2025/12/14 by Enso O. Torres Alegre, Alegre, Enso O. Torres
Computer Science · Physics and Astronomy · #81P10 #81P15 #81P40 #FOS: Physical sciences #Noncommutative and Quantum Gravity Theories #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2512.12636
openalex publication_date 2025/12/14 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/29
Within finite-dimensional generalized probabilistic theories (GPTs), we distinguish between the geometric transition probability tau(psi,phi), defined as the maximum probability of accepting phi when the state is psi, and the predictive probability P(phi|psi) assigned to measurement outcomes. We ask what functional relationship P = Phi(tau) is compatible with relativistic causality. We prove that in any GPT satisfying purification, and therefore admitting steering, the only such relationship consistent with no-signaling is the identity Phi(p) = p. Any strictly convex or concave deviation from linearity enables superluminal signaling through steering scenarios. We provide an explicit qubit example showing how nonlinear probability rules generate detectable signaling channels. Combined with standard reconstruction results, this yields the Born rule ||2 as the unique causally consistent probability assignment. Our analysis clarifies the distinction between geometric structure and probabilistic prediction in quantum theory, and identifies steering as the mechanism enforcing the Born rule.