Partial correlations in multipartite quantum systems

Information Sciences 289 (2014) 262–272

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Partial correlations in multipartite quantum systems Zhihua Guo a, Huaixin Cao a,⇑, Shixian Qu b a b

College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China

a r t i c l e

i n f o

Article history: Received 20 June 2014 Received in revised form 21 July 2014 Accepted 8 August 2014 Available online 19 August 2014 Keywords: Partial correlation Multipartite quantum system Measure

a b s t r a c t Quantum correlations are more general than entanglement. A problem on how to characterize and quantify quantum correlations has attracted substantial attention. Quantum correlations in bipartite systems have been researched extensively. In this paper, partial correlations in a multipartite quantum system a1 a2 . . . an are discussed in detail. In order to reveal correlations of a given state with respect to some subsystems, for a nonempty subset D of the index set f1; 2; . . . ; ng, D-classical correlations (D-CC), D-quantum correlations (D-QC), partially classical correlations (PCC) and genuinely quantum correlations (GQC) are introduced in a multipartite system. Subsequently, characterizations of D-CC are obtained. Secondly, a measure function GD ðqÞ of partial correlations of a state q is defined and called the D-quantum discord. It is proved that GD ðqÞ is nonnegative, independent of the choice of basis and invariant under local unitary transformations. It is also proved that a state q is D-CC if and only if GD ðqÞ ¼ 0, and then it is D-QC if and only if GD ðqÞ > 0. Moreover, some relationships among partial correlations with respect to different D are discussed in light of the measure function. Finally, an illustrative example with 6-qubit GHZ state is given. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction The quantum features of a composite quantum system are usually described by entanglement [25,12], or quantum correlations [14] of states of the system, while it has been observed that quantum correlations are more general than entanglement in a bipartite system. In a bipartite system, quantum correlations have been widely accepted in some quantum computing tasks without entanglement [13]. Hereafter, characterization and quantification of quantum correlations have received much more attention [14,26,15,29,8,6,7]. In order to quantify quantum correlations, the quantum discord (QD) was introduced by Ollivier and Zurek [24], which was used in the detection of quantum correlations [16,34] and the study of quantum key distribution [28], continuous variable systems [5], three-spin XXZ chain [33], and the two-qubit composite system subject to a common finite-temperature reservoir [32]. In a multipartite quantum system, the classification of states based on entanglement is much richer than in the bipartite case. Indeed, in multipartite quantum systems, apart from fully separable states and fully entangled states, there also exist partially separable states [11,20,9,21]. Similar to the quantum discord, the multiple entropy measure has been proposed in [17] where the averaged von-Neumann entropies of a subset of constituent particles are used to measure the quantum ⇑ Corresponding author. E-mail addresses: [email protected] (Z. Guo), [email protected] (H. Cao), [email protected] (S. Qu). http://dx.doi.org/10.1016/j.ins.2014.08.029 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.

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entanglement of the many-body system. This measure has been applied to the discussions about extremal entangled fourqubit pure states [18] and entanglement of linear cluster states [2]. Furthermore, attention is also being paid to the measurement of quantum correlations in multipartite systems [27,1,30,22]. Explicitly, Rulli and Sarandy [27] introduced a global measure for quantum correlations in multipartite systems, which is obtained by suitably recasting the quantum discord in terms of relative entropy and local von Neumann measurements. Xu [30] gave analytical expressions of global quantum discord induced by [27] for two classes of multi-qubit states. Ma et al. [22] proposed a measure of quantum correlations for a multipartite system, which is defined as the sum of the correlations for all possible partitions. Bai et al. [1] explored multipartite quantum correlations with the square of quantum discord. These previous studies considered only global correlations of multipartite quantum states and were found well applicable to many cases. However, the situation in multipartite systems is even more complicated as there exist states that are not globally classically correlated but have partially classical correlations (this term will be defined later), just as there exist states that are bi-separable with respect to some fixed partition but not fully separable (for some examples, see Ref. [9]). Indeed, it was observed [7] that tripartite states of the system a1 a2 a3 have several types of correlations, including CCC, GQC, CCX, CXC, XCC, CXX, XCX and XXC-states. For example,

q¼

X  a ED a  pk1 ak11 ak11   rak12 a3 k1

is a CXX-state where fjaak11 ig is an orthonormal basis for the Hilbert space of the system a1 and rak12 a3 ’s are states of the system a2 a3 . If there exists a rank-one orthogonal projective channel U3 on the subsystem a3 such that ðid2  U3 Þðrak12 a3 Þ ¼ rak12 a3 for all possible k1 , then q becomes a CXC-state:

q¼

X k1 ;k3

 ED   ED      pk1 ;k3 aak11 aak11   rak12 ;k3  aak33 aak33 ;

n o where jaak33 i is an orthonormal basis for the Hilbert space of the system a3 and rak12 ;k3 ’s are states of the system a2 . If there exists a rank-one orthogonal projective channel U2 on the subsystem a2 such that ðU2  id3 Þðrak12 a3 Þ ¼ rak12 a3 for all possible k1 , then q becomes a CCX-state:

q¼

X k1 ;k2

 ED   ED      pk1 ;k2 aak11 aak11   aak22 aak22   rak13 ;k2 :

