Exploring multipartite quantum correlations with the square of quantum discord
We explore the quantum correlation distribution in multipartite quantum states based on the square of quantum discord (SQD). For tripartite quantum systems, we derive the necessary and sufficient condition for the SQD to satisfy the monogamy relation. Particularly, we prove that the SQD is monogamous for three-qubit pure states, based on which a genuine tripartite quantum correlation measure is introduced. In addition, we also address the quantum correlation distributions in four-qubit pure states. As an example, we investigate multipartite quantum correlations in the dynamical evolution of multipartite cavity-reservoir systems.
pacs:03.65.Ud, 03.65.Yz, 03.67.Mn
Besides quantum entanglement, quantum correlation is also a key resource in quantum information processing knill98prl (); datta08prl (); lanyon08prl (); piani08prl (); piani10prl (); madhok11pra (); cav11pra (); roa11prl (); dakic12np (); cui12natc (); datta13ijmpb (). As a basic tool to characterize the quantum advantage modi12rmp (), quantum discord (QD) is a prominent bipartite quantum correlation measure ollivier01prl (); vedral01jpa (). Recently, generalization of the QD to multipartite systems has received much attention modi10prl (); rulli11pra (); pati10arxiv (); giorgi11prl (); walczak11epl (). However, characterization of quantum correlation structure in multipartite systems is still very challenging. Monogamy relation ckw00pra (); osb06prl (); horodecki09rmp () is an important property in multipartite quantum systems. As quantified by the square of concurrences wootters98prl (), entanglement is monogamous in multiqubit systems osb06prl () i.e.,
and this property can be used to construct genuine multipartite entanglement measures ckw00pra (); byw07pra (). Therefore, it is natural to ask whether or not the quantum correlation is monogamous, especially for the QD.
Prabhu et al. found that the QD is not monogamous and the monogamy relation
is not satisfied even for the three-qubit state prabhu12pra (). Giorgi giorgi11pra () and Fanchini et al fanchini11pra (); fanchi11arxiv () related the monogamy condition of QD to the entanglement of formation, while Ren and Fan showed that QD is not monogamous under the same measurement party ren11arxiv (). Recently, Streltsov et al further showed that the monogamy relation does not hold in general for quantum correlation measures which are nonzero for separable states str12prl (). However, these results do not imply that quantum correlation is still not monogamous in a specific case (for example, the geometric measure of discord dakic10prl () is monogamous in three-qubit pure states str12prl ()). Since the QD is accepted as a basic tool for quantum correlation, it is desirable to find a kind of monogamous QD even in several qubit systems, which on the one hand gives a clear correlation structure but on the other hand allows the characterization of genuine multipartite quantum correlation.
In this paper, we are motivated by the following two questions: (i) whether or not the QD is monogamous in certain form, and (ii) in what degree the discord is monogamous and can characterize the genuine multipartite quantum correlation. To answer these two questions, we explore the monogamy property of the square of quantum discord (SQD) in multipartite quantum systems. The paper is organized as follows. In Sec. II, we derive the necessary and sufficient condition for the SQD to be monogamous in tripartite quantum states. In three-qubit pure states, we prove that the SQD is monogamous and define a genuine tripartite quantum correlation measure. In Sec. III, we analyze the correlation distribution in multiqubit pure states and construct multipartite quantum correlation indicators. As an application, we address the dynamics of quantum correlation in multipartite cavity-reservoir systems. Finally, we present discussions and a conclusion in Sec. IV.
Ii Monogamy property and correlation measure in tripartite quantum states
ii.1 Definitions and monogamous condition
In a bipartite quantum system , the total correlation can be quantified by quantum mutual information with being von Neumann entropy ollivier01prl (), while the classical correlation is given by , in which is a positive operator-valued measure (POVM) performed on the subsystem and with vedral01jpa (). The QD is used to characterize bipartite quantum correlation, which is defined as the difference between and , and is expressed as ollivier01prl ()
where the minimum runs over all the POVMs, and is referred to as the discord of system with the measurement on subsystem . The QD can also be written in the form of quantum conditional entropy cav11pra ():
where the non-negative quantity is the measurement-induced quantum conditional entropy and is the direct quantum generalization of conditional entropy.
