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Relative entropy of quantum and classical correlations

Kavan Kishore Modi, Tomasz Paterek, Wonmin Son, Vlatko Vedral, Mark S. Williamson

Published Nov 29, 2009• 1 citation
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Abstract

Clarendon Laboratory, University of Oxford, Oxford UK(Dated: November 29, 2009)We discuss the problem of separation of total correlations in a given quantum state into entan-glement, dissonance, and classical correlations using the concept of relative entropy as a distancemeasure of correlations. This allows us to put all correlations on an equal footing. Entanglementand dissonance, whose de nition is introduced here, jointly belong to what is known as quantumdiscord. Our methods are completely applicable for multipartite systems of arbitrary dimensions.We investigate additivity relations between di erent correlations and show that dissonance may bepresent in pure multipartite states.

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Clarendon Laboratory, University of Oxford, Oxford UK(Dated: November 29, 2009)We discuss the problem of separation of total correlations in a given quantum state into entan-glement, dissonance, and classical correlations using the concept of relative entropy as a distancemeasure of correlations. This allows us to put all correlations on an equal footing. Entanglementand dissonance, whose de nition is introduced here, jointly belong to what is known as quantumdiscord. Our methods are completely applicable for multipartite systems of arbitrary dimensions.We investigate additivity relations between di erent correlations and show that dissonance may bepresent in pure multipartite states.
We discuss the problem of the separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states.
We give a pedagogical introduction to quantum discord and discuss the problem of separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy. This allows us to put all correlations on an equal footing. Entanglement and dissonance jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We finally show, using relative entropy, how different notions of quantum correlations are related to each other. This gives a single theory that incorporates all correlations, quantum and classical, and different methods of quantifying them.
A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.
The pairwise correlations in a multi-qubit state are quantified through a linear variant of relative entropy. In particular, we derive the explicit expressions of total, quantum and classical bipartite correlations. Two different bi-partioning schemes are considered. We discuss the derivation of closest product, quantum–classical and quantum–classical product states. We also investigate the additivity relation between the various pairwise correlations existing in pure and mixed states. As illustration, some special cases are examined.
We study quantum coarse-grained entropy and demonstrate that the gap in entropy between local and global coarse-grainings is a natural generalization of entanglement entropy to mixed states and multipartite systems. This ``quantum correlation entropy'' Sqc{S}^{q}cSqc is additive over independent systems, is invariant under local unitary operations, measures total nonclassical correlations (vanishing on states with strictly classical correlation), and reduces to the entanglement entropy for bipartite pure states. It quantifies how well a quantum system can be understood via local measurements and ties directly to nonequilibrium thermodynamics, including representing a lower bound on the quantum part of thermodynamic entropy production. We discuss two other measures of nonclassical correlation to which this entropy is equivalent and argue that together they provide a unique thermodynamically distinguished measure.