Tight Bell inequalities from polytope slices

TL;DR

This paper advances the study of Bell nonlocality by introducing a polytope-slicing method to derive new tight bipartite Bell inequalities and to enumerate facets for several high-dimensional scenarios. It delivers (often complete) facet lists for $(6,3,2,2)$, $(3,3,3,2)$, $(3,2,3,3)$, and $(2,2,3,5)$, with extensive facet inventories for $(4,4,2,2)$, $(3,3,4,2)$ and $(4,3,3,2)$, and analyzes quantum bounds via the NPA hierarchy and seesaw methods across $d=2,3,4$. For each inequality, the work reports quantum violations, noise resistance, and minimal detection efficiencies to close the detection loophole, connecting geometric structure to performance metrics. The results identify scenarios that outperform CHSH in terms of visibility or robustness to noise, with implications for device-independent quantum communication and potential DI-QKD protocols. Overall, the study provides a systematic, scalable framework to explore Bell polytopes and informs the design of quantum communication tasks leveraging nonlocal correlations.

Abstract

We derive new tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario $(X,Y,A,B)$ is defined by the numbers of settings and outcomes per party, $X$, $A$ and $Y$, $B$ for Alice and Bob, respectively. We derive the complete set of facets of the local polytopes of $(6,3,2,2)$, $(3,3,3,2)$, $(3,2,3,3)$, and $(2,2,3,5)$. We provide extensive lists of facets for $(2,2,4,4)$, $(3,3,4,2)$ and $(4,3,3,2)$. For each inequality we compute the maximum quantum violation, the resistance to noise, and the minimal symmetric detection efficiency required to close the detection loophole, for qubits, qutrits and ququarts. Based on these results, we identify scenarios which perform better in terms of visibility, resistance to noise, or both, when compared to CHSH. Such scenarios could find important applications in quantum communication.

Figures (1)

• Figure 1: Representation of the polytope slices. By selecting different hyperplanes and bounds we can create subpolytopes with subsets of vertices, each containing some of the facets of the original polytope.