Monogamy relation is an important property in multipartite quantum systems. Coffman et al. first showed that the monogamy relation of concurrence is satisfied in three-qubit quantum states and the residual entanglement can characterize the genuine tripartite entanglement ckw00pra (). It should be noted that, in the monogamy relation, the square of concurrence is monogamous other than the concurrence itself which is not monogamous. Previous studies indicated that the QD is not monogamous even in three-qubit pure states prabhu12pra (); giorgi11pra (); fanchini11pra (); fanchi11arxiv (); ren11arxiv (), which does not imply that the square of QD is not monogamous either.
Here, we explore the monogamy property of SQD in multipartite systems. The SQD can be written as
which satisfies all the standard requirements for quantum correlation measure str12prl (); brodutch12qic () and can characterize effectively quantum correlation in bipartite systems. Particularly, in a tripartite pure state , the measurement-induced quantum conditional entropies are related to the entanglement of formation wootters98prl () by the Koashi-Winter formula koashi04pra ()
where and are the conditional entropies with measurement on the subsystem , and is the entanglement of formation in the subsystem with the minimum taking over all the pure state decompositions and . Using the formula in Eq. (6), the SQD has the form
where the measurement is performed on subsystem , and . Moreover, in a tripartite pure state , we have the relation in which is the entanglement of formation under the bipartite partition ollivier01prl (); vedral01jpa (). Combining this relation with Eq. (7), we can derive the quantum correlation distribution of SQD
In the distribution, the first term is an entanglement distribution relation quantified by the square of entanglement of formation and the second term is a function of entanglement of formation and conditional entropy . According to Eq. (8), the necessary and sufficient condition for the monogamous SQD is
ii.2 Monogamy property in three-qubit pure states
We now look into the quantum correlation distribution in two-level (qubit) systems.
Theorem I. In any three-qubit pure state , the square of quantum discord obeys the monogamy relation
Proof. The theorem will hold when the monogamy condition in Eq. (10) is satisfied for all three-qubit pure states. In two-qubit quantum states, the entanglement of formation has an analytical expression in which is the binary entropy and is the concurrence with the decreasing non-negative s being the eigenvalues of matrix wootters98prl (). As a function of the square of concurrence, the entanglement of formation obeys the following relations:
where the Coffman-Kundu-Wootters (CKW) relation ckw00pra () and the monotonically increasing property of is used in the first equation, and the property that is a convex function of is used in the second equation. According to Eq. (12), we can obtain the first term in the monogamy condition.
For the second term , we first show that has the same sign as that of . It is straightforward to derive the following relations:
where we have used the entanglement distributions and with being the three-tangle ckw00pra (), and the monotonically increasing property of . Similarly, if , we can obtain the relation . Therefore and have the same sign, and thus the second term in the monogamy condition has the form
As a result, the non-negative property of is equivalent to
which is proven to be valid as follows.
On one hand, if , the left-hand side of Eq.(15) can be written as
where we have used in tripartite pure states. On the other hand, we have
where is a non-negative constant. In addition, we have used the monotonic property of in the second inequality and the concave property of giorgi11pra () in the third inequality which means that along with the increase of concurrence the increment of will decrease. When we choose , the entanglement of formation is
where the CKW relation has been used. Similarly, the relation can be derived. Substituting the results into Eq. (17), we have the relation
Combining Eq. (19) with Eq. (16), we can obtain . In the other case, if , the left-hand side of Eq. (15) becomes
Moreover, we have
where and with , and the concave property of is used. Combining Eq. (20) with Eq. (21), we get . Therefore, we have proven that is non-negative, namely, is non-negative. Due to and , the monogamy condition holds, and the proof is completed.
