Random quantum correlations are generically non-classical

TL;DR

This paper demonstrates that most random bipartite quantum correlations near the quantum boundary are non-classical, providing Bell inequalities to certify their non-classicality through norm estimations of random matrices.

Contribution

It introduces a probabilistic framework showing that generic quantum correlations are non-classical and provides explicit Bell inequalities for certification.

Findings

Most boundary quantum correlations are non-classical with high probability.

A new method to estimate quantum and classical norms of random matrices.

Derived Bell inequalities certifying non-classicality of random correlations.

Abstract

It is now a well-known fact that the correlations arising from local dichotomic measurements on an entangled quantum state may exhibit intrinsically non-classical features. In this paper we delve into a comprehensive study of random instances of such bipartite correlations. The main question we are interested in is: given a quantum correlation, taken at random, how likely is it that it is truly non-explainable by a classical model? We show that, under very general assumptions on the considered distribution, a random correlation which lies on the border of the quantum set is with high probability outside the classical set. What is more, we are able to provide the Bell inequality certifying this fact. On the technical side, our results follow from (i) estimating precisely the "quantum norm" of a random matrix, and (ii) lower bounding sharply enough its "classical norm", hence proving a gap between the two. Along the way, we need a non-trivial upper bound on the $\infty{\rightarrow}1$ norm of a random orthogonal matrix, which might be of independent interest.