Violations of Bell inequalities from random pure states

TL;DR

This paper analyzes how random pure quantum states violate Bell inequalities, specifically the CGLMP inequality, revealing that violations occur for larger outcome numbers N, with results derived using random matrix theory.

Contribution

It provides an analytical and numerical study of Bell inequality violations from random pure states across different N values, connecting quantum information with random matrix theory.

Findings

Bell violations are absent for small N on average

Violations increase and are observed for larger N

Analytical results connect random matrix theory to quantum violations

Abstract

We consider the expected violations of Bell inequalities from random pure states. More precisely, we focus on a slightly generalised version of the CGLMP inequality, which concerns Bell experiments of two parties, two measurement options and N outcomes and analyse their expected quantum violations from random pure states for varying N, assuming the conjectured optimal measurement operators. It is seen that for small N the Bell inequality is not violated on average, while for larger N it is. Both ensembles of unstructured as well as structured random pure states are considered. Using techniques from random matrix theory this is obtained analytically for small and large N and numerically for intermediate N. The results show a beautiful interplay of different aspects of random matrix theory, ranging from the Marchenko-Pastur distribution and fixed-trace ensembles to the O(n) model.