Avenues to generalising Bell inequalities

TL;DR

This paper extends methods for characterizing Bell inequalities to scenarios with multiple measurement outcomes, providing new families of inequalities and insights into their structure and applications in quantum nonlocality detection.

Contribution

It generalizes existing Bell inequality frameworks to multi-outcome scenarios using complex correlation functions, and extends the CGLMP inequality to three parties.

Findings

New families of Bell inequalities with understandable structure

Numerical testing shows their effectiveness in detecting nonclassicality

Generalization of CGLMP inequality to three-party scenarios

Abstract

Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site - for arbitrary number of parties - was given in Refs. [Phys. Rev. A 64, 010102(R) (2001)] and [Phys. Rev. A 64, 032112 (2001)]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than 2 multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure.We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.