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

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.

💡 Research Summary

The paper investigates the relationship between two distinct notions in finite‑dimensional generalized probabilistic theories (GPTs): the geometric transition probability τ(ψ, ϕ), defined as the maximal acceptance probability of a test that is certain to accept state ϕ when the actual state is ψ, and the predictive probability P(ϕ | ψ) that an experimenter assigns to obtaining outcome ϕ in a measurement on a system prepared in ψ. While standard quantum mechanics identifies these two quantities via the Born rule (P = τ = |⟨ϕ|ψ⟩|²), the authors ask whether a more general functional relationship P = Φ(τ) can be compatible with relativistic causality.

They first formalize τ within the GPT framework, showing that under reasonable axioms (continuous reversibility, spectrality, etc.) τ satisfies natural properties: τ(ψ, ψ)=1, τ(ψ, ϕ)=0 iff ψ and ϕ are perfectly distinguishable, and for any binary measurement {e_ϕ, e_ϕ⊥} one has τ(ψ, ϕ)+τ(ψ, ϕ⊥)=1. Predictive probabilities are then allowed to be any function Φ: