Strong Locality as a Tetrahedron: A Symmetry-Reduced Geometric Representation of the (3,3,2,2) Bell Scenario

• URL: http://arxiv.org/abs/2605.03104v2
• Date: Sun, 10 May 2026 20:23:40 GMT
• Title: Strong Locality as a Tetrahedron: A Symmetry-Reduced Geometric Representation of the (3,3,2,2) Bell Scenario
• Abstract summary: We show that strongly-local ($mathcalSL$) models separate from their complement, non-strongly-local ($mathcaloverlineSL$) models.<n>We also clarify the relation between the symmetry-reduced pyramid representation and the full (3,3,2,2) Bell polytope.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a geometric characterisation of strongly-local models in the bipartite Bell scenario with three measurement settings per site and binary outcomes, i.e.\ the (3,3,2,2) case. Restricting attention to indistinguishable sites, we introduce a three-dimensional mixed-moment space in which the mixed moments are calculated under off-diagonal measurement settings. In this reduced representation, the strongly-local region assumes the remarkably simple form of a regular tetrahedron - the 'pyramid'. We prove that only three independent linear inequalities are required to characterise this region. We call them the pyramid inequalities that separate strongly-local ($\mathcal{SL}$) models from their complement, non-strongly-local ($\mathcal{\overline{SL}}$) models. We also clarify the relation between the symmetry-reduced pyramid representation and the full (3,3,2,2) Bell polytope in the 36-dimensional conditional-probability space, which possesses 684 facet-defining inequalities. The reduction from 684 to three reflects normalisation, symmetry reduction, and projection to the mixed-moment space. In the pyramid representation, the hierarchy $\mathcal{SL} \subsetneq \mathcal{Q} \subsetneq \mathcal{NS}$ appears geometrically as a tetrahedron embedded in a somewhat larger curved body of quantum models, $\mathcal{Q}$, which in turn is embedded in a cube of no-signalling models, $\mathcal{NS}$. The qualitative and quantitative advantages of the pyramid representation over the standard CSHS representation for the (2,2,2,2) case are discussed.

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