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arXiv:quant-ph/0207124 · Journal of Physics A Mathematical and General

Classical three-box paradox

K A Kirkpatrick

Published Apr 14, 2003• 38 citations
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Abstract

A simple classical probabilistic system (a simple card game) classically exemplifies Aharonov and Vaidman's ‘three-box ‘paradox’’ (1991 J. Phys. A: Math. Gen. 24 2315 ), implying that the three-box example is neither quantal nor a paradox and leaving one with less difficulty to busy the interpreters of quantum mechanics. An ambiguity in the usual expression of the retrodiction formula is shown to have misled Albert et al (1985 Phys. Rev. Lett. 54 5) to a result not, in fact, ‘curious’; the discussion illustrates how to avoid this ambiguity.

DOI
10.1088/0305-4470/36/17/315

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The three-box paradox revisited

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The classical three-box paradox of Kirkpatrick ( 2003 J. Phys. A: Math. Gen. 36 4891 ) is compared to the original quantum three-box paradox of Aharonov and Vaidman ( 1991 J. Phys. A: Math. Gen. 24 2315 ). It is argued that the quantum three-box experiment is a 'quantum paradox' in the sense that it is an example of a classical task which cannot be accomplished using classical means, but can be accomplished using quantum devices. It is shown that Kirkpatrick's card game is analogous to a different game with a particle in three boxes which does not contain paradoxical features.

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Reply to ‘The three-box paradox revisited’ by T Ravon and L Vaidman

K A Kirkpatrick

I reply to Ravon and Vaidman's criticism ( 2007 J. Phys. A: Math. Theor. 40 2873 (this issue)) of my classical implementation of a three-box system as a card-game.

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•Nov 13, 2019•4 citations
A simple classical probabilistic system (a simple card game) classically exemplifies Aharonov and Vaidman's ‘three-box ‘paradox’’ (1991 J. Phys. A: Math. Gen. 24 2315 ), implying that the three-box example is neither quantal nor a paradox and leaving one with less difficulty to busy the interpreters of quantum mechanics. An ambiguity in the usual expression of the retrodiction formula is shown to have misled Albert et al (1985 Phys. Rev. Lett. 54 5) to a result not, in fact, ‘curious’; the discussion illustrates how to avoid this ambiguity.
The classical three-box paradox of Kirkpatrick ( 2003 J. Phys. A: Math. Gen. 36 4891 ) is compared to the original quantum three-box paradox of Aharonov and Vaidman ( 1991 J. Phys. A: Math. Gen. 24 2315 ). It is argued that the quantum three-box experiment is a 'quantum paradox' in the sense that it is an example of a classical task which cannot be accomplished using classical means, but can be accomplished using quantum devices. It is shown that Kirkpatrick's card game is analogous to a different game with a particle in three boxes which does not contain paradoxical features.
The classical three-box paradox of Kirkpatrick [J. Phys. A 36 4891 (2003)] is compared to the original quantum Three-Box paradox of Aharonov and Vaidman [J. Phys. A 24 2315 (1991)]. It is argued that the quantum Three-Box experiment is a "quantum paradox" in the sense that it is an example of a classical task which cannot be accomplished using classical means, but can be accomplished using quantum devices. It is shown that Kirkpatrick's card game is analogous to a different game with a particle in three boxes which does not contain paradoxical features.
Which theories lead to a contradiction between simple reasoning principles and modelling observers’ memories as physical systems? Frauchiger and Renner have shown that this is the case for quantum theory (Frauchiger and Renner 2018 Nat. Commun. 9 3711). Here we generalize the conditions of the Frauchiger–Renner result so that they can be applied to arbitrary physical theories, and in particular to those expressed as generalized probabilistic theories (GPTs) (Hardy 2001 arXiv: quant-ph/0101012 ; Barrett 2007 Phys. Rev. A 75 032304). We then apply them to a particular GPT, box world, and find a deterministic contradiction in the case where agents may share a PR box (Popescu and Rohrlich 1994 Found. Phys. 24 379–85), which is stronger than the quantum paradox, in that it does not rely on post-selection. Obtaining an inconsistency for the framework of GPTs broadens the landscape of theories which are affected by the application of classical rules of reasoning to physical agents. In addition, we model how observers’ memories may evolve in box world, in a way consistent with Barrett’s criteria for allowed operations (Barrett 2007 Phys. Rev. A 75 032304; Gross et al 2010 Phys. Rev. Lett. 104 080402). Ordinary readers, forgive my paradoxes: one must make them when one reflects; and whatever you may say, I prefer being a man with paradoxes than a man with prejudices. Jean-Jacques Rousseau, Emile or On Education
An apparent paradox proposed by Aharonov and Vaidman in which a single particle can be found with certainty in two (or more) boxes is analyzed. It is found that the paradox arises from a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule. The paradox is resolved by observing that the counterfactual usage of the rule is, in general, not valid. 1. Background. The Aharonov-Bergmann-Lebowitz (ABL) rule is a well-known formula for calculating the probabilities of the possible outcomes of observables measured at an intermediate time t between pre- and post-selection measurements at times t1 and t2, respectively [Aharonov, Bergmann, and Lebowitz, 1964]. If an intermediate measurement of nondegenerate observable C with eigenvalues {ci} is performed at time t, the ABL rule states that the probability of outcome cj in the case of a preselection for the state |ψ1 〉 and a post-selection for the state |ψ2 〉 is given by: 1 PABL(cj|ψ1, ψ2) = |〈ψ2|cj〉 | 2 |〈cj||ψ1〉 | 2 i |〈ψ2|ci〉 | 2 |〈ci||ψ1〉 | 2 (1) It has recently been shown 2 that the ABL rule cannot be used in a counterfactual sense: i.e., in general, it cannot be used to calculate the probabilities of possible outcomes of observables that have not actually been measured at time t. 2. The three-box example.
Which theories lead to a contradiction between simple reasoning principles and modelling observers' memories as physical systems? Frauchiger and Renner have shown that this is the case for quantum theory (Frauchiger and Renner 2018 Nat. Commun. 9 3711). Here we generalize the conditions of the Frauchiger–Renner result so that they can be applied to arbitrary physical theories, and in particular to those expressed as generalized probabilistic theories (GPTs) (Hardy 2001 arXiv:quant-ph/0101012; Barrett 2007 Phys. Rev. A 75 032304). We then apply them to a particular GPT, box world, and find a deterministic contradiction in the case where agents may share a PR box (Popescu and Rohrlich 1994 Found. Phys. 24 379–85), which is stronger than the quantum paradox, in that it does not rely on post-selection. Obtaining an inconsistency for the framework of GPTs broadens the landscape of theories which are affected by the application of classical rules of reasoning to physical agents. In addition, we model how observers' memories may evolve in box world, in a way consistent with Barrett's criteria for allowed operations (Barrett 2007 Phys. Rev. A 75 032304; Gross et al 2010 Phys. Rev. Lett. 104 080402).