Quantum weak values and logic, an uneasy couple

TL;DR

This paper examines the potential logical inconsistencies arising from using quantum weak values to infer particle paths in quantum systems, questioning their reliability in certain quantum paradoxes.

Contribution

It demonstrates that weak values can lead to logical contradictions in quantum path questions, contrasting with the consistent results from strong measurements.

Findings

Weak values may produce inconsistent answers for composite questions.

Strong measurements do not exhibit such logical inconsistencies when properly specified.

Weak values' utility in quantum paradoxes is questionable due to these inconsistencies.

Abstract

Quantum mechanical weak values of projection operators have been used to answer which-way questions, e.g. to trace which arms in a multiple Mach-Zehnder setup a particle may have traversed from a given initial to a prescribed final state. I show that this procedure might lead to logical inconsistencies in the sense that different methods used to answer composite questions, like Has the particle traversed the way X or the way Y? , may result in different answers depending on which methods are used to find the answer. I illustrate the problem by considering some examples: the quantum pigeonhole framework of Aharonov et al, the three-box problem, and Hardys paradox. To prepare the ground for my main conclusion on the incompatibility in certain cases of weak values and logic, I study the corresponding situation for strong/projective measurements. In this case, no logical inconsistencies occur provided one is always careful in specifying exactly to which ensemble or sample space one refers. My results cast doubts on the utility of quantum weak values in treating cases like the examples mentioned.