Furthermore, if ½rak13 ;k2 ; ra‘13;‘2  ¼ 0 for all possible k1 ; k2 ; ‘1 ; ‘2 , then q becomes a CCC-state:

q¼

X

k1 ;k2 ;k3

 ED   ED   ED        pk1 ;k2 ;k3 aak11 aak11   aak22 aak22   aak33 aak33 :

Motivated by the previous studies, the aim of this paper is to discuss partial correlations in a multipartite quantum system. The paper is structured as follows. In Section 2, we introduce the concepts of D-classical correlations, D-quantum correlations, partially classical correlations and genuinely quantum correlations in a multipartite system and explore related characterizations of these correlations. In Section 3, we present a measure of D-quantum correlations by using mutual information of multipartite quantum states. An illustrative example with the 6-qubit GHZ state is given. 2. Partial correlations in multipartite quantum systems In quantum information, a quantum system a is described by a complex Hilbert space H and states of the system a are described by unit vectors in H, called pure states, or density operators (positive operators of trace 1 in BðHÞ (the set of all bounded linear operators on H)), called mixed states, simply, states. The set of all states of a is denoted by DðHÞ. The elements of H are denoted by dirac notations jxi and the (right-linear) inner product of two vectors jxi and jyi in H is written as hxjyi. Also, the notation jxihyj means the operator that maps a vector jzi to the vector hyjzijxi. A quantum channel on the system a is described by a completely positive trace-preserving map on the C  -algebra BðHÞ. It was proved by Choi [4] that every quanP tum channel U on a finite dimensional system a is of the form: UðqÞ ¼ i M i qM yi , where M i ’s are called Kraus operators of U P y and satisfy the trace-preserving constraint i M i M i ¼ I, hereafter X y denotes the Hermitian adjoint of an operator X. Let a1 ; a2 ; . . . ; an be n quantum systems with state spaces H1 ; H2 ; . . . ; Hn whose dimensions are d1 ; d2 ; . . . ; dn , respectively. Denote by Ok the set of all orthonormal bases for Hk . For every ek ¼ fjek1 i; jek2 i; . . . ; jekdk ig 2 Ok , we define dk rank-one projeck

tions Pj ðek Þ ¼ jekj ihekj j ðj ¼ 1; 2; . . . ; dk Þ and then obtain a von Neumann measurement Pe ¼ fPj ðek Þ : j ¼ 1; 2; . . . ; dk g, which induces a quantum channel on the system ak as follows: k

Pe ðXÞ ¼

dk X

Pj ðek ÞX Pj ðek Þ;

8X 2 BðHk Þ;

ð2:1Þ

j¼1 k

k

it is called a projective channel on the system ak . Clearly, for every unitary operator U k on Hk , we have U k Pe U yk ¼ PUk e , which is also a projective channel on the system ak , where U k ek :¼ fU k jek1 i; U k jek2 i; . . . ; U k jekdk ig.

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In what follows, we consider the composite system a :¼ a1 a2 . . . an with state space H :¼ H1  H2  . . .  Hn and use D to denote the set of all labels i of the subsystems ai what we concerned with. Equivalently, D is a nonempty subset of the set X ¼ f1; 2; . . . ; ng. Definition 2.1. (1) A quantum channel U on a is said to be a D-channel if for each k 2 D there exists an orthonormal basis ek for Hk such k that U ¼ U1  U2  . . .  Un with Uk ¼ Pe when k 2 D and Uk ¼ idHk (the identity map on BðHk Þ) when c k 2 D :¼ X n D. (2) A state q 2 DðHÞ is said to be D-classically correlated (D-CC) if there exists a D-channel U on a such that UðqÞ ¼ q and q is said to be D-quantum correlated (D-QC) if it is not D-CC. Especially, a state q 2 DðHÞ is said to be globally classically correlated (GCC) if it is X-CC. (3) A state q 2 DðHÞ is said to be partially classically correlated (PCC) if it is D-CC for some D  X and q is said to be genuinely quantum correlated (GQC) if it is not PCC. From this definition, we see that for a bipartite system (where n ¼ 2), the f1g-CC, f2g-CC, and f1; 2g-CC states are just the classical–quantum (CQ), quantum–classical (QC) and classical–classical (CC) states [14], respectively. For a tripartite system (where n ¼ 3), a state qABC is CXX (resp. XCC, XCX) if and only if it is f1g-CC (resp. f2; 3g-CC, f2g-CC) [7]. The following theorem offers a general form of D-CC states. Theorem 2.1. Let D ¼ f1; 2; . . . ; mgðm < nÞ. Then an n-partite state q is D-CC if and only if there exist orthonormal bases ek ¼ fjek1 i; jek2 i; . . . ; jekdk ig 2 Ok ðk ¼ 1; 2; . . . ; mÞ and positive operators gj1 j2 ...jm 2 BðHmþ1  . . .  Hn Þ such that

X

q¼

m je1j1 ihe1j1 j  je2j2 ihe2j2 j  . . .  jem jm ihejm j  gj1 j2 ...jm ;

