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 gamma = ( < u_i , v_j > )_{i,j=1}^n , where the vectors u_i and v_j 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 alpha = m/n , where the previous vectors are independent and uniformly distributed in the unit sphere of R^m. In this situation, our main result shows the existence of two completely different regimes: There exists an alpha_0 > 0 such that if alpha leq alpha_0, then gamma is nonlocal with probability tending to 1 as n rightarrow infty. On the other hand, if alpha geq 2 then gamma is local with probability tending to 1 as n rightarrow infty.