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NeuroQuantology

Sleeping Beauty in Quantumland

Na’ama Hallakoun, Berry Groisman, Lev Vaidman A

Published Mar 2, 2013• 0 citations
Publisher

Abstract

It is argued that thirder resolution of the Lewis -Elga controversy about Sleeping Beauty is more clear when the coin toss is replaced by a quantum measurement and the analysis is performed in the framework of the Many-Worlds Interpretation.
DOI
10.14704/nq.2013.11.1.649

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The Sleeping Beauty Problem (SBP) is a long-standing puzzle in classical probability theory and has been used to challenge the Many-Worlds Interpretation (MWI) of quantum mechanics, since both involve objective determinacy combined with subjective uncertainty about certain events. A common concern is that MWI yields a different answer to the quantum version of SBP than the widely supported Thirder position in the classical case. We argue that this concern is unwarranted. We show that in both the quantum and classical versions of SBP, the correct credence is given by the Halfer position. In the quantum (MWI) SBP, we show that if no unjustified renormalization is introduced, the correct credence is 1/2. We then extend this result to the classical SBP by refuting four major arguments for 1/3. First, we reject the Proportion Argument by distinguishing event weight from probability. Second, we rebut Elga's Variant Argument by extending an earlier critique that identifies the implicit introduction of additional information; we further clarify this point by constructing a new variant and explaining why the Principle of Indifference is inapplicable, drawing an analogy with a mistake by d'Alembert in the history of probability theory. Third, we identify a flaw in the Technicolor Beauty Variant Argument, which arises from treating overlapping events as disjoint. Finally, we argue that causal decision theory is inappropriate for SBP, rendering the Thirders vulnerable to a Dutch Book. Our results support the consistency of MWI under the challenge posed by SBP and suggest that the dominant position on SBP needs careful reconsideration.

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Two works related to the concept of probability in the framework of the many-worlds interpretation are presented. The first deals with recent controversy in classical probability theory. Elga and D. Lewis argues that Sleeping Beauty should have different credences for the result of a fair coin toss in a particular situation. It is argued that when the coin is replaced by a quantum coin, the credence is unambiguous and since one should not expect a difference between classical and quantum coins, this provides a particular resolution of the controversy. Second work is an analysis of a recent criticism by Byrne and Hall of Everett-type approach presented by Chalmers. It is shown that the criticism has no universal validity and that the Byrne and Hall rejection of any Everett-type interpretation of quantum mechanics is unfounded.
Since the publication of Elga's seminal paper in 2000, the Sleeping Beauty paradox has been the source of much discussion, particularly in this journal. Over the past few decades the Everettian interpretation of quantum mechanics1 has also been much debated. There is an interesting connection between the way these two topics raise issues about subjective probability assignments. This connection is often alluded to, but as far as we know Peter J. Lewis's ‘Quantum Sleeping Beauty’ (2007a) is the first attempt to examine it explicitly. Lewis claims that the two debates are not independent: to be specific, he argues that accepting the Everettian interpretation of quantum mechanics requires you to be a ‘halfer’ about Sleeping Beauty, in opposition to the more widely accepted ‘thirder’ solution. This paper will argue that Lewis is wrong. Everettians do not have to be halfers. It is perfectly cogent to be both an Everettian and a thirder.
The Sleeping Beauty Problem (SBP) is a long-standing puzzle in classical probability theory and has been used to challenge the Many-Worlds Interpretation (MWI) of quantum mechanics, since both involve objective determinacy combined with subjective uncertainty about certain events. A common concern is that MWI yields a different answer to the quantum version of SBP than the widely supported Thirder position in the classical case. We argue that this concern is unwarranted. We show that in both the quantum and classical versions of SBP, the correct credence is given by the Halfer position. In the quantum (MWI) SBP, we show that if no unjustified renormalization is introduced, the correct credence is 1/2. We then extend this result to the classical SBP by refuting four major arguments for 1/3. First, we reject the Proportion Argument by distinguishing event weight from probability. Second, we rebut Elga's Variant Argument by extending an earlier critique that identifies the implicit introduction of additional information; we further clarify this point by constructing a new variant and explaining why the Principle of Indifference is inapplicable, drawing an analogy with a mistake by d'Alembert in the history of probability theory. Third, we identify a flaw in the Technicolor Beauty Variant Argument, which arises from treating overlapping events as disjoint. Finally, we argue that causal decision theory is inappropriate for SBP, rendering the Thirders vulnerable to a Dutch Book. Our results support the consistency of MWI under the challenge posed by SBP and suggest that the dominant position on SBP needs careful reconsideration.