OPEN
SUBJECT AREAS:
APPLIED MATHEMATICS
Quantum correlation exists in any
non-product state
Yu Guo1 & Shengjun Wu2,3
QUANTUM INFORMATION
1
Received
29 September 2014
Accepted
5 November 2014
Published
1 December 2014
Correspondence and
requests for materials
should be addressed to
Y.G. (guoyu3@aliyun.
com)
School of Mathematics and Computer Science, Shanxi Datong University, Datong, Shanxi 037009, China, 2Kuang Yaming
Honors School, Nanjing University, Nanjing, Jiangsu 210093, China, 3Department of Modern Physics and the Collaborative
Innovation Center for Quantum Information and Quantum Frontiers, University of Science and Technology of China, Hefei, Anhui
230026, China.
Simultaneous existence of correlation in complementary bases is a fundamental feature of quantum
correlation, and we show that this characteristic is present in any non-product bipartite state. We propose a
measure via mutually unbiased bases to study this feature of quantum correlation, and compare it with other
measures of quantum correlation for several families of bipartite states.
Q
uantum systems can be correlated in ways inaccessible to classical objects. This quantum feature of
correlations not only is the key to our understanding of quantum world, but also is essential for the
powerful applications of quantum information and quantum computation1–19. In order to characterize
the correlation in quantum state, many approaches have been proposed to reveal different aspects of quantum
correlation, such as the various measures of entanglement6–10 and the various measures of discord and related
measures17–21, etc. It is believed that some aspects of quantum correlation could still exist without the presence of
entanglement and these aspects could be revealed via local measurements with respect to some basis of a local system.
The simultaneous existence of complementary correlations in different bases is revealed very early by the Bell’s
inequalities22. Bell’s inequalities quantify quantum correlation via expectation values of local complementary
observables. In Ref. 23, the feature of genuine quantum correlation is revealed by defining measures based on
invariance under a basis change: for a bipartite quantum state, the classical correlation is the maximal correlation
present in a certain optimum basis, while the quantum correlation is characterized as a series of residual
correlations in the bases mutually unbiased (MU) to the optimum basis. In this paper, we use the fact that the
essential feature of the quantum correlation is that it can be present in any two mutually unbiased bases (MUBs)
simultaneously. Thus, one of the two bases is not necessarily the optimum basis to reveal the maximal classical
correlation in this paper. With respect to the measure proposed here, we shall show that only the product states do
not contain quantum correlation. A product state contains neither any quantum correlation nor any classical
correlation; while any non-product bipartite state contains correlation that is fundamentally quantum! We shall
also reveal interesting properties of this measure by comparing this measure to other measures of quantum
correlation for several families of bipartite states.
The MUBs constitute now a basic ingredient in many applications of quantum information processing:
quantum state tomography24, quantum cryptography25, discrete Wigner function26, quantum teleportation27,
quantum error correction codes28, and the mean king’s problem29. Two orthonormal bases {jyiæ} and {jwjæ} of
a d-dimensional Hilbert space H are said to be mutually unbiased if and only if
D E
1
ð1Þ
yi wj ~ pffiffiffi , V 1 ƒ i, jƒd:
d
In a d-dimensional Hilbert space, there exist at least 3 MUBs (when d is a power of a prime number, a full set of d
1 1 MUBs exists, more details can be found in Ref. 30).
We recall the quantity defined in Ref. 23. Let Hab 5 Ha fl Hb with dim Ha 5 da and dim Hb 5 db be the state
space of the bipartite system A1B shared by Alice and Bob. Let {jiæ} and jj9æ be the orthonormal bases of Ha and
Hb respectively. Alice selects a basis {jiæ} of Ha and performs a measurement projecting her system onto the basis
states. The Holevo quantity x{rabj{jiæ}} of rab with respect
to Alice’s
local projective measurement onto the basis
X
X
b
b
{jiæÆij}, is defined as xfrab jfjiigg~x pi ; ri :S
pr {
p S rbi . A basis {jiæ} that achieves the
i i i
i i
maximum (denoted as C1(rab)) of the Holevo quantity is called a C1-basis of rab. There could exist many C1-
SCIENTIFIC REPORTS | 4 : 7179 | DOI: 10.1038/srep07179
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bases for a state rab, and the set of these bases is denoted as Crab . Let
VPa be the set of all bases that are mutually unbiased to Pa,
Pa [ Crab . The quantity of quantum correlation in Ref. 23, denoted
by Q2(rab), is defined as
a
~ :
ð2Þ
Q2 ðrab Þ: amax max x rab P
~ a [ VPa
P [ Crab P
In other words, Q2 is defined as the Holevo quantity of Bob’s accessible information about Alice’s results, maximized over Alice’s
projective measurements in the bases that are mutually unbiased to
a C1-basis Crab , and further maximized over all possible C1-bases (if
not unique). Thus, Q2 is actually the maximum correlation present
simultaneously in any set of two MUBs of which one is a C1-basis.
