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arXiv:quant-ph/9807015

Closing a Loophole in the Case Against the Counterfactual Usage of the ABL Rule

Ruth E. Kastner

Published Jul 5, 1998• 1 citation

Abstract

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.

DOI: 10.48550/arxiv.quant-ph/9807015

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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.

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.

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.

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.

In 1964, Aharonov, Bergmann, and Lebowitz introduced their well-known ABL rule with the intention of providing a time-symmetric formalism for computing novel kinds of conditional probabilities in quantum theory. Later papers attached additional significance to the ABL rule, including assertions that it supported violations of the uncertainty principle. The present work challenges these claims, as well as subsequent attempts to salvage the original interpretation of the ABL rule. Taking a broader view, this paper identifies a subtle category error at the heart of the ABL rule that consists of confusing observables that belong to a single system with emergent observables that arise only for physical ensembles. Along the way, this paper points out other problems and fallacious reasoning in the research literature surrounding the ABL rule, including the misuse of post-selection, a reliance on pattern matching to classical formulas, and a posture of measurementism that takes experimental data as providing answers to interpretational questions.