Random and free observables saturate the Tsirelson bound for CHSH inequality

TL;DR

This paper demonstrates that random and free observables can nearly saturate the Tsirelson bound for the CHSH inequality, challenging the notion that maximal violation requires anticommuting observables.

Contribution

It introduces the concept of free observables and uses free probability theory to analyze their properties, showing near-maximal CHSH violations with random matrices.

Findings

Random observables exhibit near-maximal CHSH violation.

Free observables reproduce random matrix statistics in the infinite limit.

Constructed steering inequality shows large violation.

Abstract

Maximal violation of the CHSH-Bell inequality is usually said to be a feature of anticommuting observables. In this work we show that even random observables exhibit near-maximal violations of the CHSH-Bell inequality. To do this, we use the tools of free probability theory to analyze the commutators of large random matrices. Along the way, we introduce the notion of "free observables" which can be thought of as infinite-dimensional operators that reproduce the statistics of random matrices as their dimension tends towards infinity. We also study the fine-grained uncertainty of a sequence of free or random observables, and use this to construct a steering inequality with a large violation.