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 has 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 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).
Highlights
The extent of nonlocality is commonly determined by a set of correlations
This matrix, which may be defined in any statistical theory, implies a bound on two-point correlators analogous to that of quantum mechanics
We prove that all potential theories having a covariance structure similar to that of quantum mechanics have a similar set of realizable correlators
Summary
The extent of nonlocality is commonly determined by a set of correlations. In the simplest bipartite scenario, the four two-point correlators c1 , c2 , c3 and c4 , corresponding to the four pairs of possible outcomes of Alice and Bob, may render the theory classical, quantum, or stronger-than-quantum. We tell the richer story provided by a certain covariance matrix presented This matrix, which may be defined in any statistical theory, implies a bound on two-point correlators analogous to that of quantum mechanics. We prove that all potential theories having a covariance structure similar to that of quantum mechanics have a similar set of realizable correlators. While the positive semi-definiteness property plays a role in both, the particular covariance here leads to the identification of fundamental relations between the entries in this matrix These relations alone are shown to govern the set of realizable binary bipartite correlators in quantum mechanics but in any nonlocal theory, and to imply new tighter bounds on this set
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