Proof gap in "Sufficient conditions for uniqueness of the Weak Value" by J. Dressel and A. N. Jordan, J. Phys. A 45 (2012) 015304

TL;DR

The paper critiques a proof claiming to establish conditions for the convergence of a quantum weak value, identifying unstated assumptions and a critical gap, but suggests the conclusion may still hold under certain conditions.

Contribution

It reveals unstated assumptions and a proof gap in the original theorem, clarifying the conditions under which the weak value convergence claim may be valid.

Findings

Identified unstated assumptions in the proof

Found a critical gap or error in the proof

Conjecture that the conclusion holds for linear POVMs with pseudoinverse

Abstract

The article of the title attempts to prove a "General theorem" (GT) giving sufficient conditions under which a previously introduced "general conditioned average" "converges uniquely to the quantum weak value in the minimal disturbance limit." The "general conditioned average" is obtained from a positive operator valued measure (POVM) depending on a small "weakness" parameter g. We point out that unstated assumptions in the presentation of the "sufficient conditions" make them appear much more general than they actually are. Indeed, the stated "sufficient conditions" strengthened by these unstated assumptions seem very close to an assumption that the POVM operators be linear polynomials in g. Moreover, there appears to be a critical error or gap in the attempted proof, even assuming a linear POVM. A counterexample to the proof of the GT (though not to its conclusion) is given. Nevertheless, I conjecture that the conclusion is actually true for linear POVM's whose contextual values are chosen by the article's "pseudoinverse prescription".