Causal Rigidity of Born-Type Probability Rules in Infinite-Dimensional Operational Theories

TL;DR

This paper proves that in infinite-dimensional operational theories, the Born rule is uniquely determined by natural physical assumptions, establishing a form of causal rigidity that excludes nonlinear deviations and supports the standard quantum probability rule.

Contribution

It demonstrates that under specific operational conditions, the Born rule is the only consistent probability assignment, extending the foundational understanding of quantum probabilities in infinite-dimensional settings.

Findings

Born rule is uniquely fixed by operational assumptions

Nonlinear deviations enable signaling, thus are excluded

Continuity and sigma affinity enforce affine probability rules

Abstract

We establish an operational rigidity result for a broad class of probability rules in infinite-dimensional settings, applicable under normality and steering assumptions. Starting from a topological generalization of generalized probabilistic theories, we consider probability assignments defined as functions of an operational transition probability between pure states. We show that under three operationally motivated requirements: no superluminal signaling, availability of normal steering via purification in a sigma additive sense, and sigma affinity of probabilities under countable preparation mixtures, any admissible rule within this class must reduce to the identity. In particular, nonlinear deviations generically enable operational signaling distinctions in steering scenarios, while continuity combined with sigma affinity excludes non affine alternatives. This identifies a unique causal fixed point. Within this class of probability rules, the Born rule emerges as the only assignment compatible with no signaling in operational theories admitting normal steering. We connect the operational result to standard infinite-dimensional quantum mechanics through the normal state space of von Neumann algebras and the GNS representation, recovering the conventional Born rule for projective and generalized measurements. We discuss the scope of the assumptions and implications for proposed post quantum modifications in continuous variable and quantum field theoretic regimes.