As examples, we consider the quantum correlation distribution of SQD in generalized state prabhu12pra ()
and the two-parameter state giorgi11pra ()
In Fig.1, we plot the distribution (blue solid line) in comparison to the distribution (red dash-dotted line) for the two quantum states, where although the QD is not monogamous as pointed out in Refs. prabhu12pra (); giorgi11pra (), we can see that the SQD is monogamous.
For the further verification on the theorem, we analyze the standard form of three-qubit pure states acin00prl ()
where the real number ranges in with the condition , and the relative phase changes in . Without loss of generality, we set , , , , and , respectively. In Fig.2, the quantum correlation distribution of SQD is plotted as a function of parameters , and (the relative phase is set to ), where ranges in with equal interval being . Again, we can see that the SQD is monogamous.
ii.3 A genuine three-qubit quantum correlation measure with the hierarchy structure
A quantum correlation measure should satisfy the following necessary criteria: (i) it should be a non-negative real number; (ii) it is invariant under local unitary operations str12prl (); brodutch12qic (); and (iii) it is zero in an -partite quantum state if and only if the state is a product state in any bipartite cut ben11pra ().
Based on our previous analysis on the quantum correlation distribution of SQD, we define a tripartite quantum correlation measure as
which characterizes the genuine three-qubit quantum correlation in a pure state . The non-negative property of is satisfied due to the SQD being monogamous. The tripartite correlation is invariant under local unitary operations because the SQDs are unchanged under the transformation.
For the third requirement, we first prove that the measure is zero if a three-qubit state is a product state in any bipartite cut. When the quantum state has the form , the SQD due to the product property under this partition. The SQD because we have with being the projector composed of the eigenvector of . The case for is similar. So, the genuine tripartite quantum correlation . For the product state , we also have , since and . Similarly, we can derive for . Therefore, is zero when the three-qubit pure state is a product state in any bipartite cut.
Next, we prove that when the three-qubit pure state is not a bipartite product under any partition, the measure is always nonzero. Based on the correlation distribution in Eq. (8), it is sufficient to prove the term since the second term is nonnegative. For a non-product state , its bipartite concurrence is a positive value and we have the CKW relation . When and , we can obtain that because the entanglement is a monotonically increasing and convex function of the concurrence . When one of the two-qubit concurrence is zero, for example , the CKW relation is . According to the monotonic property, we have . It should be noted that should be removed simply because it corresponds to the case that the three-qubit pure state is a product one under the partition . Therefore, if ever the three-qubit state is of nonproduct, implying that the measure is positive.
So far, we have shown that the introduced tripartite quantum correlation measure satisfies all the three necessary criteria. Furthermore, the measure may be understood as the monogamy score difference of SQD between the given state and a bipartite product state, i.e.,
where monogamy score is . When is nonzero, the quantum state is not a product state and its monogamy score is larger than that of any bipartite product state. The score difference is just the residual SQD. The larger the value of , the farther the monogamy distance between the give state and the bipartite product state. Therefore the measure can characterize the genuine three-qubit quantum correlation and has a physical explanation in terms of the monogamy score difference.
In addition, for a three-qubit pure state , we can obtain a hierarchy structure of quantum correlations. As depicted schematically in Fig.3, Eq.(25) can be rewritten as
where quantifies the total quantum correlation in the partition , and quantify two-qubit quantum correlations, and characterizes the genuine three-qubit quantum correlation under the partition .
As an application, we consider generalized Greenberger-Horne-Zeilinger (GHZ) and states, which are two inequivalent classes under stochastic local operations and classical communication dur00pra (). The generalized state has the form . Its two-qubit quantum correlations are zero because the reduced density matrices are classical states. Therefore, there is only the genuine three-qubit quantum correlation in the generalized state. For the generalized state , both two-qubit and three-qubit quantum correlations are nonzero when parameters , , and are nonzero. When and , the tripartite quantum correlation has the maximal value .
