A simple proof that anomalous weak values require coherence

TL;DR

This paper provides a straightforward proof that anomalous weak values in quantum mechanics necessitate coherence in the eigenbasis of the observable, linking them to generalized contextuality without relying on weak measurement procedures.

Contribution

It offers a simple proof connecting anomalous weak values to coherence in the eigenbasis and identifies conditions where they witness generalized contextuality.

Findings

Anomalous weak values require coherence in the eigenbasis of the observable.

Conditions are identified under which weak values witness generalized contextuality.

The proof dispenses with the need for operational weak measurement setups.

Abstract

The quantum mechanical weak value $A_w=\left\langle \phi|A|\psi \right \rangle / \left\langle \phi | \psi \right\rangle$ of an observable $A$ is a measurable quantity associated with an observable $A$ and pre- and post-selected states $\vert\psi \rangle, \vert \phi \rangle$. 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.