Results
Our approach - Correlation measure based on MUBs. We now
present our approach in a more general way.
Definition. Let D denote the set of all two-MUB sets, i.e.,
D~fffji1 ig,fjj2 igg : fji1 ig is MU to fjj2 igg:
We define
C2 ðrab Þ:
max min x rab Pa1 ,x rab Pa2 :
ð
Þ[D
Pa1 ,Pa2
ð3Þ
ð4Þ
The quantity C2 represents the maximal amount of correlation that is
present simultaneously in any two MUBs. Similar to the other usual
measures of quantum
correlation, C2 is local unitary invariant, that is,
C2 ðrab Þ~C2 Ua 6Ub rab Ua{ 6Ub{ for any unitary operators Ua and
Ub acting on Ha and Hb respectively.
C2 versus Q2 . Although both C2 and Q2 represent quantum
correlation (here the symbol C2 is associated with Correlation in
two MUBs) instead of classical correlation (represented by C1),
they are actually quite different. As C2 is defined as the maximum
correlation present simultaneously in any two MUBs while Q2 is the
maximum correlation present simultaneously in two MUBs of which
one is a C1-basis, it is obvious that
C2 ðrab Þ§Q2 ðrab Þ
ð5Þ
for any rab. In a sense, C2 is more essential than Q2 since the
maximum in the former one is taken over arbitrarily two MUBs.
Thus, C2 may reveal more quantum correlation than Q2.
From the following Theorem and Examples, one knows that there
do exist states such that C2 wQ2 (Example 4) and C2 ~Q2 (Examples
1,2,3). A clear illustration of the difference between C2 and Q2 is also
given in Fig. 3. We know that Q2 does not exceed quantum discord D
for all the known examples23. However, one can easily find states such
that C2 exceeds quantum discord D (See Fig. 3).
The nullity of C2 . Now we show that any bipartite quantum state
contains nonzero correlation simultaneously in two mutually
unbiased bases unless it is a product state, this result is stated as
the following theorem, while the proof is left to the Method.
Theorem. C2 ðrab Þ~0 if and only if rab is a product state.
In a sense, this theorem implies that, any non-product bipartite
state contains genuine quantum correlation, and C2 reveals the
amount of quantum correlation in the state. In addition, we know
that C2 is different from the quantity Q2 in Ref. 23 since Q2(rcq) 5 0
for any classical-quantum state rcq while C2 ~0 only for product
states. The difference between the measure C2 and other measures
of quantum correlation shall be discussed below for several families
of bipartite states in more details.
Now, we shall calculate the quantity for several families of bipartite
states, and see how our measure in terms of MUBs is well justified as a
measure of quantum correlation.
Example 1 - Pure states.
Forpaffiffiffiffibipartite pure state with the Schmidt
X
li jai ijbi i, C2 ~Q2 ~Sðra Þ~Sðrb Þ~
decomposition jyi~
i
X
{l
log
l
.
It
can
be
easily
checked that C2 coincides with the
i
2 i
i
entropy of either reduced state for any pure state, which is also the
usual measure of entanglement in a pure state.
