The Trouble with Weak Values

TL;DR

This paper scrutinizes interpretational claims about weak values, emphasizing that they do not generally represent single-system properties. By analyzing the formal definition $A_w = \frac{\langle \Psi'|A|\Psi\rangle}{\langle \Psi'|\Psi\rangle}$ within the Dirac–von Neumann framework and detailing the AAV protocol, it argues that weak values are ensemble observables—often complex—and require post-selection, not a direct property of an individual system. It identifies three core fallacies—ensemble fallacy, post-selection fallacy, and measurementist fallacy—that lead to misleading interpretations, using examples like quantum Cheshire cats and Bohmian trajectories to illustrate the problems. The work calls for caution in drawing metaphysical or physical conclusions from weak values and emphasizes that their practical extraction does not justify single-system ontologies without a robust foundational justification.

Abstract

In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of state vectors. Weak values first appeared in the research literature in a pair of papers in 1987 and 1988, and were originally defined as the results of a special kind of experimental protocol involving non-disturbing measurements combined with an explicit form of post-selection. In the years since, subsequent papers on weak values have produced a number of important practical spin-offs, including new methods for signal amplification and quantum-state tomography. The present work is not concerned with those practical spin-offs, but with historical and ongoing attempts to assign weak values a transparent, single-system interpretation, as well as efforts that invoke weak values to make a number of exotic claims about the properties and behavior of individual quantum systems. This paper challenges these interpretational claims by arguing that they involve several forms of fallacious reasoning.