Exact values of quantum violations in low-dimensional Bell correlation inequalities

TL;DR

This paper analytically calculates the exact quantum violation values for Bell inequalities with up to four measurements, extending beyond the well-known CHSH case, and provides structural insights into Bell polytopes.

Contribution

It provides the first analytical computation of quantum violations for low-dimensional Bell inequalities beyond CHSH, including detailed polytope structures.

Findings

Exact quantum violation values for Bell inequalities with up to four measurements.

All violations are smaller than the Tsirelson bound of √2.

Tables summarizing the facial structure of low-dimensional Bell polytopes.

Abstract

The famous Clauser-Horne-Shimony-Holt (CHSH) inequality certifies a quantum violation, by a factor $\sqrt{2}$, of correlations predicted by the classical view of the world in the simplest possible nontrivial measurement setup (two systems with two dichotomic measurements each). In such setting, this is the largest possible violation, which is known as the \emph{Tsirelson bound}. In this paper we calculate the exact values of quantum violations for the other Bell correlation inequalities that appear in the setups involving up to four measurements; they are all smaller than $\sqrt{2}$. While various authors investigated these inequalities via numerical methods, our approach is analytic. We also include tables summarizing facial structure of Bell polytopes in low dimensions.