Proof gap in "Sufficient conditions for uniqueness of the Weak Value" by J. Dressel and A. N. Jordan, J. Phys. A 45 (2012) 015304
arXiv:1202.5604
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".
31 pages. The authors of "Sufficient conditions" have recently published a Corrigendum acknowledging a proof gap and including a Lemma apparently intended to repair it. Version 7 includes a 7-page Appendix pointing out an error which invalidates the Lemma's proof and including a simple proof of an important special case