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arXiv:1101.5937 · Physical Review A

Entanglement in the classical limit: Quantum correlations from classical probabilities

A. Matzkin

Published Aug 24, 2011• 21 citations
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Abstract

We investigate entanglement for a composite closed system endowed with a scaling property which allows the dynamics to be kept invariant while the effective Planck constant ${\ensuremath{\hbar}}_{\mathit{eff}}$ of the system is varied. Entanglement increases as ${\ensuremath{\hbar}}_{\mathit{eff}}\ensuremath{\rightarrow}0$. Moreover, for sufficiently low ${\ensuremath{\hbar}}_{\mathit{eff}}$ the evolution of the quantum correlations, encapsulated, for example, in the quantum discord, can be obtained from the mutual information of the corresponding classical system. We show this behavior is due to the local suppression of path interferences in the interaction that generates the entanglement.

DOI
10.1103/physreva.84.022111

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We investigate entanglement for a composite closed system endowed with a scaling property which allows the dynamics to be kept invariant while the effective Planck constant \ensuremathℏeff{\ensuremath{\hbar}}_{\mathit{eff}}\ensuremathℏeff​ of the system is varied. Entanglement increases as \ensuremathℏeff\ensuremath→0{\ensuremath{\hbar}}_{\mathit{eff}}\ensuremath{\rightarrow}0\ensuremathℏeff​\ensuremath→0. Moreover, for sufficiently low \ensuremathℏeff{\ensuremath{\hbar}}_{\mathit{eff}}\ensuremathℏeff​ the evolution of the quantum correlations, encapsulated, for example, in the quantum discord, can be obtained from the mutual information of the corresponding classical system. We show this behavior is due to the local suppression of path interferences in the interaction that generates the entanglement.
Interacting quantum systems evolving from an uncorrelated composite initial state generically develop quantum correlations---entanglement. As a consequence, a local description of interacting quantum systems is impossible as a rule. A unitarily evolving (isolated) quantum system generically develops extensive entanglement: the magnitude of the generated entanglement will increase without bounds with the effective Hilbert space dimension of the system. It is conceivable that coupling of the interacting subsystems to local dephasing environments will restrict the generation of entanglement to such extent that the evolving composite system may be considered as approximately disentangled. This conjecture is addressed in the context of some common models of a bipartite system with linear and nonlinear interactions and local coupling to dephasing environments. Analytical and numerical results obtained imply that the conjecture is generally false. Open dynamics of the quantum correlations is compared to the corresponding evolution of the classical correlations and a qualitative difference is found.
We show a relation between entanglement and correlations of any form. The internal entanglement of a bipartite system, and its correlations with another system, limit each other. A measure of correlations, of any nature, cannot increase under local operations. Examples are the entanglement monotones, the mutual information that quantifies total correlations, and the Henderson-Vedral measure of classical correlations. External correlations, evaluated by such a measure, set a tight upper bound on the internal entanglement that decreases as they increase, and so does quantum discord.
How strongly can one system be correlated with another? In the classical world, this basic question concerning correlative capacity has a very satisfying answer: The ``effective size'' of the marginal system, as quantified by the Shannon entropy, sets a tight upper bound to the correlations, as quantified by the mutual information. Although in the quantum world bipartite correlations, like their classical counterparts, are also well quantified by mutual information, the similarity ends here: The correlations in a bipartite quantum system can be twice as large as the marginal entropy. In the paradigm of quantum discord, the correlations are split into classical and quantum components, and it was conjectured that both the classical and quantum correlations are (like the classical mutual information) bounded above by each subsystem's entropy. In this work, by exploiting the interplay between entanglement of formation, mutual information, and quantum discord, we disprove that conjecture. We further indicate a scheme to restore harmony between quantum and classical correlative capacities. The results illustrate dramatically the asymmetric nature of quantum discord and highlight some subtle and unusual features of quantum correlations.
75 years after the term "entanglement" was coined to a peculiar feature inherent to quantum systems, the connection between quantum and classical mechanics remains an open problem. Drawing on recent results obtained in semiclassical systems, we discuss here the fate of entanglement in a closed system as Planck's constant becomes vanishingly small. In that case the generation of entanglement in a quantum system is perfectly reproduced by properly defined correlations of the corresponding classical system. We speculate on what these results could imply regarding the status of entanglement and of the ensuing quantum correlations.
Abstract. 75 years after the term "entanglement " was coined to a peculiar feature inherent to quantum systems, the connection between quantum and classical mechanics remains an open problem. Drawing on recent results obtained in semiclassical systems, we discuss here the fate of entanglement in a closed system as Planck’s constant becomes vanishingly small. In that case the generation of entanglement in a quantum system is perfectly reproduced by properly defined correlations of the corresponding classical system. We speculate on what these results could imply regarding the status of entanglement and of the ensuing quantum correlations.
It is well known that quantum entanglement makes certain tasks in quantum information theory possible. However, there are also quantum tasks that display a quantum advantage without entanglement. Distinguishing classical and quantum correlations in quantum systems is therefore of both practical and fundamental importance. Realistic quantum systems are not closed, and therefore it is important to study the various correlations when the system loses its coherence due to interactions with the environment. In this paper, we study in detail the dynamics of different kinds of correlations, classical correlation, quantum discord and entanglement in open quantum systems, in particular, a two-qubit system evolving under Kossakowski-type quantum dynamical semigroups of completely positive maps. In such an environment, classical and quantum correlations can even persist asymptotically. By analytic and numerical analysis, we find that the quantum discord is larger than the classical correlation for asymptotic states. Furthermore, we show that the quantum discord is more resistant to the action of the environment than quantum entanglement, and it can persist even in the asymptotic long-time regime.