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Zeitschrift für Naturforschung A

A Remark on Joint Probability Distributions for Quantum Observables

Slawomir Bugajski

Published Nov 1, 1978• 4 citations
Publisher

Abstract

A simple formal result concerning joint probability distributions in quantum mechanics is obtained. It is shown that some relatively weak properties of the joint distributions cannot be satisfied in the scope of standard quantum mechanics.
DOI
10.1515/zna-1978-1118

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A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the observables must commute. This has consequences for some historical and recent quantum nonlocal models: they are analytically disallowed without the need for experiment, as they imply that all local observables must commute among themselves.
The topic of probabilty in quantum mechanics is rather vast, and in this article, we shall choose to discuss it from the perspective of whether and in what sense quantum mechanics requires a generalisation of the usual (Kolmogorovian) concept of probability. We shall focus on the case of finite-dimensional quantum mechanics (which is analogous to that of discrete probability spaces), partly for simplicity and partly for ease of generalisation. While we shall largely focus on formal aspects of quantum probability (in particular the non-existence of joint distributions for incompatible observables), our discussion will relate also to notorious issues in the interpretation of quantum mechanics. Indeed, whether quantum probability can or cannot be ultimately reduced to classical probability connects rather nicely to the question of 'hidden variables' in quantum mechanics. An abridged version of this essay will appear in A. Hajek and C. Hitchcock (eds.), 'The Oxford Handbook of Probability and Philosophy', Oxford University Press. The present version includes in particular many more footnotes and references and the Appendix with the proofs of the Lemma and Proposition of Section 5.