discord:

Eof is always larger

than Qd in the speci-

fied dynamical process

Summary

The intrinsic relation is revealed between quantum discord and entanglement of formation in unified purifications

Quantification of quantum discord for the systems with rank two is obtained

Analytical expression of Eof for a sort of mixed states of systems is achieved

Application to describe dynamical behavior of quantum correlations of physical systems under decoherence

2n

24

Acknowledgement

XinQi Li (Beijing Normal Univ.)

JiuShu Shao (Beijing Normal Univ.)

YiJing Yan (HKUST, Hong Kong)

Dr. JianWei Xu (Sichuan Univ., Chengdu)

Thanks!