How Many Quantum Correlations Are Not Local?

We study how generic is the property of nonlocality among the set of quantum correlations for bipartite dichotomic measurements. To do so, we consider the characterization of these quantum correlations as those of the form γ = (〈ui, vj〉)i,j=1, where the vectors ui and vj are in the unit sphere of a real Hilbert space. The important parameters in this description are the number of vectors n and the dimension of the Hilbert space m. Thus, it is natural to study the probability of a quantum correlation being nonlocal as a function of α = mn , where the previous vectors are independent and uniformly distributed in the unit sphere of R. In this situation, our main result shows the existence of two completely different regimes: There exists an α0 > 0 such that if α ≤ α0, then γ is nonlocal with probability tending to 1 as n → ∞. On the other hand, if α ≥ 2 then γ is local with probability tending to 1 as n→∞. 1998 ACM Subject Classification H.1.1 Systems and Information Theory