From Bell Inequalities to Tsirelson's Theorem: A Survey

TL;DR

This survey explains Bell inequalities and Tsirelson's theorem, provides an explicit optimal construction, and discusses methods for obtaining upper bounds on quantum violations, including recent advances using semidefinite programming.

Contribution

It offers an accessible introduction, an explicit optimal construction related to Tsirelson's theorem, and surveys recent techniques for bounding quantum violations of Bell inequalities.

Findings

Explicit optimal construction for Tsirelson's theorem

Methods to derive upper bounds on quantum violations

Recent results on solving infinite semidefinite programs

Abstract

The first part of this paper contains an introduction to Bell inequalities and Tsirelson's theorem for the non-specialist. The next part gives an explicit optimum construction for the "hard" part of Tsirelson's theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson's theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.