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arXiv:1302.2984 · Physical Review A

Generalized entropy and global quantum discord in multiparty quantum systems

Dong Pyo Chi, Jeong San Kim, Kyungjin Lee

Published Jun 28, 2013• 21 citations
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Abstract

Using Tsallis-$q$ entropy, we introduce the generalized concept of global quantum discord, namely the $q$-global quantum discord, and provide its analytic evaluation for two classes of multiqubit states. We also provide a sufficient condition, for which the pairwise quantum correlations in terms of $q$-global quantum discord is monogamous in multiparty quantum systems.

DOI
10.1103/physreva.87.062339

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Using Tsallis-qqq entropy, we introduce the generalized concept of global quantum discord, namely the qqq-global quantum discord, and provide its analytic evaluation for two classes of multiqubit states. We also provide a sufficient condition, for which the pairwise quantum correlations in terms of qqq-global quantum discord is monogamous in multiparty quantum systems.
Due to some ambiguity in defining mutual Tsallis entropy in the classical probability theory, its generalization to quantum theory is discussed and, as a consequence, two types of generalized quantum discord, called qqq-discords, are defined in terms of quantum Tsallis entropy. qqq-discords for two-qubit Werner and isotropic states are calculated and it is shown that one of them is positive, at least for states under investigation, for all q>0q>0q>0. Finally, an analytical expression for qqq-discord of certain family of two-qubit X states is presented.
Due to some ambiguity in the definition of mutual Tsallis entropy in classical probability theory, its generalization to quantum theory is discussed and, as a consequence, two types of generalized quantum discords, called q-discords, are defined in terms of quantum Tsallis entropy. Both q-discords for two-qubit Werner and isotropic states are determined and compared and it is shown that one of them is non-negative, at least for states under investigation, for all q > 0. Finally, an analytical expression for q-discord of certain family of two-qubit X-states is presented. Using this example, we show that both types of q-discords can take negative values for some q > 1, hence their use as correlations measures is rather limited.
We show that the mutually exclusive nature of classical and quantum correlations distributed in multiparty quantum systems can be characterized in terms of qqq-expectation value. Using Tsallis qqq entropy and qqq-expectation value, we first provide generalized definitions of classical and quantum correlations, and establish their trade-off relations in three-party quantum systems of arbitrary dimension with respect to qqq-expectation value for $q\ensuremath{\ge}1$. We also provide equivalence conditions for monogamy and polygamy inequalities of quantum entanglement and quantum discord distributed in three-party quantum systems of arbitrary dimension with respect to qqq-expectation value for $q\ensuremath{\ge}1$.
Abstract The original quantum discord (QD) is shown to be not monogamous except for the three-qubit states. Recently, a complete monogamy relation for multiparty quantum system was established for entanglement in (Guo and Zhang 2020 Phys. Rev. A 101 032301), and in addition, a new multipartite generalization of QD was proposed in (Radhakrishnan et al 2020 Phys. Rev. Lett. 124 110401). In this work, we firstly define the complete monogamy for this multipartite quantum discord (MQD) and the global quantum discord (GQD). MQD, with the same spirit as the complete monogamy of entanglement, is said to be completely monogamous (i) if it does not increase under coarsening of subsystems and (ii) if some given combination of subsystems reach the total amount of the correlation, then all other combination of subsystems that do not include all the given subsystems do not contain such a correlation any more. Here, coarsening of subsystems means discarding or combining some subsystems up to the given partition. Simultaneously, the complete monogamy of GQD is also defined with slight modification on the coarsening relation. Consequently, we explore all the coarsening relations of MQD and show that it is completely monogamous with a modicum assumption. In addition, with the same spirit, we investigate all the coarsening relations for GQD and show that GQD is not completely monogamous. That is, in the sense of the complete monogamy relation, MQD as a generalization of the original QD captures the nature of such a quantum correlation, and thus it is nicer than GQD as a generalization.
We propose 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. The measure is symmetric with respect to subsystem exchange and is shown to be nonnegative for an arbitrary state. As an illustration, we consider tripartite correlations in the Werner-GHZ (Greenberger-Horne-Zeilinger) state and multipartite correlations at quantum criticality. In particular, in contrast with the pairwise quantum discord, we show that the global quantum discord is able to characterize the infinite-order quantum phase transition in the Ashkin-Teller spin chain.
We establish a unified view of the polygamy of multiparty quantum entanglement in arbitrary dimensions. Using quantum Tsallis-qqq entropy, we provide a one-parameter class of polygamy inequalities of multiparty quantum entanglement. This class of polygamy inequalities reduces to the known polygamy inequalities based on tangle and entanglement of assistance for a selective choice of the parameter qqq. We further provide one-parameter generalizations of various quantum correlations based on Tsallis-qqq entropy. By investigating the properties of the generalized quantum correlations, we provide a sufficient condition on which the Tsallis-qqq polygamy inequalities hold in multiparty quantum systems of arbitrary dimensions.