Sampling quantum nonlocal correlations with high probability

Abstract

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form γ=(⟨ui,vj⟩)ni,j=1, where the vectors ui and vj are in the unit ball of a real Hilbert space. In this work we study the probability of the nonlocal nature of these correlations as a function of α=mnα=mn, where the previous vectors are sampled according to the Haar measure in the unit sphere of RmRm. In particular, we prove the existence of an α0>0 such that if α≤α0, γ is nonlocal with probability tending to 1 as n→∞, while for α>2, γ is local with probability tending to 1 as n→∞.