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Quantum Probability from Decision Theory?

H. Barnum, Carlton M. Caves, J. Finkelstein, Christopher A. Fuchs, R. Schack

Published Jan 1, 2000• 101 citations
OpenAlex

Abstract

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.

Related papers

Quantum probability from decision theory?

H. Barnum, Carlton M. Caves, J. Finkelstein +2 more

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.

arXiv:quant-ph/9907024•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•May 8, 2000•78 citations

A Note on Deutsch’s “Quantum Theory of Probability and Decisions”

Malcolm R. Forster

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

•Jan 1, 1999•0 citations

Deutsch on quantum decision theory

Peter J. Lewis

A major problem facing no-collapse interpretations of quantum mechanics in the tradition of Everett is how to understand the probabilistic axiom of quantum mechanics (the Born rule) in the context of a deterministic theory in which every outcome of a measurement occurs. Deutsch claims to derive a decision-theoretic analogue of the Born rule from the non-probabilistic part of quantum mechanics and some non-probabilistic axioms of classical decision theory, and hence concludes that no probabilistic axiom is needed. I argue that Deutsch’s derivation begs the question.

•Jan 1, 2003•5 citations

Quantum Probability from Decision Theory?

J. Finkelstein

Deutsch has recently (in quant-ph/9906015) offered a justification, based only on the non-probabilistic axioms of quantum theory and of classical decision theory, for the use of the standard quantum probability rules. In this note, this justification is examined.

arXiv:quant-ph/9907004•Jul 1, 1999•0 citations

Quantum theory of probability and decisions

David Deutsch

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.

arXiv:quant-ph/9906015•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 8, 1999•563 citations

ON AN ARGUMENT OF DAVID DEUTSCH

Richard David Gill

We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born's law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born's law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason's theorem under stronger assumptions. However, for some special cases the proof method gives positive results while using different assumptions to Gleason. This leads to the conjecture that the proof could be improved to give the same conclusion as Gleason under unitary invariance together with a much weaker functional invariance condition.

arXiv:quant-ph/0307188•Quantum Probability and Infinite Dimensional Analysis•Jan 1, 2005•13 citations
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.
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.
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
A major problem facing no-collapse interpretations of quantum mechanics in the tradition of Everett is how to understand the probabilistic axiom of quantum mechanics (the Born rule) in the context of a deterministic theory in which every outcome of a measurement occurs. Deutsch claims to derive a decision-theoretic analogue of the Born rule from the non-probabilistic part of quantum mechanics and some non-probabilistic axioms of classical decision theory, and hence concludes that no probabilistic axiom is needed. I argue that Deutsch’s derivation begs the question.
We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born's law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born's law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason's theorem under stronger assumptions. However, for some special cases the proof method gives positive results while using different assumptions to Gleason. This leads to the conjecture that the proof could be improved to give the same conclusion as Gleason under unitary invariance together with a much weaker functional invariance condition.