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

TL;DR

This work extends Born-rule derivations to infinite-dimensional operational theories by formulating a topological GPT that cleanly separates normal from singular states. Under three operational principles—no superluminal signaling, normal steering via purification, and $\\sigma$-affinity for countable mixtures—the authors prove a causal rigidity result: any probability rule of the form $P(\\phi|\\psi)=\\Phi(\\tau(\\psi,\\phi))$ must satisfy $\\Phi(p)=p$, recovering the Born rule as a unique fixed point. In the quantum setting, the operational transition probability $\\tau$ aligns with the standard quantum transition probability, so the Born rule $P(\\phi|\\psi)=|\\langle\\phi|\\psi\\rangle|^2$ emerges for projective and generalized measurements via the GNS construction on von Neumann algebras. The results constrain post-quantum modifications in continuous-variable and quantum-field regimes by showing that nonlinear deviations generically enable signaling under normal steering, thereby reinforcing the universality and causality of the Born rule in infinite dimensions.

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.

Figures (3)

• Figure 1: State Space Structure in Infinite Dimensions. The full state space $\Omega$ decomposes into normal (physically realizable) and singular (pathological) components. Pure states lie on the boundary, while mixed states admit multiple decompositions into pure states via normal ensembles.
• Figure 2: Normal Steering and Signaling Test. Alice's measurement choice ($M_1$ or $M_2$) steers Bob's system into different ensembles ($\mu_1$ or $\mu_2$) with the same average state $\omega_B$. Under $\sigma$-affinity, if probabilities follow $P=\Phi(\tau)$ with nonlinear $\Phi$, Bob's statistics depend on Alice's choice, enabling superluminal signaling.
• Figure 3: Functional Forms and Signaling. The identity $\Phi(p)=p$ corresponds to the Born rule. Convex or concave deviations create a Jensen gap $\Delta$ between $\Phi(\lambda p_1+(1-\lambda)p_2)$ and $\lambda\Phi(p_1)+(1-\lambda)\Phi(p_2)$; via steering this becomes operationally observable, enabling signaling.

Theorems & Definitions (21)

• Definition 1: Topological Operational System
• Definition 2: Normal and Singular Components
• Definition 3: Operational Purity
• Definition 4: Normal Ensemble
• Definition 5: Operational Transition Probability
• Lemma 1: Basic Properties of $\tau$
• proof
• Remark 1: Quantum Specialization
• Remark 2: Physical Necessity of $\sigma$-Affinity
• Definition 6: Generalized Probability Rule