ð2:2Þ

j1 ;j2 ;...;jm

where 1 6 jk 6 dk . Proof. Necessity. Suppose that q is D-CC. Then it follows from Definition 2.1 that there exists a D-channel U such that

UðqÞ ¼ q, where 1

m

U ¼ Pe  . . .  Pe  idHmþ1  . . .  idHn : By writing q ¼

P

1 iXi

 X 2i  . . .  X ni , we have

q ¼ UðqÞ ¼

X

1

m

mþ1 Pe ðX 1i Þ  . . .  Pe ðX m  . . .  X ni i Þ  Xi

i

! d1 X X 1 1 1 1 1 ¼ jej1 ihej1 jX i jej1 ihej1 j  . . .  i

X

¼

j1 ¼1

dm X

! m m m m jem jm ihejm jX i jejm ihejm j

 X mþ1  . . .  X ni i

jm ¼1

m je1j1 ihe1j1 j  je2j2 ihe2j2 j  . . .  jem jm ihejm j  gj1 j2 ...jm ;

j1 ;j2 ;...;jm

where

gj1 j2 ...jm ¼

X

m m mþ1 he1j1 jX 1i je1j1 i . . . hem  . . .  X ni : jm jX i jejm iX i

i

For all 1 6 jk 6 dk ðk ¼ 1; 2; . . . ; mÞ and jwi 2 Hmþ1  . . .  Hn , put jui ¼ je1j1 i  je2j2 i  . . .  jem Then jm i  jwi. 0 6 hujqjui ¼ hwjgj1 j2 ...jm jwi. This shows that gj1 j2 ...jm are positive. k Sufficiency. Suppose that (2.2) holds. For 1 6 k 6 m, we let Uk ¼ Pe , which is the quantum channel with Kraus operators Pjk ðek Þ ¼ jekj ihekj j ðjk ¼ 1; 2; . . . ; dk Þ; and when m þ 1 6 k 6 n, take Uk ¼ idHk . Then U ¼ nk¼1 Uk is a D-channel satisfying k

k

UðqÞ ¼ q. By Definition 2.1, q is D-CC. h From the proof of Theorem 2.1, we can easily obtain a general form of GCC states as follows. Theorem 2.2. An n-partite state q 2 DðHÞ is ek ¼ fjek1 i; jek2 i; . . . ; jekdk ig 2 Ok ðk ¼ 1; 2; . . . ; nÞ such that

q¼

X

j1 ;j2 ;...;jn

je1j1 ihe1j1 j  je2j2 ihe2j2 j  . . .  jenjn ihenjn j:

GCC

if

and

only

if

there

exist

orthonormal

bases

ð2:3Þ

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Now let us discuss a general form of general D-CC states. To this end, we let D ¼ fi1 ; i2 ; . . . ; im g be any nonempty proper subset of X with 1 6 i1 < . . . < im 6 n and put Dc ¼ f‘1 ; ‘2 ; . . . ; ‘nm g with 1 6 ‘1 < ‘2 < . . . < ‘nm 6 n, and HD ¼ ns¼1 Ks , where Ks ¼ His ð1 6 s 6 mÞ and Kmþs ¼ H‘s ð1 6 s 6 n  mÞ. Then we can define a linear isomorphism UD : BðHÞ ! BðHD Þ as

UD ðX 1  X 2  . . .  X n Þ ¼ Y 1  Y 2  . . .  Y n ; where Y s ¼ X is ð1 6 s 6 mÞ; Y mþs ¼ X ‘s ð1 6 s 6 n  mÞ. Clearly, UD is a completely positive trace-preserving isomorphism. With these notations, we have the following theorem.

Theorem 2.3. Let q 2 DðHÞ. Then the following are equivalent. (1) q is D-CC. (2) UD ðqÞ is f1; 2; . . . ; mg-CC. (3) There exist an orthonormal basis ek ¼ ðek1 ; ek2 ; . . . ; ekdk Þ 2 Ok ðk ¼ 1; 2; . . . ; mÞ and positive operators gj1 j2 ...jm BðHmþ1  . . .  Hn Þ such that

q ¼ ðUD Þ

X

1

in

! m je1j1 ihe1j1 j  . . .  jem jm ihejm j  gj1 j2 ...jm :

ð2:4Þ

j1 ;j2 ;...;jm

Equivalently,

UD ðqÞ ¼

X

m je1j1 ihe1j1 j  . . .  jem jm ihejm j  gj1 j2 ...jm :

j1 ;j2 ;...;jm

Proof

ð1Þ () ð2Þ: Let U ¼ U1  U2  . . .  Un be a local quantum channel. If U is a D-channel, then it can be written as Uk ¼ Pe when k 2 D and Uk ¼ idHk when k 2 Dc . Clearly, i1

i2

k

im

UD  U ¼ Pe  Pe  . . .  Pe  idH‘1  idH‘2  . . .  idH‘nm ; which is a f1; 2; . . . ; mg-channel. Conversely, if UD  U is a f1; 2; . . . ; mg-channel, then we note that U is a D-channel since UD is invertible. Combining this observation with the fact that UD ðUðqÞÞ ¼ ðUD  UÞðUD ðqÞÞ, we see that q is D-CC if and only if UD ðqÞ is f1; 2; . . . ; mg-CC. ð2Þ () ð3Þ: We use Theorem 2.1 and the equivalence of ð1Þ and ð2Þ. h It is easy to obtain the following results from Theorem 2.3. Corollary 2.1. Let m  2 and q be a D-CC state. Then trDc ðqÞ 2 DðHi1  Hi2  . . .  Him Þ is GCC.

