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arXiv:quant-ph/0603065 · Journal of Mathematical Physics

Bayesian analog of Gleason’s theorem

Thomas Marlow

Published Dec 1, 2006• 7 citations
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Abstract

We introduce a novel notion of probability within quantum history theories and give a Gleasonesque proof for these assignments. This involves introducing a tentative novel axiom of probability. We also discuss how we are to interpret these generalized probabilities as partially ordered notions of preference, and we introduce a tentative generalized notion of Shannon entropy. A Bayesian approach to probability theory is adopted throughout; thus the axioms we use will be minimal criteria of rationality rather than ad hoc mathematical axioms.

DOI
10.1063/1.2390658

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We introduce a novel notion of probability within quantum history theories and give a Gleasonesque proof for these assignments. This involves introducing a tentative novel axiom of probability. We also discuss how we are to interpret these generalized probabilities as partially ordered notions of preference, and we introduce a tentative generalized notion of Shannon entropy. A Bayesian approach to probability theory is adopted throughout; thus the axioms we use will be minimal criteria of rationality rather than ad hoc mathematical axioms.
Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason's theorem.
A quantum state can be understood in a loose sense as a map that assigns a value to every observable. Formalizing this characterization of states in terms of generalized probability distributions on the set of effects, we obtain a simple proof of the result, analogous to Gleason's theorem, that any quantum state is given by a density operator. As a corollary we obtain a von Neumann-type argument against noncontextual hidden variables. It follows that on an individual interpretation of quantum mechanics the values of effects are appropriately understood as propensities.
In a recent paper, Deutsch claims to derive the ‘probabilistic predictions of quantum theory’ from the ‘non–probabilistic axioms of quantum theory’ and the ‘nonprobabilistic part of classical decision theory.’ We show that his derivation includes a crucial hidden assumption that vitiates the force of his argument. Furthermore, we point out that in classical decision theory a standard set of non–probabilistic axioms is already sufficient to endow possible outcomes with a natural probability structure. Within that context we argue that Gleason's theorem, relying on fewer assumptions than Deutsch, provides a compelling derivation of the quantum probability law.
Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment. We identify the class of general probabilistic theories which also admit Gleason-type theorems. It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes. Our result also implies that the standard no-restriction hypothesis applied to effects is not equivalent to the dual no-restriction hypothesis applied to states which is found to be less restrictive.
In a recent paper, Deutsch claims to derive the `probabilistic predictions of quan-tum theory ’ from the `non-probabilistic axioms of quantum theory ’ and the `non-probabilistic part of classical decision theory. ’ We show that his derivation includes a crucial hidden assumption that vitiates the force of his argument. Furthermore, we point out that in classical decision theory a standard set of non-probabilistic axioms is already su ¯ cient to endow possible outcomes with a natural probability structure. Within that context we argue that Gleason’s theorem, relying on fewer assump-tions than Deutsch, provides a compelling derivation of the quantum probability law.
This paper reports three almost trivial theorems that nevertheless appear to have significant import for quantum foundations studies. 1) A Gleason-like derivation of the quantum probability law, but based on the positive operator-valued measures as the basic notion of measurement (see also Busch, quant-ph/9909073). Of note, this theorem also works for 2-dimensional vector spaces and for vector spaces over the rational numbers, where the standard Gleason theorem fails. 2) A way of rewriting the quantum collapse rule so that it looks almost precisely identical to Bayes rule for updating probabilities in classical probability theory. And 3) a derivation of the tensor-product rule for combining quantum systems (and with it the very notion of quantum entanglement) from Gleason-like considerations for local measurements on bipartite systems along with classical communication.