Designing Bell inequalities from a Tsirelson bound

TL;DR

This paper introduces an analytic bound on the quantum value of correlation Bell inequalities, based on the maximal singular value of their coefficient matrix, aiding in the design and analysis of such inequalities.

Contribution

It provides a new simple bound similar to Tsirelson's bound, with criteria for tightness, applicable to many well-known Bell inequalities, and helps in constructing inequalities that reveal measurement dimensions.

Findings

The bound is tight for many famous Bell inequalities.

The bound is based on the maximal singular value of the coefficient matrix.

It facilitates the construction of Bell inequalities that witness measurement dimensions.

Abstract

We present a simple analytic bound on the quantum value of general correlation type Bell inequalities, similar to Tsirelson's bound. It is based on the maximal singular value of the coefficient matrix associated with the inequality. We provide a criterion for tightness of the bound and show that the class of inequalities where our bound is tight covers many famous examples from the literature. We describe how this bound helps to construct Bell inequalities, in particular inequalities that witness the dimension of the measured observables.