Criteria for measures of quantum correlations

Abstract

Entanglement does not describe all quantum correlations and several authors have shown the need to go beyond entanglement when dealing with mixed states. Various different measures have sprung up in the literature, for a variety of reasons, to describe bipartite and multipartite quantum correlations; some are known under the collective name {\it quantum discord}. Yet, in the same sprit as the criteria for entanglement measures, there is no general mechanism that determines whether a measure of quantum and classical correlations is a proper measure of correlations. This is partially due to the fact that the answer is a bit muddy. In this article we attempt tackle this muddy topic by writing down several criteria for a ``good" measure of correlations. We breakup our list into \emph{necessary}, \emph{reasonable}, and \emph{debatable} conditions. We then proceed to prove several of these conditions for generalized measures of quantum correlations. However, not all conditions are met by all measures; we show this via several examples. The reasonable conditions are related to continuity of correlations, which has not been previously discussed. Continuity is an important quality if one wants to probe quantum correlations in the laboratory. We show that most types of quantum discord are continuous but none are continuous with respect to the measurement basis used for optimization.

Measurement-induced geometric measures of correlations based on the Hellinger distance and their local decoherence

“…In this section, we apply the modified Brodutch and Modi method of constructing geometric measures of correlations [22,23] to obtain analytical expressions for measurement-induced geometric classical and quantum correlations based on the Hellinger distance for two-qubit Bell diagonal states.…”

“…• M is a non-selective rank-1 projective measurement performed on one subsystem of the multipartite system in a state ρ, and M minimizes the quantum correlations Q(ρ) [22], but if the classical correlations C(ρ) are not uniquely determined by the minimization procedure, then the classical correlations C(ρ) are additionally maximized over the all measurements M that minimize the quantum correlations Q(ρ) [23], • M is a non-selective rank-1 projective measurement performed on one subsystem of the multipartite system in a state ρ, and M maximizes the classical correlations C(ρ) [22], but if the quantum correlations Q(ρ) are not uniquely determined by the maximization procedure, then the quantum correlations Q(ρ) are additionally minimized over the all measurements M that maximize the classical correlations C(ρ) [23].…”

“…The problems with geometric quantum discord based on the Schatten 2-norm have highlighted the need for a general method of constructing bona fide measures of correlations within the information-theoretic paradigm. Recently, Brodutch and Modi [22] proposed a method in which quantum correlations present in composite quantum systems are quantified by a distance between a given multipartite quantum state and the classical-quantum state emerging from a measurement performed on the considered state, where the measurement is chosen according to a specific strategy. Moreover, classical correlations are quantified by a distance between the classical-quantum state and the completely separable state resulting from the same measurement performed on the tensor product of the states of the individual subsystems.…”

“…Moreover, classical correlations are quantified by a distance between the classical-quantum state and the completely separable state resulting from the same measurement performed on the tensor product of the states of the individual subsystems. Furthermore, Brodutch and Modi [22] identified two strategies that provide bona fide measures of classical and quantum correlations that satisfy the following necessary conditions: (i) product states have no correlations, (ii) all correlations are invariant under local unitary operations, (iii) all correlations are non-negative, and (iv) classical states have no quantum correlations. However, it has been shown that the Brodutch and Modi method should be modified as in the case of measurement-induced classical and quantum correlations based on the trace distance for two-qubit Bell diagonal states one of the two possible strategies results in the non-uniqueness of classical correlations [23].…”

On the quantumness of correlations in nuclear magnetic resonance

“…In other words, we just have to distinguish between classical and quantum correlations. In the same spirit as what happens to entanglement, some non-classical correlation criteria and observable witnesses were proposed in [35][36][37][38][39][40] and experimentally verified in [13,41,42]. A classicality (or non-classicality) witness is regarded as an observable (or a set of observables) that can be directly measured in an experimental setup.…”