Betting on the outcomes of measurements: a Bayesian theory of quantum probability

doi:10.1016/S1355-2198(03)00035-2

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

Volume 34, Issue 3, September 2003, Pages 395-414

https://doi.org/10.1016/S1355-2198(03)00035-2 Get rights and content

Abstract

We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance, the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the other measurements is returned to the agent. We show how the rules of rational betting imply all the interesting features of quantum probability, even in such finite gambles. These include the uncertainty principle and the violation of Bell's inequality among others. Quantum gambles are closely related to quantum logic and provide a new semantics for it. We conclude with a philosophical discussion on the interpretation of quantum mechanics.

Section snippets

The gamble

The Bayesian approach takes probability to be a measure of ignorance, reflecting our state of knowledge and not merely the state of the world. It follows Ramsey's contention that “we have the authority both of ordinary language and of many great thinkers for discussing under the heading of probability … the logic of partial belief” (Ramsey, 1926, p. 55). Here we shall assume, furthermore, that probabilistic beliefs are expressed in rational betting behavior: “The old-established way of

Uncertainty relations

Consider the following quantum gamble $M$ consisting of seven incompatible measurements (Boolean algebras), each generated by its three possible outcomes: 〈E₁,E₂,F₂〉, 〈E₁,E₃,F₃〉, 〈E₂,E₄,E₆〉, 〈E₃,E₅,E₇〉, 〈E₆,E₇,F〉, 〈E₄,E₈,F₄〉, 〈E₅,E₈,F₅〉. Note that some of the outcomes are shared by two measurements; these are denoted by the letter E. The other outcomes each belong to a single algebra and are denoted by F. As before, when two algebras share an event, they also share its complement so that, for

Semantics for quantum logic and structural realism

The line we have taken has some affinity with Bohr's approach—or more precisely, with the view often attributed to Bohr⁴—in that we treat the outcomes of future measurements as mere possibilities and do not associate them with properties that exist prior to the act of

Acknowledgements

I would like to thank Harvey Brown, Jeremy Butterfield, William Demopoulos, Wayne Myrvold, and Simon Saunders for helpful comments and suggestions. This research is supported by an Israel Science Foundation grant number 879/02.

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Betting on the outcomes of measurements: a Bayesian theory of quantum probability

Abstract

Section snippets

The gamble

Uncertainty relations

Semantics for quantum logic and structural realism

Acknowledgements

Studies in the History and Philosophy of Modern Physics

Studies in the History and Philosophy of Modern Physics

Quantum probability from decision theory?

Proceedings of the Royal Society of London A

On the Einstein–Podolsky–Rosen Paradox

Physics

Quantum dialogue

The logic of quantum mechanics

Annals of Mathematics

The interpretation of quantum mechanics

Interpreting the quantum world

Quantum probabilities as Bayesian probabilities

Physical Review A

Probability induction and statistics