The ABL Rule and the Perils of Post-Selection

TL;DR

The paper argues that the ABL rule does not furnish a time-symmetric formulation of quantum mechanics and does not enable real violations of the uncertainty principle. It identifies four key fallacies—ensemble fallacy, post-selection fallacy, pattern-matching fallacy, and measurementism—that skew interpretations of the ABL rule and post-selection. The work clarifies the correct operational meaning of the ABL rule through a careful derivation and emphasizes that ensemble observables do not directly inform single-system properties. It concludes with a call for rigorous justification when post-selection is used, while acknowledging legitimate spin-offs and proposing directions for future research on related ideas such as weak values.

Abstract

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.