Born Rule and Noncontextual Probability

TL;DR

This paper presents a new proof of the Born rule in quantum mechanics based on the noncontextuality of probability, addressing gaps in previous derivations and linking it to physical principles like invariance and no-signaling.

Contribution

It introduces a novel proof of the Born rule using noncontextuality of probability and demonstrates its continuity, critiquing and extending prior approaches by Zurek and Deutsch.

Findings

Proof of the Born rule based on noncontextuality of probability

Demonstration of probability continuity with respect to amplitudes

Identification of implicit assumptions in previous derivations

Abstract

The probabilistic rule that links the formalism of Quantum Mechanics (QM) to the real world was stated by Born in 1926. Since then, there were many attempts to derive the Born postulate as a theorem, Gleason's being the most prominent. The Gleason derivation, however, is generally considered rather intricate and its physical meaning, in particular in relation with the noncontextuality of probability (NP), is not quite evident. More recently, we are witnessing a revival of interest in possible demonstrations of the Born rule, like Zurek's and Deutsch's based on the decoherence and on the theory of decisions, respectively. Despite an ongoing debate about the presence of hidden assumptions and circular reasonings, these have the merit of prompting more physically oriented approaches to the problem. Here we suggest a new proof of the Born rule based on the noncontextuality of probability. Within the theorem we also demonstrate the continuity of probability with respect to the amplitudes, which has been suggested to be a gap in Zurek's and Deutsch's approaches, and we show that NP is implicitly postulated also in their demonstrations. Finally, physical motivations of NP are given based on an invariance principle with respect to a resolution change of measurements and with respect to the principle of no-faster-than-light signalling.