Also noting that the QD is asymmetric for different measurement parties, the tripartite quantum correlation under qubit permutation is not equivalent to each other: for a generic quantum state. From this consideration, we may define a new tripartite quantum correlation measure:
where , and the measure may be referred to as the three-qubit mean-SQD. This mean-SQD not only satisfies all three conditions for a multipartite correlation measure, but also is independent of bipartite partitions, reflecting really the global tripartite quantum correlation in a three-qubit pure state .
ii.4 Tripartite correlation indicator in mixed states
In three-qubit mixed states, the quantum correlation distribution of SQD is not always monogamous. As an example, we analyze the quantum state
where the non-normalized pure state components are and , respectively. Using the Koashi-Winter formula, we have the discord
where subsystem is equivalent to a logic qubit and the subsystem is the environment degree of freedom purifying the mixed state. Because is a rank-2 quantum state, the environment subsystem is equivalent to a logic qubit. In Eq. (29), we set the parameters , , , and . When the parameters , we can get by using the Wootters formula wootters98prl (), which results in . Similarly, we have and . Substituting these SQDs into the correlation distribution , we can determine the value of the distribution is .
Although the quantum correlation distribution can be negative, we can still introduce a tripartite quantum correlation indicator whenever the distribution in a mixed state is always monogamous (an example of this case will be presented in the next section). In this case, we may define the indicator as
where . Furthermore, we can introduce a symmetric tripartite correlation indicator
which indicates the global tripartite quantum correlation in a three-qubit mixed state.
Iii Multipartite quantum correlation indicators in four-qubit systems
In four-qubit pure states, the structure of quantum correlation distributions is more complicated than that in three-qubit states. In general, these distributions are not monogamous. However, if the distributions of SQD are monogamous in a given four-qubit system, we can also construct an indicator of the four-body correlation with the components
where the superscript means that the correlation distribution lies in the partition between one qubit and the other three qubits and the case for is the distribution between two two-qubit subsystems. Under qubit permutations, and have four and six inequivalent components, respectively. The nonzero component indicates the genuine multipartite quantum correlation in the designated partition of a given state. For example, in the generalized four-qubit state , the correlation distribution is always non-negative, and we have . Another example is the cluster state rau01prl (), in which we have and .
At this stage, as an interesting example, we consider the dynamical property of quantum correlations in a real quantum system. As is known, the dynamical property of a two-qubit quantum correlation has been widely investigated both theoretically and experimentally (see, for example, Refs. maz09pra (); maz10prl (); jxu10nc (); auc11prl (); cen12pra (); rong12prb (); fran13ijmpb () and references therein). However, the dynamical property of multipartite quantum correlations is still very challenging. We now use the multipartite correlation indicator to analyze the dynamical evolution in four-partite cavity-reservoir systems. The system is composed of two entangled cavity photons being affected by the dissipation of two individual -mode reservoirs, where the interaction of a single cavity-reservoir system is described by Hamiltonian lop08prl ()
The initial state is , where the dissipative reservoirs are in the vacuum state. In the limit of for a reservoir with a flat spectrum, the output state of the cavity-reservoir system has the form lop08prl ()
where with the amplitudes being and . For the output state, we analyze its relevant components of the three- and four-partite quantum correlation indicators and given in Eqs. (31) and (33). Here, we use the method introduced by Chen et al. for calculating the quantum discord of two-qubit states (see the calculation in the Appendix) che11pra ().
In Fig.4, we plot different components of multipartite quantum correlation indicators as a function of the time evolution parameter and the initial state amplitude . It is noted that all the correlation distributions are non-negative and we have and for these components. When the time , the quantum state is a product state and these indicators are zero. Along with the time evolution, they first increase to their maxima, and then decay asymptotically. When the parameter , the output state evolves to a product state again and all the multipartite quantum correlations disappear.