3. A measure of partial correlations of a multipartite state In this section, we extend quantum discord to define a measure of D-quantum correlations of a multipartite state. For any n-partite states q; r 2 DðHÞ, we use SðqÞ and SðqkrÞ to denote the von Neumann entropy of q and the quantum relative entropy of q with respect to r, respectively. Namely,

SðqÞ ¼ trðq log qÞ; SðqkrÞ ¼ trðq log qÞ  trðq log rÞ: P Now, let us define the mutual information of q as IðqÞ ¼ ni¼1 Sðqai Þ  SðqÞ, where qai means the reduced state of q on the subsystem ai , that is, the partial trace of q with respect to the subsystems aj ðj – iÞ. The following lemmas will produce some basic properties. Lemma 3.1. (1) Let U ¼ ni¼1 Ui be a local quantum channel on the system a. Then ðUðqÞÞai ¼ Ui ðqai Þ for any 1 6 i 6 n. P (2) Let UðXÞ ¼ t P t XPt ð8X 2 BðHÞÞ, where fPt g is any projective measurement for the system SðqjjUðqÞÞ ¼ SðUðqÞÞ  SðqÞ, moreover, SðUðqÞÞ P SðqÞ.

a.

Then

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Z. Guo et al. / Information Sciences 289 (2014) 262–272

Proof. (1) We write q ¼

P

X Y

ðUðqÞÞai ¼

j

¼

X Y j

¼ Ui

1 jXj

 X 2j  . . .  X nj . Then UðqÞ ¼

!

j

U1 ðX 1j Þ  U2 ðX 2j Þ  . . .  Un ðX nj Þ, so for any 1 6 i 6 n, we compute

trðUk ðX kj ÞÞ Ui ðX ij Þ

k–i

!

trðX kj Þ

Ui ðX ij Þ

k–i

XY trðX kj ÞX ij j

P

!

k–i

¼ Ui ðqai Þ: (2) By P i UðqÞ ¼ UðqÞP i ¼ P i qP i , we see P i logðUðqÞÞ ¼ logðUðqÞÞP i and

SðqjjUðqÞÞ ¼ SðqÞ  tr½q logðUðqÞÞ ! X Pi q logðUðqÞÞ ¼ SðqÞ  tr i

¼ SðqÞ  tr

X

! Pi q logðUðqÞÞPi

i

¼ SðqÞ  tr

X

! Pi qPi logðUðqÞÞ

i

¼ SðUðqÞÞ  SðqÞ: 

Lemma 3.2. Let q 2 DðHÞ. Then (1) IðqÞ ¼ Sðqjjqa1  qa2  . . .  qan Þ. (2) IðqÞ is non-negative, continuous in q on DðHÞ, and local unitary invariant, i.e.,

IððU 1  U 2  . . .  U n ÞqðU 1  U 2  . . .  U n Þy Þ ¼ IðqÞ; for all unitary operators U k on Hk . Moreover, IðqÞ ¼ 0 if and only if q ¼ qa1  qa2  . . .  qan . (3) IðUðqÞÞ 6 IðqÞ for any quantum channel U on the system a.

Proof. (1) We show this property for a tripartite q 2 DðHA  HB  HC Þ with reduced states qA ; qB and qC . Suppose that the specP P P trum decompositions of qA ; qB and qC are qA ¼ i ki jiihij, qB ¼ j lj jjihjj and qC ¼ k mk jkihkj. Then we can denote

q¼

X

0

00

0

00

0

00

pi0 i00 j0 j00 k0 k00 ji ihi j  jj ihj j  jk ihk j:

0 00 0 00 0 00

ii jj kk

Since

P

jk piijjkk

¼ hijqA jii ¼ ki ;

P

ik piijjkk

¼ hjjqB jji ¼ lj and

P

ij piijjkk

¼ hkjqC jki ¼ mk , we have

SðqjjqA  qB  qC Þ ¼ SðqÞ  tr½q logðqA  qB  qC Þ ¼ SðqÞ 

! X X hijkj q logðkr ls mt Þjrihrj  jsihsj  jtihtj jijki rst

ijk

X ¼ SðqÞ  piijjkk logðki lj mk Þ ijk

X X X ki log ki  lj log lj  mk log mk ¼ SðqÞ  i A

j B

C

¼ Sðq Þ þ Sðq Þ þ Sðq Þ  SðqÞ:

k

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This shows that the first conclusion holds. The others are from the first one. (2) We use conclusion (1) and the related properties of the relative entropy. (3) By using (1) and Lemma 3.1(1) as well as the fact that SðXjjYÞ P SðEðXÞjjEðYÞÞ for any quantum channel E on a, we have

