The signaling dimension of two-dimensional and polytopic systems

The signaling dimension of any given physical system represents its classical simulation cost, that is, the minimum dimension of a classical system capable of reproducing all the input/output correlations of the given system. The signaling dimension landscape is vastly unexplored; the only non-trivial systems whose signaling dimension is known – other than quantum systems – are the octahedron and the composition of two squares. Building on previous results by Matsumoto, Kimura, and Frenkel, our first result consists of deriving bounds on the signaling dimension of any system as a function of its Minkowski measure of asymmetry. We use such bounds to prove that the signaling dimension of any two-dimensional system (i.e. with two-dimensional set of admissible states, such as polygons and the real qubit) is two if and only if such a set is centrally symmetric, and three otherwise, thus conclusively settling the problem of the signaling dimension for such systems. Guided by the relevance of symmetries in the two dimensional case, we propose a branch and bound division-free algorithm for the exact computation of the symmetries of any given polytope, in polynomial time in the number of vertices and in factorial time in the dimension of the space. Our second result then consist of providing an algorithm for the exact computation of the signaling dimension of any given system, that outperforms previous proposals by exploiting the aforementioned bounds to improve its pruning techniques and incorporating as a subroutine the aforementioned symmetries-finding algorithm. We apply our algorithm to compute the exact value of the signaling dimension for all rational Platonic, Archimedean, and Catalan solids, and for the class of hyper-octahedral systems up to dimension five.

💡 Research Summary

The paper investigates the “signaling dimension” of physical systems within the framework of Generalized Probabilistic Theories (GPTs). The signaling dimension quantifies the minimal dimension of a classical system that can reproduce all input‑output correlations generated by a given GPT system. While the concept is straightforward, computing it for arbitrary GPTs has remained largely open, with exact results known only for a few non‑trivial cases such as the octahedron and the composition of two squares.

The authors first derive new bounds on the signaling dimension in terms of the Minkowski measure of asymmetry, a geometric quantity that captures how far a convex state space deviates from central symmetry. They show that for any system S, if S is centrally symmetric then its signaling dimension never exceeds the affine dimension of S, whereas if S lacks central symmetry the signaling dimension is at least three. By combining these bounds with earlier results of Matsumoto‑Kimura and Frenkel, they obtain a complete characterization for all two‑dimensional systems (i.e., systems whose state space has affine dimension two, such as regular polygons and the real qubit). Specifically, the signaling dimension equals two exactly when the state space is centrally symmetric, and equals three otherwise. This settles a long‑standing open problem and yields a simple parity rule for regular polygons: even‑sided polygons have signaling dimension two, odd‑sided polygons have signaling dimension three.

The second major contribution is an exact, division‑free algorithm for finding the symmetry group of any polytope. Existing symmetry‑finding methods either rely on exact real arithmetic (which is not realistic on physical computers) or enumerate all permutations of the vertices, leading to factorial complexity in the number of vertices and the ambient dimension. The authors adopt a combinatorial viewpoint: a symmetry corresponds to a permutation of vertex labels that preserves all pairwise inner products, i.e., the Gram matrix. This condition can be checked using only integer arithmetic. They embed the permutation search in a branch‑and‑bound tree, pruning branches that cannot satisfy the Gram‑matrix condition. The resulting algorithm runs in polynomial time in the number of vertices and factorial time in the dimension, but the factorial factor is dramatically reduced in practice compared with naïve exhaustive search.

Armed with the asymmetry bounds and the symmetry‑finding subroutine, the authors design a new exact algorithm for computing the signaling dimension of any polytope‑based GPT system. The algorithm first evaluates the Minkowski asymmetry to obtain tight lower and upper bounds on the signaling dimension. It then iteratively tests candidate classical dimensions within this interval, using the symmetry group to prune the search space of possible correlation matrices. This combined approach yields a substantial speed‑up over previous methods while guaranteeing exact results.

The authors apply their framework to a broad class of systems: all rational regular (Platonic), Archimedean, and Catalan solids, as well as hyper‑octahedral systems up to dimension five. Their computations recover known results (e.g., the octahedron’s signaling dimension of three, the composition of two squares yielding five) and provide new exact values for many previously unresolved solids. Notably, even centrally symmetric polytopes can have signaling dimension larger than two when they are not simplices, confirming that central symmetry is necessary but not sufficient for attaining the lower bound.

In summary, the paper makes three key advances: (1) it establishes a precise, geometry‑based criterion for the signaling dimension of all two‑dimensional GPTs; (2) it introduces an efficient, integer‑only algorithm for exact symmetry detection of arbitrary polytopes; and (3) it integrates these tools into a powerful exact method for computing signaling dimensions of high‑dimensional polytope‑based systems. The work closes several gaps in the theory of GPT simulation costs and opens avenues for future research on higher‑dimensional, non‑rational, or composite systems, as well as on connections between signaling dimension, communication complexity, and cryptographic resources.