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International Journal of Quantum Information

QUANTUM DISCORD DERIVED FROM TSALLIS ENTROPY

Jacek Jurkowski

Published Feb 1, 2013• 17 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749913500135

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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.
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.
Several ways have been proposed in the literature to define a coherence measure based on Tsallis relative entropy. One of them is defined as a distance between a state and a set of incoherent states with Tsallis relative entropy taken as a distance measure. Unfortunately, this measure does not satisfy the required strong monotonicity, but a modification of this coherence has been proposed that does. We introduce three new Tsallis coherence measures coming from a more general definition that also satisfy the strong monotonicity, and compare all five definitions between each other. Using three coherence measures that we discuss, one can also define a discord. Two of these have been used in the literature, and another one is new. We also discuss two correlation measures based on Tsallis relative entropy. We provide explicit expressions for all three discord and two correlation measure on pure states. Lastly, we provide tight upper and lower bounds on two discord and correlations measures on any quantum state, with the condition for equality.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called XXX states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.
Quantum discord is the quantitative difference between two alternative expressions for bipartite mutual information, given respectively in terms of two distinct definitions for the conditional entropy. By constructing a stochastic model of shared states, classical discord can be similarly defined, quantifying the presence of some stochasticity in the measurement process. Therefore, discord can generally be understood as a quantification of the system's state disturbance due to local measurements, be it quantum or classical. We establish an operational meaning of classical discord in the context of state merging with noisy measurement and thereby show the quantum-classical separation in terms of a negative conditional entropy.
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 $q\ensuremath{\ge}1$.
Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. 88, 017901 (2001)], is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.