Sampling Quantum Nonlocal Correlations with High Probability

Abstract

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form ${\gamma=(\langle u_i,v_j\rangle)_{i,j=1}^n}$, where the vectors u _i and v _j 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 ${\alpha=\frac{m}{n}}$, where the previous vectors are sampled according to the Haar measure in the unit sphere of ${\mathbb R^m}$. In particular, we prove the existence of an ${\alpha_0 > 0}$ such that if ${\alpha\leq \alpha_0}$, ${\gamma}$ is nonlocal with probability tending to 1 as ${n\rightarrow \infty}$, while for ${\alpha > 2}$, ${\gamma}$ is local with probability tending to 1 as ${n\rightarrow \infty}$.