Quantifying quantum discord
and Entanglement of Formation
via Unified Purifications
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!