A simple proof that anomalous weak values require coherence

Abstract

The quantum mechanical weak value $A_w=⟨\varphi |A|\psi ⟩/⟨\varphi |\psi ⟩$ of an observable $A$ is a measurable quantity associated with an observable $A$ and pre- and post-selected states $|\psi ⟩,|\varphi ⟩$. Much has been discussed about the meaning and metrological uses of anomalous weak values, lying outside of the range of eigenvalues of $A$. We present a simple proof that anomalous weak values require that the (possibly mixed) pre- and post- selection states have coherence in the eigenbasis of $A$. We also present conditions under which anomalous $A_w$ are witnesses of generalized contextuality, dispensing with the operational weak measurement set-up.

6 pages, 2 figures