Quantum Weak Values and Logic: An Uneasy Couple

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 Hardy's 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.

Understanding quantum mechanics: a review and synthesis in precise language

“…Such approaches are known as weak measurement [249,250], protective measurement [224,251] and interaction-free measurement [252,253]. The understanding of such processes, and their results, depends on which approach is taken to understanding quantum mechanics generally and, within that, how conventional measurement is understood [206,[254][255][256][257][258]. These approaches are not considered further in this review.…”

Hardy’s paradox according to non-classical semantics

“…Even so, an interpretation of the weak truth degrees, uncontroversial in a sense, can be based on the observation made in [14] and the comments in [15]: According to them, non-zeroness (respectively, zeroness) of the value v w (P ⋄ ) can be interpreted as the divergence (conformity) of the proposition ⋄ from (to) the false, i.e., a trivial proposition whose truth value being always false.…”

Response to Comment on ‘Non-representative Quantum Mechanical Weak Values’ by Ben-Israel and Vaidman

“…Put otherwise, a weak valuesupposed to characterize the system independently of how the weak value is measuredcould in fact show properties that depend non-trivially on the very way it is measured. I have investigated this ambiguity from a different angle in [3].…”