A simple proof that anomalous weak values require coherence

The quantum mechanical weak value $A_w=⟨\phi \mathrm{\mid }A\mathrm{\mid }\psi ⟩\mathrm{/}⟨\phi \mathrm{\mid }\psi ⟩$ of an observable $A$ is a measurable quantity associated with an observable $A$ and pre- and post-selected states $\mathrm{\mid }\psi ⟩,\mathrm{\mid }\phi ⟩$. 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.