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

Abstract

We present a geometric characterisation of 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, thus resulting in a `symmetric local' scenario, 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 symmetric local region assumes the remarkably simple form of a regular tetrahedron - the 'pyramid'. We prove that only three inequalities are required to characterise the interior of this region. We call them the pyramid inequalities that separate symmetric local ($\mathcal{SL}$) models from their complement, non-symmetric local ($\mathcal{\overline{SL}}$) models. Remarkably, in this representation, the 'symmetric quantum' region forms an elliptope defined by a single inequality. 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 $\mathcal{SL} \subsetneq \mathcal{SQ} \subsetneq \mathcal{NS}$ appears geometrically as a tetrahedron embedded in a somewhat larger curved body of symmetric quantum models, $\mathcal{SQ}$, which in turn is embedded in a cube of no-signalling models, $\mathcal{NS}$. This provides a unified three-dimensional visualisation of symmetric local, symmetric quantum, and symmetric post-quantum correlations. The qualitative advantages of the pyramid representation over the standard one-dimensional CSHS representation of the $(2,2,2,2)$ case are also discussed.