On the Significance of the Quantum Mechanical Covariance Matrix

TL;DR

This paper introduces a covariance matrix framework to characterize quantum correlations and nonlocal theories, revealing new Bell inequalities and linking non-classicality to Tsallis entropy, with implications for experimental validation.

Contribution

It presents a novel covariance matrix approach to characterize quantum and nonlocal correlations, uncovering new Bell inequalities and operational measures of non-classicality.

Findings

Quantum correlations can be characterized by a specific covariance matrix.

New Bell-type inequalities relate nonlocality to non-additive entropy.

Proposes experimental tests with weak measurements in Bell scenarios.

Abstract

The characterization of quantum correlations, being stronger than classical, yet weaker than those appearing in non-signaling models, still poses many riddles. In this work we show that the extent of binary correlations in a general class of nonlocal theories can be characterized by the~existence of a certain covariance matrix. The set of quantum realizable two-point correlators in the~bipartite case then arises from a subtle restriction on the structure of this general covariance matrix. We also identify a class of theories whose covariance does not have neither a quantum nor an "almost quantum" origin, but which nevertheless produce the accessible two-point quantum mechanical correlators. Our approach leads to richer Bell-type inequalities in which the extent of nonlocality is intimately related to a non-additive entropic measure. In particular, it suggests that the Tsallis entropy with parameter q=1/2 is a natural operational measure of non-classicality. Moreover, when~generalizing this covariance matrix we find novel characterizations of the quantum mechanical set of correlators in multipartite scenarios. All these predictions might be experimentally validated when adding weak measurements to the conventional Bell test (without adding postselection).