IðUðqÞÞ ¼ SðUðqÞjjni¼1 ðUðqÞÞai Þ ¼ SðUðqÞjjUðni¼1 qai ÞÞ 6 Sðqjjni¼1 qai Þ ¼ IðqÞ:  k

k

For any two D-channels U ¼ U1  . . .  Un and W ¼ W1  . . .  Wn with Uk ¼ Pe , Wk ¼ Pf if k 2 D and Uk ¼ Wk ¼ idHk if i1 im i1 im k k 2 Dc , we write U ¼ Uðe ;...;e Þ ; W ¼ Wðf ;...;f Þ . For each k 2 D, choose a unitary operator U k such that U k ek ¼ f and put U k ¼ Ik c when k 2 D . It can be computed that for all q 2 DðHÞ,

ðU 1  . . .  U n ÞUðqÞðU 1  . . .  U n Þy ¼ WððU 1  . . .  U n ÞqðU 1  . . .  U n Þy Þ: It follows from Lemma 3.2(2) that

IðUðqÞÞ ¼ IðWððU 1  . . .  U n ÞqðU 1  . . .  U n Þy ÞÞ; 8q 2 DðHÞ: Equivalently,

IðWðqÞÞ ¼ IðUððU 1  . . .  U n Þy qðU 1  . . .  U n ÞÞÞ; 8q 2 DðHÞ: Put

UðUi1 ;...;Uim Þ ðqÞ ¼ ðU 1  . . .  U n Þy qðU 1  . . .  U n Þ: Then UðU i1 ;...;Uim Þ is a quantum channel on a with the following property

IðWðqÞÞ ¼ IðUðUðUi1 ;...;Uim Þ ðqÞÞÞ; 8q 2 DðHÞ: k

This shows that for any fixed D-channel U ¼ U1  . . .  Un with Uk ¼ Pe if k 2 D and Uk ¼ idHk if k 2 Dc , we have

fIðWðqÞÞ : W 2 Dchan ðHÞg ¼ fIðUðUðUi1 ;...;Uim Þ ðqÞÞÞ : U ik 2 UðHik Þg;

ð3:1Þ

for all q 2 DðHÞ, where Dchan ðHÞ is the set of all D-channels on a and UðHik Þ stands for the group of all unitary operators on Hik . The continuity of the mutual information and the compactness of UðHi1 Þ . . . UðHim Þ imply that the set on the righthand side of (3.1) has its maximal element. This allows us to define

ID ðqÞ ¼ max IðUðUðUi1 ;...;Uik Þ ðqÞÞÞ ¼ U i 2UðHi Þ k

k

max IðWðqÞÞ:

W2Dchan ðHÞ

ð3:2Þ

We are now at position to extend quantum discord to multipartite systems. For any nonempty proper subset D ¼ fi1 ; i2 ; . . . ; im g of X with 1 6 i1 < . . . < im 6 n, and every n-partite state q 2 DðHÞ, we define

GD ðqÞ :¼ IðqÞ  ID ðqÞ;

ð3:3Þ

and call it the D-quantum discord of q. By (3.3) and Lemma 3.2(1) as well as Lemma 3.1(2), we see that

GD ðqÞ ¼ min

ei1 ;...;eim

¼ min

ei1 ;...;eim

¼ min

ei1 ;...;eim

  i1 im IðqÞ  IðUðe ;...;e Þ ðqÞÞ

SðqjjUðe

i1

;...;eim Þ

ðqÞÞ 

m X Sðqak jjUk ðqak ÞÞ

!

k¼1

  i1 im Sðqkqa1  . . .  qan Þ  SðUðe ;...;e Þ ðqÞkU1 ðqa1 Þ  . . .  Un ðqan ÞÞ ;

where the minimum was taken for all of bases ei1 ; . . . ; eim for Hi1 ; . . . ; Him , respectively. In order to discuss properties of GD ðqÞ, we firstly prove the following lemma. Lemma 3.3. Let q 2 DðHÞ. Then (1) SðWðUðqÞÞÞ 6 SðWðqÞÞ þ SðUðqÞÞ  SðqÞ provided that W and U are quantum channels on a induced by projective measurements with WðUðqÞÞ ¼ UðWðqÞÞ. P ik i1 i2 im (2) SðUðe ;e ;...;e Þ ðqÞÞ 6 m SðUðe Þ ðqÞÞ  ðm  1ÞSðqÞ. Pk¼1 m ðei1 ;ei2 ;...;eim Þ ðeik Þ (3) IðU ðqÞÞ P k¼1 IðU ðqÞÞ  ðm  1ÞIðqÞ.

