Complete facet descriptions for Bell polytopes

Problem

Determine complete facet descriptions for local-behavior polytopes beyond the presently solved Bell scenarios, either for a specified unresolved finite scenario or for a nontrivial infinite family with additional structure. For positive integers \(N\), \(M\), and \(K\), let \(\mathbf{x}\in\{1,\ldots,M\}^{N}\) denote the measurement settings and \(\mathbf{a}\in\{1,\ldots,K\}^{N}\) the outcomes. The relevant local polytope is

\begin{equation} \mathcal{L}_{N,M,K} :=\operatorname{conv}\!\left\{ p_{\mathbf f}:p_{\mathbf f}(\mathbf a\mid\mathbf x) =\prod_{i=1}^{N}\mathbf{1}\!\left\{a_i=f_i(x_i)\right\},\quad f_i:\{1,\ldots,M\}\to\{1,\ldots,K\} \right\}. \tag{1} \end{equation}

Here \(\mathbf 1\{\cdot\}\) is the indicator function. The task is to characterize all facet-defining inequalities of the polytope in Eq. (1) in a chosen unresolved regime, modulo permutations of parties, settings, and outcomes and modulo liftings obtained by adjoining redundant settings or outcomes. A solution must carry a proof of completeness, not merely generate a large collection of facets.

Source

Krüger and Werner explicitly pose the search for completeness-certified facet descriptions of Bell polytopes in tractable finite scenarios and structured families [KW05].

Progress

Fine proved that positivity, normalization, and the Clauser–Horne–Shimony–Holt inequalities completely describe the \((N,M,K)=(2,2,2)\) local scenario. This settles only the smallest nontrivial case [Fin82].
Werner and Wolf classified all full-correlation facets for arbitrary \(N\) when every party has two dichotomic observables. Their theorem concerns a correlator projection of Eq. (1), not the full conditional-probability polytope [WW01].
Staufenbiel’s polyhedral-sampling method found more than \(1.29\times10^8\) inequivalent facet classes in one unresolved scenario, but the method intentionally sacrifices completeness. Thus the computation enlarges the known catalog without supplying the required certificate [Sta26].

Comment

The source problem explicitly replaces an implausible classification uniform in all \((N,M,K)\) by completeness-certified finite cases and structured families [KW05]. The unresolved gap is a new theorem of completeness for one such nontrivial regime; heuristic completeness or sampling without a certificate is insufficient.

References

[Fin82]: A. Fine, “Hidden Variables, Joint Probability, and the Bell Inequalities,” Physical Review Letters 48, 291–295 (1982).DOI
[WW01]: R. F. Werner and M. M. Wolf, “All Multipartite Bell Correlation Inequalities for Two Dichotomic Observables per Site,” Physical Review A 64, 032112 (2001).DOI arXiv
[Sta26]: C. Staufenbiel, “Bell Inequalities from Polyhedral Sampling,” arXiv:2604.22859 (2026).DOI arXiv
[KW05]: O. Krüger and R. F. Werner (eds.), “Some Open Problems in Quantum Information Theory,” arXiv:quant-ph/0504166 (2005).DOI arXiv

Complete facet descriptions for Bell polytopes

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