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arXiv:1607.06585

Maximal mutual correlation: an algebraic measure of correlations in bipartite quantum systems

Lech Jakóbczyk

Published Jul 22, 2016• 0 citations
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Abstract

We introduce an algebraic measure of correlations in bipartite quantum systems. The proposed quantity, called maximal mutual correlation, provides the information how much a given state differs from the product state of its marginals. In contrast to the entanglement or quantum discord, maximal mutual correlation can be non - zero for some classically correlated quantum states.

DOI
10.48550/arxiv.1607.06585

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We introduce an algebraic measure of correlations in bipartite quantum systems. The proposed quantity, called maximal mutual correlation, provides the information how much a given state differs from the product state of its marginals. In contrast to the entanglement or quantum discord, maximal mutual correlation can be non - zero for some classically correlated quantum states.
A generalization of quantum discord to multipartite systems is proposed. A key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems. It is by construction zero only for systems with classically correlated subsystems and is a non-negative quantity, giving a measure of the total nonclassical correlations in the multipartite system with respect to a fixed measurement ordering. For the tripartite case, we show that the discord can be decomposed into contributions resulting from changes induced by nonclassical correlation breaking measurements in the conditional mutual information and tripartite mutual information. The former gives a measure of the bipartite nonclassical correlations and is a non-negative quantity, while the latter is related to the monogamy of the nonclassical correlations.
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Mutual information is the reciprocal information that is common to or shared by two or more parties. Quantum mutual information for bipartite quantum systems is non-negative, and bears the interpretation of total correlation between the two subsystems. This may, however, no longer be true for three or more party quantum systems. In this paper, we propose an alternative definition of multipartite information, taking into account the shared information between two and more parties. It is non-negative, observes monotonicity under partial trace as well as completely positive maps, and equals the multipartite information measure in literature for pure states. We then define multiparty quantum discord, and give some examples. Interestingly, we observe that quantum discord increases when a measurement is performed on a large number of subsystems. Consequently, the symmetric quantum discord, which involves a measurement on all parties, reveals the maximal quantumness. This raises a question on the interpretation of measured mutual information as a classical correlation.
Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well-known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these non-classical correlations is discord. Here we introduce a new measure of quantum correlations which we call entropic accord that fits between entanglement and discord. It is defined as the optimised minimax mutual information of the outcome of the projective measurements between two parties. We show a strict hierarchy exists between entanglement, entropic accord and discord for two-qubit states. We study two-qubit states which shows the relationship between the three entropic quantities. In addition to revealing a class of correlations that are distinct from discord and entanglement, the entropic accord measure can be inherently more intuitive in certain contexts.
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.
Two classically identical expressions for the mutual information generally differ when the systems involved are quantum. This difference defines the quantum discord. It can be used as a measure of the quantumness of correlations. Separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord, thus showing that absence of entanglement does not imply classicality. We relate this to the quantum superposition principle, and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system, i.e., the pointer states.