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Proof. (1) Since any quantum channel does not increase relative entropy, we have SðUðqÞjjUðWðqÞÞÞ 6 SðqjjWðqÞÞ. Thus, by using the condition that WðUðqÞÞ ¼ UðWðqÞÞ and Lemma 3.1(2), we see that

SðWðUðqÞÞÞ  SðUðqÞÞ 6 SðWðqÞÞ  SðqÞ: In other words, SðWðUðqÞÞÞ 6 SðUðqÞÞ þ SðWðqÞÞ  SðqÞ. (2) By using (1), we compute i1

SðUðe

;ei2 ;...;eim Þ ðei1 Þ

¼ SðU

ðei1 Þ

6 SðU

ðqÞÞ

ðei2 ;...;eim Þ

ðU

ðqÞÞÞ

ðqÞÞ þ SðUðe

ðei1 Þ

i2

;...;eim Þ

ðqÞÞ  SðqÞ

ðei2 Þ

6 SðU ðqÞÞ þ SðU ðqÞÞ þ SðUðe m X ik 6 SðUðe Þ ðqÞÞ  ðm  1ÞSðqÞ:

i3 ;...;eim Þ

ðqÞÞ  2SðqÞ

k¼1

(3) Using (2), one has i1

i2

im

IðUðe ;e ;...;e Þ ðqÞÞ n X i1 i2 im SðUi ðqai ÞÞ  SðUðe ;e ;...;e Þ ðqÞÞ ¼ i¼1 n X

SðUi ðqai ÞÞ 

P

i¼1

m X ik SðUðe Þ ðqÞÞ þ ðm  1ÞSðqÞ k¼1

" # m nm m X X X X X ik ¼ SðUik ðqaik ÞÞ  SðUðe Þ ðqÞÞ þ Sðqas Þ þ SðU‘t ðqa‘t ÞÞ  Sðqas Þ þ ðm  1ÞSðqÞ t¼1

s–ik

k¼1

k¼1 s–ik

m nm m nm X X X X ik IðUðe Þ ðqÞÞ þ Sðqa‘t Þ  ðm  1Þ Sðqik Þ  m Sðqa‘t Þ þ ðm  1ÞSðqÞ ¼ t¼1

k¼1

t¼1

k¼1

" # m n X X ðeik Þ ai IðU ðqÞÞ  ðm  1Þ Sðq Þ  SðqÞ ¼ i¼1

k¼1

m X ik ¼ IðUðe Þ ðqÞÞ  ðm  1ÞIðqÞ:



k¼1

Next, we will discuss some properties of the D-quantum discord GD ðqÞ. Theorem 3.1. (1) GD ðqÞ  0; GD ðqÞ ¼ 0 if and only if q is D-CC. (2) GD is local unitary invariant, i.e., for all unitary operators U 1 ; . . . ; U n on H1 ; . . . ; Hn , we have

GD ððU 1  . . .  U n ÞqðU 1  . . .  U n Þy Þ ¼ GD ðqÞ: (3) If D0  D, then GD0 ðqÞ 6 GD ðqÞ. S P (4) If D ¼ sj¼1 Dj , then 1s sj¼1 GDj ðqÞ 6 GD ðqÞ. Pm P 1 (5) m j¼1 Gfij g ðqÞ 6 GD ðqÞ 6 m j¼1 Gfij g ðqÞ, where D ¼ fi1 ; i2 ; . . . ; im g.

Proof. (1) By definition and Lemma 3.1. (2) By (3.2) and (3.3). (3) Without loss of generality, we can let D ¼ fi1 ; i2 ; . . . ; im g and D0 ¼ fi1 ; i2 ; . . . ; ir gðr < mÞ. Noticing the fact that i

i

im

Uðe 1 ;e 2 ;...;e Þ ðqÞ ¼ Uðe

irþ1

;eirþ2 ;...;eim Þ

ðUðe

i1 i2 ;e ;...;eir Þ

ðei1 ;ei2 ;...;eim Þ

ðqÞÞ; i1

i2

ir

we see from Lemma 3.2(3) that IðU ðqÞÞ 6 IðUðe ;e ;...;e Þ ðqÞÞ. This shows that ID ðqÞ 6 ID0 ðqÞ and so GD0 ðqÞ 6 GD ðqÞ. P (4) By (3), we know that GDj ðqÞ 6 GD ðqÞ for all 1 6 j 6 s, and so 1s sj¼1 GDj ðqÞ 6 GD ðqÞ.

Z. Guo et al. / Information Sciences 289 (2014) 262–272

269

S (5) The use of D ¼ m j¼1 fij g and (4) imply that the first inequality holds. In the following, we will prove the second inequalðe Þ ity holds. Taking ðei1 ; ei2 ; . . . ; eim Þ 2 Oi1 . . . Oim such that IðqÞ  IðU ij ðqÞÞ ¼ Gfij g ðqÞ for j ¼ 1; 2; . . . ; m, we see from Lemma 3.3(3) that i1

i2

im

GD ðqÞ 6 IðqÞ  IðUðe ;e ;...;e Þ ðqÞÞ m X ðe Þ IðU ij ðqÞÞ þ ðm  1ÞIðqÞ 6 IðqÞ  j¼1

¼

m  X

ðei Þ

IðqÞ  IðU

j

 ðqÞÞ

j¼1 m X Gfij g ðqÞ:  ¼ j¼1

As an application of Theorem 3.1, we obtain the following result. Corollary 3.1. (1) (2) (3) (4)

Let Let Let Let

D0  D  X. If q is D-CC, then it is D0 -CC. D ¼ fi1 ; i2 ; . . . ; im g with 1 6 i1 < i2 < . . . < im 6 n. Then q is D-CC if and only if it is fik g-CC for all k ¼ 1; 2; . . . ; m. S D ¼ sj¼1 Dj . Then q is D-CC if and only if it is Dj -CC for all j ¼ 1; 2; . . . ; m. D be a nonempty proper subset of X. Then q is GCC if and only if it is both D-CC and Dc -CC.