Example 2 - Werner states. Next, we consider the Werner states of a
d fl d dimensional system5,
1
ðI{aPÞ,
ð6Þ
rw ~
dðd{aÞ
where 21 # a # 1, I is the identity operator in the d2-dimensional
Xd
jiih jj6j jihij is the operator that
Hilbert space, and P~
i,j~1
exchanges A and B. For a local measurement with respect to basis
1
states {jeiæ} of Ha, with probability pi ~ , Alice will obtain the k-th
d
basis state jekæ, and Bob will be left with the state
X
1
a jj’i with akj 5
ðI{aje’k ihe’k jÞ, where je’k i~
rbk ~
j kj
d{a
Æekjjæ. It is straightforward to show that
d
1{a
z
log ð1{aÞ~Q2 ðrw Þ:ð7Þ
C2 ðrw Þ~x pi ; rBi ~log2
d{a
d{a 2
The entanglement
formation Ef for the Werner
0 0 ofs
11states is given as
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1
da{1
AA, with h(x) ; 2x
Ef ðrw Þ~h@ @1z 1{ max 0,
2
d{a
log2 x 2 (1 2 x) log2(1 2 x)31. The three different measures of
quantum correlation, i.e., C2 , the quantum discord D and the
entanglement of formation Ef, are illustrated in Fig. 1 for
comparison. From this figure, we see that the curve for
Figure 1 | Measures of quantum correlation for the Werner states as functions of a when d 5 2 (left) and d 5 3 (right). The red curve represents our
measure C2 , the green curve represents the quantum discord D and the blue curve represents the entanglement of formation Ef.
SCIENTIFIC REPORTS | 4 : 7179 | DOI: 10.1038/srep07179
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Figure 2 | Measures of quantum correlation for the isotropic states as functions of b when d 5 2 (left) and d 5 3 (right). The red curve represents our
measure C2 , the blue curve represents the quantum discord D and the green curve represents the entanglement of formation Ef.
entanglement of formation intersects the other two curves; thus, Ef
can be larger or smaller than C2 .
Example 3 - Isotropic states. For the d fl d isotropic states
1 r~ 2
ð8Þ
ð1{bÞIz d 2 b{1 Pz , b [ ½0,1,
d {1
1 X
where P1 5 jW1æ ÆW1j, jWz j~ pffiffiffi
jiiji’i is the maximally
i
d
entangled pure state in Cd 6Cd . Let {jekæ Æekj} be an arbitrarily
given projective measurement on Alice’s part. Bob’s state after
after Alice gets the k-th measurement result is
1 rbk ~ 2
ð9Þ
d ð1{bÞIz d 2 b{1 je’k ihe’k j ,
d {1
X
where je’k i~
a jj’i with akj 5 Æekjjæ. As the eigenvalues of rbk
j kj
does not depend on the basis for Alice’s measurement, one can easily
show that
dbz1
dbz1
log2
dz1
dz1
d{db
d{db
z
log2 2
:
dz1
d {1
C2 ðrÞ~Q2 ðrÞ~log2 dz
ð10Þ
The entanglement of formation Ef for the isotropic states is given
as32,33
8
0,
bƒ d1 ,
>
>
>
<
4ðd{1Þ
1
d vbv d 2 ,
Ef ðrÞ~ hðcÞzð1{cÞlog2 ðd{1Þ,
ð11Þ
>
>
ð
b{1
Þd
log
ð
d{1
Þ
>
4
ð
d{1
Þ
2
:
zlog2 d,
d 2 ƒbƒ1,
d{2
1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
bz ðd{1Þð1{bÞ . The quantum discord of the
d
isotropic state is34
where c~
DðrÞ~b log2 bz
1{b
1{b
log
dz1 2 d 2 {1
1{b{ d1 zdb
1zdb
{
:
log2
dz1
d 2 {1
ð12Þ
The three different measures of quantum correlation, i.e., C2 , the
quantum discord D and the entanglement of formation Ef, are
illustrated in Fig. 2 for comparison. From this figure, we see that
the curve for entanglement of formation intersects the other two
curves; thus, Ef can be larger or smaller than C2 .
Example 4 - A family of two-qubit states. As the last example, we
consider a family of two-qubit states that are Bell-diagonal states.
This family of states admit the form
!
3
X
1
sab ~
rj sj 6sj :
I2 6I2 z
ð13Þ
4
j~1
We rearrange the three numbers {r1, r2, r3} according to their
absolute values and denote the rearranged set as {r1 , r2 , r3 } such
that jr1 j§jr2 j§jr 3 j. Next we show that
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi1
1z r 21 zr22 =2
A:
C2 ðsab Þ~1{h@
ð14Þ
2
Without loss of generality, we prove (14) only for the case r1 §r2 §r3 .
