Generalized Tsirelson's bound from parity symmetry considerations

TL;DR

This paper explores a family of two-outcome random games with specific symmetry constraints, demonstrating how to construct local hidden variable models that can violate Bell inequalities beyond Tsirelson's bound up to the algebraic maximum.

Contribution

It introduces a geometric framework for understanding Bell inequality violations, extending beyond Tsirelson's bound within a class of symmetric correlation functions.

Findings

Bell inequality can be violated up to the algebraic maximum of 4.

Violation amount is a geometric property of the correlation functions.

Family of models includes the standard Bell experiment as a special case.

Abstract

The Bell experiment is a random game with two binary outcomes whose statistical correlation is given by $E_0(\Theta)=-\cos(\Theta)$, where $\Theta \in [-\pi, \pi)$ is an angular input that parameterizes the game setting. The correlation function $E_0(\Theta)$ belongs to the affine space ${\cal H} \equiv \left\{E(\Theta)\right\}$ of all continuous and differentiable periodic functions $E(\Theta)$ that obey the parity symmetry constraints $E(-\Theta)=E(\Theta)$ and $E(\pi-\Theta)=-E(\Theta)$ with $E(0)=-1$ and, furthermore, are strictly monotonically increasing in the interval $[0, \pi)$. Here we show how to build explicitly local statistical models of hidden variables for random games with two binary outcomes whose correlation function $E(\Theta)$ belongs to the affine space ${\cal H}$. This family of games includes the Bell experiment as a particular case. Within this family of random games, the Bell inequality can be violated beyond the Tsirelson bound of $2\sqrt{2}$ up to the maximally allowed algebraic value of 4. In fact, we show that the amount of violation of the Bell inequality is a purely geometric feature.