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arXiv:quant-ph/0606067 · Journal of Physics A Mathematical and Theoretical

The three-box paradox revisited

Tamar Ravon, Lev Vaidman

Published Feb 28, 2007• 46 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/40/11/021

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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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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.
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.
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.
The quantum three-box paradox is a prototypical example of some bizarre predictions for intermediate measurements made on pre- and postselected systems. Although in principle those effects can be explained by measurement disturbance, it is not clear what mechanisms are required to fully account for the observed correlations. In this paper, this paradox is scrutinized from the causal point of view. We consider an array of potential causal structures behind the experiment, eliminating those without enough explanatory power. A distinction is made between the propagation in the system of just the measurement outcome and the information about the full measurement context. We also discuss the consequences of the realist position in which preexisting values are revealed by measurements. Interestingly, the answers depend crucially on whether the original version of the paradox is considered or its extension where the third box is allowed for inspection too. This illustrates the richness of the paradox which is better appreciated from the causal perspective.