Quantum probabilities as Bayesian probabilities

TLDR

Bayesian probability assigns degrees of belief to single trials, and while classical maximal information yields definite answers for all questions, quantum maximal information is inherently incomplete. The paper demonstrates that individual quantum-system probabilities, governed by a fundamental law, can be interpreted within the Bayesian framework. The authors argue that classical versus quantum probabilities differ in the information they encode, and show that Bayesian assignments in quantum mechanics must follow the quantum probability rule, yield unique state assignments from maximal information, and link probability more tightly to frequency than classically, also providing a Bayesian formulation of quantum-state tomography.

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.