A complete and operational resource theory of measurement sharpness

Quantum 8, 1235 (2024).

https://doi.org/10.22331/q-2024-01-25-1235

We construct a resource theory of $sharpness$ for finite-dimensional positive operator-valued measures (POVMs), where the $sharpness-non-increasing$ operations are given by quantum preprocessing channels and convex mixtures with POVMs whose elements are all proportional to the identity operator. As required for a sound resource theory of sharpness, we show that our theory has maximal (i.e., sharp) elements, which are all equivalent, and coincide with the set of POVMs that admit a repeatable measurement. Among the maximal elements, conventional non-degenerate observables are characterized as the canonical ones. More generally, we quantify sharpness in terms of a class of monotones, expressed as the EPR–Ozawa correlations between the given POVM and an arbitrary reference POVM. We show that one POVM can be transformed into another by means of a sharpness-non-increasing operation if and only if the former is sharper than the latter with respect to all monotones. Thus, our resource theory of sharpness is $complete$, in the sense that the comparison of all monotones provides a necessary and sufficient condition for the existence of a sharpness-non-increasing operation between two POVMs, and $operational$, in the sense that all monotones are in principle experimentally accessible.