Sufficient conditions for uniqueness of the weak value

TL;DR

This paper clarifies the conditions under which the generalized weak value is uniquely defined as a weak limit of a conditioned average, addressing criticisms and providing a formal proof of uniqueness.

Contribution

It establishes sufficient conditions for the uniqueness of the generalized weak value and responds to criticisms by analyzing counter-examples within the formalism.

Findings

Counter-example does not satisfy the conditions, thus not invalidating the uniqueness.

Proper application of the formalism yields a natural measurement interpretation.

Theorem proves uniqueness under the specified conditions.

Abstract

We review and clarify the sufficient conditions for uniquely defining the generalized weak value as the weak limit of a conditioned average using the contextual values formalism introduced in Dressel J, Agarwal S and Jordan A N 2010 Phys. Rev. Lett. 104, 240401. We also respond to criticism of our work in [arXiv:1105.4188v1] concerning a proposed counter-example to the uniqueness of the definition of the generalized weak value. The counter-example does not satisfy our prescription in the case of an underspecified measurement context. We show that when the contextual values formalism is properly applied to this example, a natural interpretation of the measurement emerges and the unique definition in the weak limit holds. We also prove a theorem regarding the uniqueness of the definition under our sufficient conditions for the general case. Finally, a second proposed counter-example in [arXiv:1105.4188v6] is shown not to satisfy the sufficiency conditions for the provided theorem.