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American Journal of Physics

Comment on “What quantum mechanics is trying to tell us,” by Ulrich Mohrhoff [Am. J. Phys. 68 (8), 728–745 (2000)]

Ruth E. Kastner

Published Aug 1, 2001• 5 citations
Publisher

Abstract

Mohrhoff proposes using the Aharonov–Bergmann–Lebowitz (ABL) rule for time-symmetric “objective” (meaning nonepistemic) probabilities corresponding to the possible outcomes of not-actually-performed measurements between specified pre- and post-selection measurement outcomes. It is emphasized that the ABL rule was formulated on the assumption that such intervening measurements are actually made and that it does not necessarily apply to counterfactual situations. The exact nature of the application of the ABL rule considered by Mohrhoff is made explicit and is shown to fall short of his stated counterfactual claim.

DOI
10.1119/1.1371918

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Mohrhoff proposes using the Aharonov–Bergmann–Lebowitz (ABL) rule for time-symmetric “objective” (meaning nonepistemic) probabilities corresponding to the possible outcomes of not-actually-performed measurements between specified pre- and post-selection measurement outcomes. It is emphasized that the ABL rule was formulated on the assumption that such intervening measurements are actually made and that it does not necessarily apply to counterfactual situations. The exact nature of the application of the ABL rule considered by Mohrhoff is made explicit and is shown to fall short of his stated counterfactual claim.
Mohrhoff proposes using the Aharonov-Bergmann-Lebowitz (ABL) rule for time-symmetric ``objective'' (meaning non-epistemic) probabilities corresponding to the possible outcomes of not-actually-performed measurements between specified pre- and post-selection measurement outcomes. It is emphasized that the ABL rule was formulated on the assumption that such intervening measurements are actually made and that it does not necessarily apply to counterfactual situations. The exact nature of the application of the ABL rule considered by Mohrhoff is made explicit and is shown to fall short of his stated counterfactual claim.
A currently discussed interpretation of quantum theory, Time-Symmetrized Quantum Theory, makes certain claims about the properties of systems between pre- and post- selection measurements. These claims are based on a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule for calculating the probabilities of measurement outcomes between such measurements. It has been argued by several authors that the counterfactual usage of the ABL rule is, in general, incorrect. This paper examines what might appear to be a loophole in those arguments and shows that this apparent loophole cannot be used to support a counterfactual interpretation of the ABL rule. It is noted that the invalidity of the counterfactual usage of the ABL rule implies that the characterization of those outcomes receiving probability 1 in a counterfactual application of the rule as `elements of reality' is, in general, unfounded.
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.
We examine some surprising results that have been obtained for pre- and postselected quantum systems. We show that these results depend on a counterfactual interpretation of the rule of Aharonov, Bergmann, and Lebowitz [Phys. Rev. 134, B1410 (1964)] (the ABL rule) and that this interpretation is not valid in general. We then argue, with the help of the consistent histories interpretation of quantum mechanics, that there is a special class of situations where application of the counterfactual interpretation of the ABL rule can be justified. We then reexamine the aforementioned surprising results.
The Aharonov-Bergmann-Lebowitz rule assigns probabilities to quantum measurement results at time t on the condition that the system is prepared in a given way at t_1 < t and found in a given state at t_2 > t. The question whether the rule can also be applied counterfactually to the case where no measurement is performed at the intermediate time t has recently been the subject of controversy. I argue that the counterfactual meaning may be understood in terms of the true value of an observable at t. Such a value cannot be empirically determined for, by stipulation, the measurement that would yield it is not performed. Nevertheless, it may or may not be well-defined depending on one's proposed interpretation of quantum mechanics. Various examples are discussed illustrating what can be asserted at the intermediate time without running into contradictions.