Criteria for measures of quantum correlations

TL;DR

This paper establishes criteria for evaluating measures of quantum correlations, including quantum discord, emphasizing the importance of continuity and analyzing whether existing measures meet these standards.

Contribution

It introduces a structured set of criteria for good quantum correlation measures and assesses existing measures against these standards, highlighting gaps and issues.

Findings

Most quantum discord measures are continuous.

Quantum discord measures are not continuous with respect to measurement basis.

Not all measures satisfy all proposed criteria.

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 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 necessary, reasonable, and 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.