Bell inequalities for three systems and arbitrarily many measurement outcomes

TL;DR

This paper introduces a new family of Bell inequalities for three-party systems with many outcomes, showing they can serve as tools to witness the dimension of quantum systems and revealing the relationship between quantum violations and Hilbert space dimensions.

Contribution

The authors generalize Mermin Bell inequalities to multiple outcomes and analyze their facets, quantum violations, and potential as multipartite dimension witnesses.

Findings

Inequalities define facets for small outcome numbers.

Quantum violations increase with Hilbert space dimension.

Maximal violations require systems with higher local dimensions.

Abstract

We present a family of Bell inequalities for three parties and arbitrarily many outcomes, which can be seen as a natural generalization of the Mermin Bell inequality. For a small number of outcomes, we verify that our inequalities define facets of the polytope of local correlations. We investigate the quantum violations of these inequalities, in particular with respect to the Hilbert space dimension. We provide strong evidence that the maximal quantum violation can only be reached using systems with local Hilbert space dimension exceeding the number of measurement outcomes. This suggests that our inequalities can be used as multipartite dimension witnesses.