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arXiv:quant-ph/9906015 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum theory of probability and decisions

David Deutsch

Published Aug 8, 1999• 563 citations
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Abstract

The probabilistic predictions of quantum theory are conventionally obtained from a special probabilistic axiom. But that is unnecessary because all the practical consequences of such predictions follow from the remaining non–probabilistic axioms of quantum theory, together with the non–probabilistic part of classical decision theory.
DOI
10.1098/rspa.1999.0443

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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.
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.
ABSTRACT: David Deutsch has a forthcoming article called “Quantum Theory of Probability and Decisions " in which he claims to derive the standard probabilistic interpretation of the wavefunction from non-probabilistic considerations. In this note, I try to develop his argument in a classical context, to see whether the same ideas can generate probabilistic conclusions from nonprobabilistic assumptions in a more general way. While my answer is negative, the attempt pinpoints the parts of Deutsch’s quantum mechanical argument that must rely on special features of the formalism (assuming that his proof is correct). 1. No Probabilities In, No Probabilities Out? You cannot derive an ‘ought ’ from an ‘is’. And I believe that it was Popper (1959) who once said, in response to those who think that randomness can be defined nonprobabilistically, that one cannot derive a probabilistic conclusion from non-probabilistic premises. Yet this is exactly what Deutsch (to appear) claims for quantum mechanical probabilities. Historically in the development of quantum theory, the Schrödinger
After briefly recalling some of the main theses of quantum probability concerning the foundational aspects of quantum mechanics and of probability theory, we describe the various categories of axioms in classical probability theory and focus our attention on the classical conditioning axioms. Four classical conditioning axioms are proposed, their probabilistic and model-independent meaning is discussed and it is proven that: (i) the only mathematical realization of these axioms is the classical Bayes rule (or equivalently the classical theorem of composite probabilities) and (ii) von Neumann's projection postulate, which can be considered one of the first examples of quantum conditioning axioms, violates two of these axioms. Finally, we construct an example showing that, even in classical probability, an uncritical application of Bayes' rule can lead to non-optimal predictions. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
In this paper we derive the complex Hilbert space formalism of quantum theory from four simple information theoretic axioms. It is shown that quantum theory is the only non classical probabilistic theory satisfying the following axioms: distinguishability, conservation, reversibility, composition. The new results of this reconstruction compared to other reconstructions by other authors are: (i) we get rid of axiom "subspace" in favor of axiom conservation eliminating mathematical requirements contained in previous axiomatics; (ii) we are able to classify all the probabilistic theories that are consistent requiring (a) only the first two axioms (b) only the first three axioms; this could be useful in experimental tests of quantum theory since it gives the possibility to understand whether or not other mathematical models could be consistent with such tests; (iii) we provide a connection between two different approaches to quantum foundations, quantum logic and the one based on information theoretic primitives showing that any theory satisfying the first two axioms given above either is classical or is a theory in which physical systems are described by a projective geometry.