Quantum probabilities as Bayesian probabilities

Carlton M. Caves, Christopher A. Fuchs, and Rüdiger Schack
Phys. Rev. A 65, 022305 – Published 4 January 2002
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Abstract

In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. In this paper, we show that, despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite answers for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally, we give a Bayesian formulation of quantum-state tomography.

  • Received 15 August 2001

DOI:https://doi.org/10.1103/PhysRevA.65.022305

©2002 American Physical Society

Authors & Affiliations

Carlton M. Caves1,*, Christopher A. Fuchs1, and Rüdiger Schack2

  • 1Bell Labs, Lucent Technologies, 600–700 Mountain Avenue, Murray Hill, New Jersey 07974
  • 2Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom

  • *Permanent address: Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131-1156.

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Vol. 65, Iss. 2 — February 2002

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