A projective measurement performed on qubit A can be written as
1
a
P+
~ ðI2 +~
n:~
sÞ, parameterized by the unit vector ~
n. When Alice
2
1 p 1{p { {
Figure 3 | Different measures of quantum correlation for two special classes of states: r1 ~ yz yz z wz wz z
jw ihw j (left) and
2
2
2
1{p z z z z {
{
y
(right). In each figure, the red curve represents our measure C2 , the green curve represents the
y zw
w
r2 ~pjy ihy jz
2
quantum discord D, the blue curve represents the measure Q2, and the dashed orange curve represents the entanglement of formation Ef.
SCIENTIFIC REPORTS | 4 : 7179 | DOI: 10.1038/srep07179
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1
obtains p6, Bob will be in the corresponding states rb+ ~ ðI2 +
2
P
1
.
The
entropy
S rb+
n
r
s
Þ,
each
occurring
with
probability
j j j j
2
1zjr1 j
reaches its minimum value h
when ~
n~ð1,0,0Þ. Let
2
ð1Þ
~
n2 ~ða,b,0Þ with ax 1 by 5 0, then P+ is
n1 ~ðx,y,0Þ and ~
1
1
ð2Þ
ð1Þ
ð2Þ
mutually unbiased to P+ , where P+ ~ ðI2 +~
sÞ, P+ ~ ðI2 +
n1 :~
2
2 !
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n o
2 r 2 zy2 r 2
1z
x
ð
1
Þ
1
2
~
n2 :~
sÞ. It is immediate that x sab P+ ~1{h
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
n o
1z a2 r12 zb2 r22
ð2Þ
. Thus C2 fsab g~1{
and x sab P+ ~1{h
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
1z ðr12 zr22 Þ=2
as desired since h(c) is a monotonic
h
2
1
decreasing function when c§ . Our quantity C2 is compared with
2
the quantum discord D and the entanglement of formation Ef for r1
and r2 in Fig. 3.
From the left figure of Fig. 3, it is clear that C2 is quite different
from both D and Q2. Unlike Q2 that does not exceed D for all known
examples, C2 can exceed D. We have C2 ðr1 ÞvDðr1 Þ when p is closed
1
to , while C2 ðr1 ÞwDðr1 Þ when p is closed to 0 or 1; we also have
2
1
C2 ðr1 Þ~Q2 ðr1 Þ when p~ , and C2 ðr1 Þ increases monotonously
2
1
while Q2(r1) decreases monotonously when p deviates from . In
2
Fig. 3, the difference between our measure C2 and the other measures
is well illustrated by the extreme cases when p 5 0 or 1 in the left
figure and when p 5 0 in the right figure. For example, for
1 1 s~ yz yz z wz wz , our measure has a finite value
2
2
while the other measures vanish.
Correlation revealed via more MUBs. In addition, we can define a
quantity based on m MUBs (3 # m # dim Ha 1 1), namely,
max
min x rab Pa1 ,
Cm ðrab Þ:
a
a
a
ðP1 ,P2 ,,Pm Þ [ Dm
ð15Þ
a
a x rab P2 , ,x rab Pm :
where
Dm ~ Pa1 ,Pa2 , ,Pam :
Pak is MU to Pal f or any
k=l :
ð16Þ
It is clear that Ckz1 ƒCk ƒC2 . The following are obvious from the
arguments in the previous examples: i) Cm ðrÞ~0 if and only if r is a
isotropic
product state, ii) Cm ~C2 for both the Werner states and the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
2
2
2
1z ðr1 þ r2 þ r3 Þ=3
states, and iii) C3 ðsab Þ~1{h
for the
2
family of two-qubit states in Eq. (13).
Discussion
We have provided a very different approach to quantify quantum
correlation in a bipartite quantum state. Our approach captures the
essential feature of quantum correlation: the simultaneous existence
of correlations in complementary bases. We have proved that the
only states that don’t have this feature are the product states, which
contains no correlation (classical or quantum) at all. Thus, any nonproduct state contains correlation that is fundamentally quantum.
This feature of quantum correlation characterized here could be the
key feature that enables quantum key distribution (QKD) with
SCIENTIFIC REPORTS | 4 : 7179 | DOI: 10.1038/srep07179
entangled states, since the quantum correlation that exists simultaneously in MUBs, which can be quantified by Cm , is the resource for
entanglement-based QKD via MUBs.
