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arXiv:1101.6057 · Europhysics Letters (EPL)

Quantum discord and multipartite correlations

M. Okrasa, Zbigniew Walczak

Published Dec 1, 2011• 70 citations
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Abstract

Recently, it was realized that quantum discord can be seen as the minimal amount of correlations which are lost when some local quantum operations are performed. Based on this formulation of quantum discord, we provide a systematical analysis of quantum and classical correlations present in both bipartite and multipartite quantum systems. As a natural result of this analysis, we introduce a new measure of the overall quantum correlations which is lower bounded by quantum discord.

DOI
10.1209/0295-5075/96/60003

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Recently, it was realized that quantum discord can be seen as the minimal amount of correlations which are lost when some local quantum operations are performed. Based on this formulation of quantum discord, we provide a systematical analysis of quantum and classical correlations present in both bipartite and multipartite quantum systems. As a natural result of this analysis, we introduce a new measure of the overall quantum correlations which is lower bounded by quantum discord.
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.
Quantum information and computation, based on quantum correlations of quantum systems, is one of the promising new fields of physics and information science. Entanglement is the key resource of quantum information processing. However, it is shown recently that quantum entanglement is not the only kind of quantum correlations useful for quantum information processing. Quantum systems without any entanglement can still be helpful for quantum information processing since unentangled systems could have another kind of quantum correlations - quantum discord. Thus, quantum discord has drawn much attention in recent years. Here, we briefly review the concepts and properties of quantum discord. Several measures of discord including the original measure of discord, relative entropy-based discord, geometric discord, global discord, Gaussian discord, are reviewed and discussed.
Quantum discord is usually referred to as a measure for quantum correlations. In the search of the fundamental resource to gain a quantum advantage in quantum information applications, quantum discord is considered a promising candidate. In this paper we present an alternative view on quantum correlations in terms of the rank of the correlation matrix as introduced by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)]. According to our analysis, information about the quantum discord does not necessarily determine the amount of quantum correlations but rather the quantumness of the state. Nonzero quantum discord is only a necessary but not a sufficient condition for correlations above the classically achievable limit. This becomes clear when we consider states of nonzero discord, which can be created from zero discord states only by a single local operation. We further show that the set of these states has measure zero.
Quantum discord, a measure of genuinely quantum correlations, is generalized to continuous variable systems. For all two-mode Gaussian states, we calculate analytically the quantum discord and a related measure of classical correlations, solving an optimization over all Gaussian measurements. Almost all two-mode Gaussian states are shown to have quantum correlations, while for separable states, the discord is smaller than unity. For a given amount of entanglement, it admits tight upper and lower bounds. Via a duality between entanglement and classical correlations, we derive a closed formula for the Gaussian entanglement of formation of a family of three-mode Gaussian states.
We introduce a monogamy inequality for quantum correlations, which implies that the sum of pairwise quantum correlations is upper limited by the amount of multipartite quantum correlations as measured by the global quantum discord. This monogamy bound holds either for pure or mixed quantum states provided that bipartite quantum discord does not increase under discard of subsystems. We illustrate the monogamy behavior for multipartite pure states with Schmidt decomposition as well as for WWW-GHZ mixed states. As a by-product, we apply the monogamy bound to investigate residual multipartite correlations.
In quantum mechanics, quantum coherence of a state relative to a quantum measurement can be identified with the quantumness that has to be destroyed by the measurement. In particular, quantum coherence of a bipartite state relative to a local quantum measurement encodes quantum correlations in the state. If one takes minimization with respect to the local measurements, then one is led to quantifiers which capture quantum correlations from the perspective of coherence. In this vein, quantum discord, which quantifies the minimal correlations that have to be destroyed by quantum measurements, can be identified as the minimal coherence, with the coherence measured by the relative entropy of coherence. To advocate and formulate this idea in a general context, we first review coherence relative to Lüders measurements which extends the notion of coherence relative to von Neumann measurements (or equivalently, orthonomal bases), and highlight the observation that quantum discord arises as minimal coherence through two prototypical examples. Then, we introduce some novel measures of quantum correlations in terms of coherence, illustrate them through examples, investigate their fundamental properties and implications, and indicate their applications to quantum metrology.