On the inequivalence of the CH and CHSH inequalities due to finite statistics

TL;DR

This paper investigates how finite sample sizes in experiments cause inequivalence between Bell inequalities like CH and CHSH, affecting their robustness and proposing a method to optimize their statistical reliability.

Contribution

It introduces a group-theoretic framework to analyze the impact of finite statistics on Bell inequality violations and suggests an optimization approach for robustness.

Findings

Finite sample sizes cause inequivalence between CH and CHSH inequalities.

The group-theoretic decomposition links distribution properties to symmetry representations.

A method to optimize the statistical robustness of Bell inequality violations is proposed.

Abstract

Different variants of a Bell inequality, such as CHSH and CH, are known to be equivalent when evaluated on nonsignaling outcome probability distributions. However, in experimental setups, the outcome probability distributions are estimated using a finite number of samples. Therefore the nonsignaling conditions are only approximately satisfied and the robustness of the violation depends on the chosen inequality variant. We explain that phenomenon using the decomposition of the space of outcome probability distributions under the action of the symmetry group of the scenario, and propose a method to optimize the statistical robustness of a Bell inequality. In the process, we describe the finite group composed of relabeling of parties, measurement settings and outcomes, and identify correspondences between the irreducible representations of this group and properties of outcome probability distributions such as normalization, signaling or having uniform marginals.