At the end of this section, let us show an illustration of the D-quantum discord GD ðqÞ of q by using the 6-qubit GHZ state

qðtÞ ¼ ð1  tÞ

i I6 t h þ ðj0ih0jÞ6 þ ðj0ih1jÞ6 þ ðj1ih0jÞ6 þ ðj1ih1jÞ6 64 2 6

of the system a1 a2 . . . a6 with state space ðC2 Þ and computing its D-quantum discord. Firstly, we calculate IðqðtÞÞ. It is easy to note that qðtÞ has a simple eigenvalue 1þ63t and an eigenvalue 1t of multiplicity 64 64 63, and admits reduced density operators qðtÞaj ¼ 2I ð1 6 j 6 6Þ, where I denotes the identity operator on C2 . Thus, we can calculate that

  1t 1  t 1 þ 63t 1 þ 63t : log log IðqðtÞÞ ¼ 6 þ 63 þ 64 64 64 64 Let U be any 2 2 unitary matrix. Then it exhibits the form U ¼ eia By taking the orthonormal basis e1 ¼ fj0i; j1ig for C2 , we have



u

v

v u

 , where u; v 2 C with juj2 þ jv j2 ¼ 1 and a 2 R.

1

ðU y  I5 ÞUðUe Þ ðqðtÞÞðU  I5 Þ y

¼ ðU y  I5 ÞðUj0ih0jU y  I5 ÞqðtÞðUj0ih0jU y  I5 Þ ðU  I5 Þ y

þ ðU y  I5 ÞðUj1ih1jU y  I5 ÞqðtÞðUj1ih1jU y  I5 Þ ðU  I5 Þ ¼ ð1  tÞ

I6 t þ 64 2

P1 0 ; 0 P2

where

⎛

⎞ ⎛ ⎞ uv¯ |u|2 0 u¯v¯ |v|2 0 −¯ 0 0 ⎠ and P2 ¼ ⎝ 0 0 0 ⎠ P1 ¼ ⎝ 0 uv 0 |v|2 −uv 0 |v|2 1

1þ31t are 32 32 rank one projection matrices. This shows that UðUe Þ ðqðtÞÞ has an eigenvalue of multiplicity 2 and an eigen64 aj ðUe1 Þ I value 1t of multiplicity 62, and admits reduced density operators ð U ð q ðtÞÞÞ ¼ ð1 6 j 6 6Þ. Therefore, 64 2

  1 1t 1t 1 þ 31t 1 þ 31t ; log log IðUðUe Þ ðqðtÞÞÞ ¼ 6 þ 62 þ2 64 64 64 64

which is independent of U. It follows from Eqs. (3.2) and (3.3) that

Gf1g ðqðtÞÞ ¼

1t 1  t 1 þ 63t 1 þ 63t 1 þ 31t 1 þ 31t þ  : log log log 64 64 64 64 32 64

270

Z. Guo et al. / Information Sciences 289 (2014) 262–272

Consider the swap gate

0

1

0

0

0

1

B0 0 1 0C C B U¼B C @0 1 0 0A 0

0

0 1

P Pm and the unitary transform C : T # CðTÞ ¼ UTU y on BðC2  C2 Þ. It is easy to check that Cð m k¼1 X k  Y k Þ ¼ k¼1 Y k  X k for all Pm 2 2 4 Pm X k ; Y k 2 BðC Þ. Thus, for all Z k 2 BððC Þ Þ; k¼1 X k  Y k  Z k and k¼1 Y k  X k  Z k are unitarily equivalent and then have the same eigenvalues. Therefore, we see that Gfig ðqðtÞÞ ¼ Gf1g ðqðtÞÞ for all i ¼ 2; 3; . . . ; 6. For any unitary matrices

U ¼ e ia



u

v

   v w r ; V ¼ eib ; u r w

where u; v ; w; r 2 C with juj2 þ jv j2 ¼ 1; jwj2 þ jrj2 ¼ 1 and a; b 2 R. Take the orthonormal basis e1 ¼ e2 ¼ ðj0i; j1iÞ for C2 , applying similar method we have

⎛

⎞ P1 0 0 0 ⎜ 0 ⎟ 1 2 I6 t ⎜ 0 P2 0 ⎟ ½U y  V y  I4 UðUe ;Ve Þ ðqðtÞÞ½U  V  I5  ¼ ð1  tÞ þ ⎝ 0 0 P3 0 ⎠; 64 2 0 0 0 P4 where

⎛

⎞ ⎛ ⎞ |u|2|w|2 0 u¯v¯w¯ |u|2|r|2 0 −¯ ¯r uv¯w¯ ¯r ⎠; P 2 ¼ ⎝ ⎠; 0 0 0 0 0 0 P1 ¼ ⎝ uvwr 0 |v|2 |r|2 −uvwr 0 |v|2|w|2 ⎛

