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Communications in Theoretical Physics

Correlations in Werner States

Luo Shun-Long, Nan Li

Published Feb 1, 2008• 3 citations
Publisher

Abstract

Werner states are paradigmatic examples of quantum states and play an innovative role in quantum information theory. In investigating the correlating capability of Werner states, we find the curious phenomenon that quantum correlations, as quantified by the entanglement of formation, may exceed the total correlations, as measured by the quantum mutual information. Consequently, though the entanglement of formation is so widely used in quantifying entanglement, it cannot be interpreted as a consistent measure of quantum correlations per se if we accept the folklore that total correlations are measured (or rather upper bounded) by the quantum mutual information.

DOI
10.1088/0253-6102/49/2/15

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Werner states are paradigmatic examples of quantum states and play an innovative role in quantum information theory. In investigating the correlating capability of Werner states, we find the curious phenomenon that quantum correlations, as quantified by the entanglement of formation, may exceed the total correlations, as measured by the quantum mutual information. Consequently, though the entanglement of formation is so widely used in quantifying entanglement, it cannot be interpreted as a consistent measure of quantum correlations per se if we accept the folklore that total correlations are measured (or rather upper bounded) by the quantum mutual information.
On the premises that total correlations in a bipartite quantum state are measured by the quantum mutual information, and that separation of total correlations into quantum and classical parts satisfies an intuitive dominance relation, we examine to what extent various entropic entanglement measures, such as the distillable entanglement, the relative entropy entanglement, the squashed entanglement, the entanglement cost, and the entanglement of formation, can be regarded as consistent measures of quantum correlations. We illustrate that the entanglement of formation often overestimates quantum correlations and thus is too big to be a genuine measure of quantum correlations. This indicates that the entanglement of formation does not quantify the quantum correlations intrinsic to a quantum state, but rather characterizes the pure entanglement needed to build the quantum state via local operations and classical communication. Furthermore, it has the consequence that, if the additive conjecture for the entanglement of formation is true (as is widely believed), then the entanglement cost, which is an operationally defined measure of entanglement with significant physical meaning, cannot be a consistent measure of quantum correlations in the sense that it may exceed total correlations. Alternatively, if the entanglement cost is dominated by total correlations, as our intuition suggests, then we can immediately disprove the additive conjecture. Both scenarios have their counterintuitive and appealing aspects, and a natural challenge arising in this context is to prove or disprove that the entanglement cost is dominated by the quantum mutual information.
The maximal violation of Bell inequality and correlations measured by mutual information and relative entropy for the Werner state are calculated.It is found that the measurements that result in the maximal violation of Bell inequality can extract the most information from the Werner state;in the two kinds of measures of correlation,quantum discord is all greater than classical correlation,whereas entanglement does not necessarily greater than classical correlation even in the case of the maximal violation of Bell inequality,which means that quantum discord can describe the nonlocality of correlation much better than entanglement so that makes it have wider application in quantum communication and quantum computation.
Abstract In Li and Luo (2007 Phys. Rev. A 76 032327), the inequality ( 1 / 2 )  ≥  was identified as a fundamental postulate for a consistent theory of quantum versus classical correlations for arbitrary measures of total  and quantum  correlations in bipartite quantum states. Besides, Hayden et al (2006 Commun. Math. Phys. 265 95) have conjectured that, in some conditions within systems endowed with infinite-dimensional Hilbert spaces, quantum correlations may dominate not only half of total correlations but total correlations itself. Here, in a two-mode Gaussian state, quantifying  and  respectively by the quantum mutual information  G and the entanglement of formation (EoF)  F G , we verify that  F , R G is always less than ( 1 / 2 )  R G when  G and  F G are defined via the Rényi-2 entropy. While via the von Neumann entropy,  F , V G may even dominate 
Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.
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.