Bell Inequalities from Polyhedral Sampling

TL;DR

This paper introduces the Adjacency Sampling method to efficiently generate Bell inequalities, significantly expanding the known classes in complex scenarios where complete enumeration is infeasible.

Contribution

The paper presents a faster, approximate sampling technique for Bell inequalities that outperforms existing partial results in complex scenarios.

Findings

Reproduces all known Bell inequality classes in tested cases.

Discovered over 1.29×10^8 classes in the ,3,3,3 scenario, surpassing previous counts.

Nearly tripled the number of known classes in the ,5,2,2 scenario.

Abstract

Bell inequalities play a central role in certifying quantum correlations and underpin protocols such as device-independent quantum key distribution. However, enumerating all Bell inequalities for a given scenario remains intractable beyond the simplest cases, as it requires solving a computationally hard facet enumeration problem on the associated Bell polytope. We propose the Adjacency Sampling method, which builds on the Adjacency Decomposition method but sacrifices completeness for speed. On previously solved Bell polytopes, the method reproduces every known class of inequalities. For scenarios where no complete enumeration exists, it greatly exceeds existing partial results: in $\mathcal{L}_{3,3,3,3}$ we obtain over $1.29 \times 10^8$ classes, more than 25 times the previous count; in $\mathcal{L}_{4,5,2,2}$ we nearly triple the known list to 49\,358 classes; and for $\mathcal{L}_{4,6,2,2}$ we report over 4.3 million classes.