The ABL Rule and the Perils of Post-Selection

TL;DR

This paper critically examines the ABL rule in quantum mechanics, revealing a fundamental category error and addressing misconceptions about its implications, especially regarding post-selection and measurement interpretation.

Contribution

It identifies a key conceptual mistake in the ABL rule's interpretation and clarifies the proper understanding of observables in quantum ensembles.

Findings

The ABL rule involves a category error confusing single-system and ensemble observables.

Misuse of post-selection leads to fallacious conclusions in quantum interpretations.

The paper clarifies the correct conceptual framework for understanding quantum conditional probabilities.

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.