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Summing to uncertainty: On the necessity of additivity in deriving the Born rule

Jiaxuan Zhang*

  • *Contact author: jiaxuan.zhang@physics.ox.ac.uk
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Phys. Rev. A 114, 042208 – Published 13 October, 2026

DOI: https://doi.org/10.1103/mqy7-s2db

Abstract

The emergence of intrinsic probability is one of the central problems in quantum mechanics, and investigating the origin of the Born rule provides the most direct approach to this problem. Many derivations of the Born rule have been proposed based on different postulates and frameworks. Identifying which postulates are genuinely necessary is therefore essential for understanding the origin of quantum probability. Here we prove that two postulates commonly employed in derivations of the Born rule, additivity and noncontextuality, are not equivalent and thus cannot be unified into a single postulate. We further show that the additivity postulate is indispensable in five major derivations of the Born rule: Gleason's theorem, Busch's extension of Gleason's theorem, the Deutsch-Wallace theorem, Zurek's envariance derivation, and the Finkelstein-Hartle theorem. We demonstrate that these derivations rely either on explicit additivity assumptions or on implicit forms of additivity embedded in the original arguments and that removing or weakening additivity gives rise to significant problems in the derivations. Our results support the view that additional postulates with intrinsic probabilistic properties should be introduced to derive the Born rule.

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Physics Subject Headings (PhySH)

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Quantum Probability Requires Probability

Published 13 October, 2026

A new analysis reveals that all major derivations of the Born rule depend on a probabilistic assumption.

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Article Text

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Phys. Rev. A 114, 042208– Published 13 October, 2026
  • Received 22 May 2026
  • Revised 28 July 2026
  • Accepted 4 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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