Sampling quantum nonlocal correlations with high probability

TL;DR

This paper investigates the probability of quantum nonlocal correlations arising from randomly sampled vectors, establishing thresholds for when such correlations are likely to be nonlocal or local as the system size grows.

Contribution

It provides probabilistic thresholds for the emergence of nonlocal quantum correlations based on the ratio of dimensions in high-dimensional sampling.

Findings

Nonlocal correlations occur with high probability when the sampling ratio is below a certain threshold.

Local correlations dominate with high probability when the sampling ratio exceeds 2.

The results quantify how high-dimensional random sampling influences quantum nonlocality.

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 $n\rightarrow \infty$.