Small violations of Bell inequalities for multipartite pure random states

TL;DR

This paper investigates the likelihood of random multipartite pure states violating Bell inequalities, showing that such violations become exponentially unlikely as the number of parts or local dimension increases.

Contribution

It provides probabilistic bounds on Bell inequality violations for random states, revealing conditions under which violations are rare in large systems.

Findings

Probability of violation decreases exponentially with the number of parts.

Violations become unlikely as local dimension increases for systems with at least 3 parts.

Results apply to states with uniformly bounded coefficients in Bell scenarios.

Abstract

For any finite number of parts, measurements and outcomes in a Bell scenario we estimate the probability of random $N$-qu$d$it pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the the local Hilbert space dimension goes to infinity.