Causal Consistency Selects the Born Rule: A Derivation from Steering in Generalized Probabilistic Theories

TL;DR

This paper demonstrates that within generalized probabilistic theories, the Born rule uniquely emerges as the causally consistent probability rule when considering steering and no-signaling constraints.

Contribution

It proves that the Born rule is the only probability assignment compatible with relativistic causality in GPTs with purification and steering, linking geometric and probabilistic structures.

Findings

Nonlinear probability rules enable superluminal signaling.

The Born rule is uniquely derived from causality and steering principles.

Explicit qubit example illustrates detectable signaling from nonlinear probabilities.

Abstract

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 |<phi|psi>|^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.