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arXiv:1203.4612

Markovian Classicality from Zero Discord for Bipartite Quantum Systems

M. Arsenijević, J. Jeknić-Dugić, M. Dugić

Published Mar 20, 2012• 1 citation
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Abstract

Modern quantum information theory provides new tools for investigating the decoherence-induced "classicality" of open quantum systems. Recent observation that almost all quantum states bear non-classical correlations [A. Ferraro {\it et al}, Phys. Rev. A {\bf 81}, 052318 (2010)] distinguishes the zero-discord classicality essentially as a pathology of the Markovian bipartite-systems realm. Nevertheless, we formally construct such a classical model and its variant that represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of the Markovian zero-discord classicality. A need for the more elaborate and more systematic search for the alternative such models reveals we are still learning about the very meaning of "classicality" in the realm of open quantum systems.

DOI
10.48550/arxiv.1203.4612

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Modern quantum information theory provides new tools for investigating the decoherence-induced "classicality" of open quantum systems. Recent observation that almost all quantum states bear non-classical correlations [A. Ferraro {\it et al}, Phys. Rev. A {\bf 81}, 052318 (2010)] distinguishes the zero-discord classicality essentially as a pathology of the Markovian bipartite-systems realm. Nevertheless, we formally construct such a classical model and its variant that represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of the Markovian zero-discord classicality. A need for the more elaborate and more systematic search for the alternative such models reveals we are still learning about the very meaning of "classicality" in the realm of open quantum systems.
Recent observation that almost all quantum states bear non-classical correlations [A. Ferraro et al, Phys. Rev. A 81, 052328 (2010)] may seem to imply that the Markovian bipartite systems are practically deprived of zero discord states. Nevertheless, complementary to the result of Ferraro et al, we construct a model of a Markovian bipartite system providing zero discord for arbitrary long time interval, that we term 'Markovian classicality'. Our model represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of Markovian classicality. Interestingly enough, we are not able to offer any alternative to the model. Physical relevance of the model is twofold. First, the model is in intimate relation to the topics of quantum information locality, quantum discord saturation and quantum decorrelation. Second, the model is of the general physical interest. It pertains to a specific structure (decomposition into parts/subsystems) of a composite system, not to a special physical kind of composite systems. Being a characteristic of a structure, by definition, the model of Markovian classicality is not a model of sudden death of discord. We emphasize wide-range implications of our results.
Quantum discord measures the total non-classical correlations in a composite quantum system. As a possible resource for the classically non-achievable tasks, amount of quantum discord is crucial for efficient performance of the quantum technology tasks. Zero discord means the absence of the required resources. For a wide class of the bipartite open quantum systems, initial nonzero discord is known practically never goes to zero, except asymptotically. However, if initial discord equals zero, there remains the possibility to remain zero for the finite time intervals for certain Markovian models of open systems-that we call Markovian classicality. In this paper we search for the models that allow for Markovian classicality. We point out a general model that represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of Markovian classicality. Physical relevance of the model is twofold. First, the model is in intimate relation to the topics of quantum information locality, quantum discord saturation and quantum decorrelation. Second, the model is of the general physical interest. It pertains to a specific structure (decomposition into parts/subsystems) of a composite system, not to a special kind of composite system. Being a characteristic of a structure, by definition, the model of Markovian classicality is not a model of sudden death of discord. We emphasize wide-range implications of our results.
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.
Knowledge of the dynamical behavior of correlations with no classical counterpart, like entanglement, nonlocal correlations and quantum discord, in open quantum systems is of primary interest because of the possibility to exploit these correlations for quantum information tasks. Here we review some of the most recent results on the dynamics of correlations in bipartite systems embedded in non-Markovian environments that, with their memory effects, influence in a relevant way the system dynamics and appear to be more fundamental than the Markovian ones for practical purposes. Firstly, we review the phenomenon of entanglement revivals in a two-qubit system for both independent environments and a common environment. We then consider the dynamics of quantum discord in non-Markovian dephasing channel and briefly discuss the occurrence of revivals of quantum correlations in classical environments.
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.