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:
Definition I:
Definition II:
Classical correlations
--conditional entropy: the residual entropy
(unknown information) of S given the state of A
The two definitions are equivalent for the classical world
Introduction to quantum discord (cont.)
For a given quantum state of bipartite systems:
Total correlations:
Classical correlations: J AB  S (  B )  S (  B | A)
Conditional
entropy
 S (  B )  min{ Ak }  pk S (  Bk | Ak )
k
Quantum discord:
Generally not identical to entanglement!
(except for pure states)
Quantum discord for particular examples
1. Classical correlated states:
 AB   pk k A k A   Bk
k
S (  AB )  S (  A )  S (  B | A)

I (  AB )  J AB
&
QAB  0
2. Werner states:
 AB
1 z

I4  z  
4
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 information
has an energy cost
(Landauer’s principle)
Fig. from Maruyama etc, RMP 2009
Information
vs
energy
1 bit  kT ln 2
Wex (  )  kT [ln 2  S (  )]
Quantum discord and Maxwell’s demon
(cont.)
Quantum demon (nonlocal):
Wexq  kT [2 ln 2  S (  AB )]
Classical demon (local):
Wexc  kT{2 ln 2  [S ( A )  S (B | A)]}
Difference between the efficiency
defines the quantum discord
Wexq  Wexc  kTQAB
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
Resources for quantum computation:
quantum discord or entanglement?
H(n)
Quantum discord
or entanglement?
Uf
Discord in deterministic quantum computation
arXiv:10062460
Both quantum discord
and entanglement are
responsible for the QC
speedup.
DQC:
See E. Knill and R. Laflamme, (1998)
Studies on quantification of
Qd and Eof
Previous results:
Eof: arbitrary two-qubit states (Wootters, PRL 2008)
Qd: Bell diagonal states (Luo etc., PRA 2008)
Two-qubit X-states (Ali etc., PRA 2010)
Our results:
Intrinsic relation between Qd and Eof
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
Qd versus Eof in Unified Purifications
C
Purification
A
Set:
dim C  rank(  AB )
Properties:
Locally equivalent ABC ~ ' ABC
B
Qd versus Eof in Unified Purifications (cont.)
C
Conditional entropy:
A
--Two different definitions:
1. von-Neumann projective
measurement
2. Positive operator-valued
measurement
B
Trilateral relationship of Qd and Eof
in Unified purifications
C
Eof of
II
AB
Q
A
II
AC
&Q
B
Consequence: quantify quantum discord via Eof & vice versa
Quantifying quantum correlations: quantum discord
versus entanglement of formation
Quantifying Qd via Eof
Quantifying Eof via Qd
Eof of two-qubit systems:
Wootters’ formula
1. Bell-diagonal States
2. Two-qubit X-states
Discord of n  2 systems with rank two
dimC=2
dimA=n
2
dimB=2
4
Entanglement for corresponding mixed
states (rank-2) of 4  2 systems
2
Qd and Eof in pure states of three qubits
Concurrence:
Entanglement of formation
Qd and Eof in pure states of three qubits
(cont.)
2
2
 ABC  CA2 ( BC )  CAB
 CAC
3-tangle:
r
AB
Q
C
2
A( BC )
r
XY
Q
 C ( BC )  C
2
 C (  XY )
2
2
AB( C )
Deriving quantum discord via
entanglement of formation
States of a 4  2 systems with no more than two nonzero eigenvalues:
 ABC
p1  p2 
1
2
(4  2  2)
(
: Bell-diagonal state)
Deriving quantum discord via
entanglement of formation (cont.)
S (  A )    i l og2 i
i
S (  AB )  1
E (  BC )   x l og2 x  (1  x ) l og2 (1  x )
1 2
1, 2  k1 (1  cos1 )
2
1
3, 4  k02 (1  cos 2 )
2
1
(1  1  C 2 )
2
C  m ax{2m   i ,0}
x
i
1
2
k12  ( 0, 13 ); ( 13 , 12 ); ( 12 ,1)
cos1  1, cos 2 
Deriving entanglement of formation
via quantum discord
Analytical expression for Eof
other than two-qubit systems!
cos1  1, cos 2 
k12  (0, 15 ); ( 15 ,1)
1
2
Comparison: Amount of Qd and Eof
C
Trilateral relation
QAB  E (  AB )  QAC  QCA
QCB  E ( CB )  QCA  QAC
A
B
Result 1:
Result 2:
QAB  E (  AB )  QCB  E ( CB )
QAB  QCB  E(  AB )  E( CB )
or
QAB  E (  AB )  QCB  E ( CB )
Fanchini etc., arXiv:10062460
Applications: dynamics under decoherence
Initial state:
Evolution under a phase-damping process
Experimentally realizable
via optical systems
J.-S. Xu, etc., Nat. Commun. 1:7 doi:
10.1038/ncomms1005 (2010).
Entanglement of formaiton:
Quantum discord:
Eof
Eof is always larger
than Qd in the specified dynamical process
Summary
 The intrinsic relation is revealed between quantum discord
and entanglement of formation in unified purifications
 Quantification of quantum discord for the n  2 systems
with rank two is obtained
 Analytical expression of Eof for a sort of mixed states of
4  2 systems is achieved
 Application to describe dynamical behavior of quantum
correlations of physical systems under decoherence
Acknowledgement
XinQi Li
(Beijing Normal Univ.)
JiuShu Shao
(Beijing Normal Univ.)
YiJing Yan
(HKUST, Hong Kong)
Dr. JianWei Xu
(Sichuan Univ., Chengdu)
Thanks!