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Reducing the Statistical Axiom

L. Polley

Published Jun 29, 1999• 2 citations

Abstract

In a recent article (quant-ph/9906015) Deutsch, using concepts of decision theory, provided a strikingly simple argument to explain why quantum-mechanical probabilities are squares of amplitudes. The decisiontheoretic reasoning was criticized by Barnum et al. (quant-ph/9907024). Taking a more conventional approach, we reinterpret a crucial substitution in Deutsch’s paper as the time evolution of an apparatus, capable of separating internal states of an “atom ” into a suitable number of “cavities”. Permutational symmetries play a central role in this analysis. PACS numbers: 03.65.BzIn a recent paper, Deutsch [1] invoked the non-probabilistic part of classical decision theory to deduce from only non-probabilistic axioms of quantum mechanics the Statistical Axiom: The probability for measuring xi in a superposition state |ψ 〉 = ∑ i ψi|xi 〉 is |ψi | 2. The paper has attracted some vehement criticism [2] relating to its decision-theoretic setting, but also to priorities. As pointed out by Ref. [2], in 1957 already Gleason [3] gave a mathematically rigorous proof for the redundance of the Statistical Axiom, showing that if normalized probabilities are to be assigned to the eigenvectors of each bounded hermitian operator, then the only solution is the usual quantum mechanical prescription. Unfortunately for a result of such fundamental importance to quantum physics, the existing proofs of Gleason’s theorem are not easily received by many physicists, although mathematically elementary [4], synoptical [5], and even textbook [6] elaborations have been presented. Ref. [1] has a strong point in this respect, suggesting that the mathematics behind amplitudes being square-roots of probabilities essentially is the Pythagorean theorem in Hilbert space. Consider a superposition state of the special form |ψ 〉 =

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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.

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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

In a recent article (quant-ph/9906015) Deutsch, using concepts of decision theory, provided a strikingly simple argument to explain why quantum-mechanical probabilities are squares of amplitudes. The decisiontheoretic reasoning was criticized by Barnum et al. (quant-ph/9907024). Taking a more conventional approach, we reinterpret a crucial substitution in Deutsch’s paper as the time evolution of an apparatus, capable of separating internal states of an “atom ” into a suitable number of “cavities”. Permutational symmetries play a central role in this analysis. PACS numbers: 03.65.BzIn a recent paper, Deutsch [1] invoked the non-probabilistic part of classical decision theory to deduce from only non-probabilistic axioms of quantum mechanics the Statistical Axiom: The probability for measuring xi in a superposition state |ψ 〉 = ∑ i ψi|xi 〉 is |ψi | 2. The paper has attracted some vehement criticism [2] relating to its decision-theoretic setting, but also to priorities. As pointed out by Ref. [2], in 1957 already Gleason [3] gave a mathematically rigorous proof for the redundance of the Statistical Axiom, showing that if normalized probabilities are to be assigned to the eigenvectors of each bounded hermitian operator, then the only solution is the usual quantum mechanical prescription. Unfortunately for a result of such fundamental importance to quantum physics, the existing proofs of Gleason’s theorem are not easily received by many physicists, although mathematically elementary [4], synoptical [5], and even textbook [6] elaborations have been presented. Ref. [1] has a strong point in this respect, suggesting that the mathematics behind amplitudes being square-roots of probabilities essentially is the Pythagorean theorem in Hilbert space. Consider a superposition state of the special form |ψ 〉 =

In 1989, Deutsch gave a basic physical explanation of why quantum-mechanical probabilities are squares of amplitudes. Essentially, a general state vector is transformed into a highly symmetric equal-amplitude superposition. The argument was recently elaborated and publicised by DeWitt. It has remained incomplete, however, inasmuch as both authors anticipate the usual normalization (sum of amplitudes squared) of state vectors. In the present paper, a thought experiment is devised in which Deutsch's idea is demonstrated independently of the normalization, exploiting further symmetries instead.

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.

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.

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.