⎞ ⎛ 2 2 ⎞ |v|2|w|2 0 −¯ |v| |r| 0 u¯v¯w¯ uv¯w¯ ¯r ¯r ⎠; P 4 ¼ ⎝ ⎠ 0 0 0 0 0 0 P3 ¼ ⎝ 2 2 2 2 −uvwr 0 |u| |r| uvwr 0 |u| |w| 1

2

which are 16 16 matrices. Then UðUe ;Ve Þ ðqðtÞÞ has an eigenvalue 1t þ 2t ðjuj2 jwj2 þ jv j2 jrj2 Þ of multiplicity 2, an eigenvalue 64 2 2 2 2 t 1t þ 2 ðjuj jrj þ ajv j jwj Þ of multiplicity 2 and an eigenvalue 64 of multiplicity 60, and admits reduced density operators 2 1 j ðUðU1 e ;Ve Þ ðqðtÞÞÞ ¼ 2I ð1 6 j 6 6Þ. Then we calculate

1t 64

1

IðUðUe

;Ve2 Þ

ðqðtÞÞÞ ¼

6 X

1

SððUðUe

;Ve2 Þ

aj

ðqðtÞÞÞ Þ  SðUðUe

1

;Ve2 Þ

ðqðtÞÞÞ

j¼1

    1t 1t 1t t 1t t ¼ 6 þ 60 log þ2 þ ðjuj2 jwj2 þ jv j2 jrj2 Þ log þ ðjuj2 jwj2 þ jv j2 jrj2 Þ þ 2 64 64 64 2 64 2     1t t 1  t t þ ðjuj2 jrj2 þ jv j2 jwj2 Þ log þ ðjuj2 jrj2 þ jv j2 jwj2 Þ : 64 2 64 2 Put x ¼ juj2 jwj2 þ jv j2 jrj2 , then juj2 jrj2 þ jv j2 jwj2 ¼ 1  x. Hence, we have

1t 1t log 64 64        

1t t 1t t 1t t 1t t þ 2 max þ x log þ x þ þ ð1  xÞ log þ ð1  xÞ x2½0;1 64 2 64 2 64 2 64 2 1t 1  t 1 þ 31t 1 þ 31t log log þ : ¼ 6 þ 62 64 64 32 64

If1;2g ðqðtÞÞ ¼ 6 þ 60

This shows that Gf1;2g ðqðtÞÞ ¼ Gfig ðqðtÞÞ for all i ¼ 1; 2; . . . ; 6. Similarly, for every nonempty set D  f1; 2; . . . ; 6g, we have

271

Z. Guo et al. / Information Sciences 289 (2014) 262–272 1 Δ−quantum discord diagonal line

0.9 0.8

GΔ(ρ(t))

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

t Fig. 1. The blue solid curve represents the relationship between the D-quantum discord GD ðqðtÞÞ and time t; the red dashed line denotes the diagonal line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

GD ðqðtÞÞ ¼ IðqðtÞÞ  If1;2g ðqðtÞÞ ¼

1t 1  t 1 þ 63t 1 þ 63t 1 þ 31t 1 þ 31t log log log þ  ; 64 64 64 64 32 64

which is just the same as the result given in [15]. Please refer to Fig. 1 for the relationship between GD ðqðtÞÞ and t. Furthermore, it is worth noting that the converse of Corollary 2.1 is not true. From Fig. 1, we see GD ðqð12ÞÞ > 0 for all D, and 1 1 I 5 this implies that qð2Þ is D-QC for all D but trai ðqð2ÞÞ ¼ 2 for all i, which is a GCC-state of the 5-qubit system. 4. Conclusions In order to reveal quantum correlations in a multipartite quantum system a1 a2 . . . an with respect to some subsystems, correlations of quantum states have been discussed by introducing the concepts of D-classical correlations, D-quantum correlations, partially classical correlations and genuinely quantum correlations and defining the measure function GD ðqÞ of partial correlations of a state q (called D-quantum discord). It has been proved that the proposed GD ðqÞ is nonnegative, independent of the basis and invariant under local unitary transformations. This measure can serve to quantify partial correlations of a multipartite state. For instance, a state q is D-classical correlated if and only if GD ðqÞ ¼ 0, and then it is D-quantum correlated if and only if GD ðqÞ > 0. Moreover, some relationships among partial correlations with respect to different D have been explored in light of the measure function. An example with the 6-qubit GHZ state illustrated the rationality of the present classification and the validity of the defined measure. With the quick development of quantum information science, quantum theory including quantum correlations has become an important tool of quantum algorithm [10,23,31], quantum communication [19], processing of multidimensional color image [3], and others. We believe that the discussion presented here may contribute to the better understanding of quantum correlations in multipartite quantum systems and support further the applications of quantum information science to applied science. Acknowledgments Zhihua Guo is supported by the National Natural Science Foundation of China (11171197, 11401359), the Fundamental Research Funds for the Central Universities (GK201402005), China Postdoctoral Science Foundation (2014M552405) and the Natural Science Research Program of Shaanxi Province (2014JQ1010). Huaixin Cao is supported by the National Natural Science Foundation of China (11371012). Especially, we would like to thank anonymous reviewers for their kind comments and valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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