In the cavity-reservoir system, its multipartite entanglement evolution was investigated in Refs. lop08prl (); byw09pra (); wen11epjd (). The genuine multipartite entanglement can be characterized by a series of entanglement indicators. Here, in our analysis, we consider the following components:
where is the square of concurrence and the subscripts in the second equation. The component can be used to characterize the genuine multipartite entanglement in the partition , and can indicate the genuine block-block entanglement in the partition byw09pra (). Moreover, the component is used to quantify the qubit-block entanglement in three-qubit mixed states loh06prl (); byw08pra (); wen11epjd ().
In Fig.5, we plot the relevant components of multipartite quantum correlation indicators and in comparison to the multipartite entanglement indicators and for the output state . As seen from the figure, the multipartite quantum correlation is correlated with the multipartite entanglement in every partition structure. However, the peaks of correlation and entanglement do not coincide completely. The reason is that quantum correlation and quantum entanglement are not equivalent in general. Particularly, in the dynamical procedure, the evolution of two-qubit entanglement can exhibit the phenomenon of entanglement sudden death hor01pra (); eis03jmo (); tyu04prl (), but the corresponding evolution of quantum correlation is always asymptotic. In addition, the peak values of quantum correlation indicators can be greater (Fig. 5a) or less (Fig. 5b-d) than those of quantum entanglement indicators. This is due to the fact that different measures of quantum states lack the same ordering virmani00pla (); lang11ijqi (); okrasa12epl (). Although the quantum correlation can be greater than entanglement in separable states, the ordering may change in a generic quantum state. For example, quantum discord is not always greater than the entanglement of formation even in two-qubit quantum states luo08pra ().
Iv Discussion and conclusion
The QD is very difficult to compute because of the minimization over all positive operator-valued measures. Till now, the analytical result of QD is still an open problem except for some specific classes of quantum states luo08pra (); lang10prl (); gio10prl (); ade10prl (); ali10pra (); cen11pra (); che11pra (); shi12pra (). However, in three-qubit pure states, we can calculate two-qubit QD via the Wootters formula wootters98prl () and Koashi-Winter relation koashi04pra (). In this case, the analytical formula of genuine tripartite quantum correlation is available and can be rewritten as
Therefore, in three-qubit pure states, not only the hierarchy structure of quantum correlation holds but also all the quantum correlations can be calculated analytically.
In conclusion, we have explored multipartite quantum correlations with the monogamy of SQD and answered the two important questions. We have proven that the SQD is monogamous in three-qubit pure states and the residual correlation is a reasonable measure for genuine three-qubit quantum correlation, which gives a clear hierarchy structure for quantum correlations. For three-qubit mixed states, although the distribution of SQD is not always monogamous, we have constructed an effective indicator which can detect the genuine tripartite quantum correlation in a specific class of states. For four-qubit pure states, the monogamy property of SQD may still be used to construct effective indicators for measuring genuine multipartite quantum correlations. As an interesting example, we have addressed the evolution of multipartite cavity-reservoir systems. The present work may shed a light on understanding of quantum correlations in multipartite systems.
This work was supported by the RGC of Hong Kong under Grant No. HKU7044/08P. Y.K.B. and N.Z. were also supported by NSF-China (Grant No. 10905016), Hebei NSF (Grant No. A2012205062), and the fund of Hebei Normal University. M.Y.Y. was also supported by NSF-China (Grant No. 11004033) and NSF of Fujian Province (Grant No. 2010J01002).
Appendix: calculation of the discord in cavity-reservoir systems
The density matrix of a two-qubit state can be written
When the elements satisfy the following relations che11pra ():
Chen et al proved that the optimal measurement for the quantum discord is . In the output state , we find the optimal measurement is for state . Then, according to the definition of the quantum discord in Eq. (4), we can get the value of . For other two-qubit quantum discords in the correlation distributions, we find that the optimal measurement is also , where we use the property that subsystem () is equivalent to a logic qubit. In a similar way, we can calculate these SQDs.
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