FCD-QM 1-A: Hyperbolic Degree Bounds Signaling Dimension in Generalized Probabilistic Theories

Geoffrey Marcellin

Published Sep 24, 2026• 0 citations

Abstract

The signaling dimension of a probabilistic system is the minimum number of classical states required to reproduce, with shared randomness, every finite prepare-and-measure input-output correlation generated by the system. Frenkel and Weiner proved the complex-quantum simulation result that is equivalently expressed as signaling dimension n for an n-level system. Building on Frenkel’s later positive-multilinear-form/supply-demand formulation, we show that hyperbolic polarizations provide a general source of the multilinear structure required by that simulation mechanism, extending the resulting bound to arbitrary proper hyperbolicity cones. If the effect cone of an ordered probabilistic model is the hyperbolicity cone of a homogeneous polynomial p of degree d, then its signaling dimension is at most d. The proof uses the non-negativity of the polar form of p on the hyperbolicity cone and an order-theoretic identity relating the smallest hyperbolic eigenvalue of an effect to its minimum value over normalized dual states. As a corollary, every canonical Euclidean Jordan model has signaling dimension exactly equal to its Jordan rank. This simultaneously covers classical systems, real, complex and quaternionic matrix Jordan models, spin factors, the exceptional Albert algebra, and direct sums. For symmetric cones this yields a numerical identity between signaling dimension, Jordan rank, the cone Carathéodory number, and the optimal self-concordant barrier parameter. The general hyperbolic bound need not be tight; the ℓ1-cone/square-bit example of Ito and Lourenço provides a strict gap when viewed through signaling dimension. The result separates exact classical prepare-and-measure simulation dimension from internal state-space geometry at fixed signaling dimension and motivates the study of which operational invariants resolve the structure that signaling dimension alone does not capture.

DOI: 10.5281/zenodo.22946777

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