Method
Proof of the Theorem in the Main Text. The ‘if’ part is obvious, and we only need to
show the
we only need to prove
if’ part. In other
words,
‘only
athata rab 5 ra fl rb if
for
any
MUB
pair
P ,P [ D. It is
either x rab Pa1 ~0 or x rab Pa2 ~0
1 2
equivalent to show that both x rab Pa1 =0 and x rab Pa2 =0 for a certain MUB
a a
pair P1 ,P2 [ D if rab is not a product state.
We assume that rab is not a product state, then the maximal classical correlation is
nonzero, i.e., C1(rab) ? 0. Let fjei ig [ Crab , we have x(rabj{jeiæ}) ? 0. Therefore, we
only need to find a second basis (MU to {jeiæ}) such that the corresponding Holevo
quantity is nonzero. We denote the projective measurement corresponding to {jeiæ} by
P~fPk ~X
jek ihek jg. Then
X ð1Þ
bð1Þ
Pk 6Ib rab Pk 6Ib ~
p jek ihek j6rk . As C1(rab) ? 0, we
P ðrab Þ~
k
k k
bð1Þ
bð1Þ
know that rk0 =rb and rl0 =rb at least for some k0 and l0. We arbitrarily choose a
basis {jfiæ} that is MU to {jeiæ}. If x{rabj{jfiæ}} ? 0, then we already obtain the second
basis and the theorem is true.
If x{rabj{jfiæ}} 5 0, we can construct the MUB pair as follows. As in this case, the
measurement corresponding to {jfiæ} yields the following output state
1
0 ð2Þ
p 1 rb
0
...
0
C
B
ð
2
Þ
B 0
p 2 rb . . .
0 C
C
B
:
ð17Þ
B .
..
.. C
C
B .
@ .
.
P
. A
0
Thus, rab can be represented as
0 ð2Þ
p1 rb
B
B B
B .
B .
@ .
ð2Þ
0
. . . p d rb
...
ð2Þ
...
..
.
p2 rb
..
.
P
ð2Þ
1
C
C
C
C
C
A
ð18Þ
. . . p d rb
with respect to the local basis {jfiæ}, and at least one of the off-diagonal blocks is not
zero (otherwise, rab is a product state). Without loss of generality we assume that the
(1,2)-block-entry of the above matrix is nonzero. It follows that there exists a 2 by 2
unitary matrix U2, such that, under the local basis {U2 › Id22jfiæ}, the state admits the
form
1
0 ð2Þ
q1 %b
...
C
B
ð2Þ
B q2 sb . . .
C
C
B
ð19Þ
B .
..
.. C
C
B .
@ .
.
P
. A
...
ð2Þ
pd rb
with %b =rb and sb ? rb. That is x{rabj{U2 › Id22jfiæ}} ? 0. This unitary matrix U2
can be chosen as
!
pffiffiffiffiffiffiffiffiffiffiffi
1{ 2
pffiffiffiffiffiffiffiffiffiffiffi :
U2 ~
ð20Þ
{
1{ 2
with a very small positive number. Even though x{rabj{U2 › Id22jfiæ}} could be very
small, it is nonzero. As is a very small and x{rabj{jeiæ}} ? 0, we also have x{rabj{U2 ›
Id22jeiæ}} ? 0. Thus, the Holevo quantity is nonzero at least for a certain MUB pair
(i.e., {U2 › Id22jeiæ} and {U2 › Id22jfiæ}), and therefore C2 ðrab Þ=0.
Thus, C2 ðrab Þ=0 for any rab that is not a product state. The proof is completed.
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Acknowledgments
Y. Guo acknowledges support from the Natural Science Foundation of China (No.
11301312, No. 11171249), the Natural Science Foundation of Shanxi (No. 2013021001-1,
No. 2012011001-2) and the Research start-up fund for Doctors of Shanxi Datong University
(No. 2011-B-01). S. Wu acknowledges support from the Natural Science Foundation of
China (No. 11275181) and the Fundamental Research Funds for the Central Universities
(No. 20620140007).
Author contributions
Y.G. and S.W. contributed equally to the paper.
Additional information
Competing financial interests: The authors declare no competing financial interests.
How to cite this article: Guo, Y. & Wu, S. Quantum correlation exists in any non-product
state. Sci. Rep. 4, 7179; DOI:10.1038/srep07179 (2014).
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