Entropic partial orderings of quantum measurements
Adam Teixidó-Bonfill, Joseph Schindler, Dominik Šafránek
TL;DR
This work analyzes four partial orderings on quantum measurements (POVMs): stochastic post-processing, measured relative entropy, observational entropy, and a linear-relations ordering. It proves a strict hierarchy $①\rightarrow ②\rightarrow ③\rightarrow ④$ for general POVMs, with projective measurements collapsing the hierarchy so that all four notions become equivalent; the equality $S_M(\rho)=S_N(\rho)$ for all $\rho$ is shown to be equivalent to post-processing equivalence $M \equiv N$. The paper develops structural tools (including disjoint convex combinations and linear-span arguments) to separate the orderings and to characterize when entropic comparisons imply or fail to imply stochastic relations. These results connect entropic monotones to the network of POVM equivalence classes and have implications for the resource theory of quantum measurements and coarse-graining in quantum information and thermodynamics.
Abstract
We investigate four partial orderings on the space of quantum measurements (i.e on POVMs or positive operator valued measures), describing four notions of coarse/fine-ness of measurement. These are the partial orderings induced by: (1) classical post-processing, (2) measured relative entropy, (3) observational entropy, and (4) linear relation of POVMs. The orderings form a hierarchy of implication, where e.g. post-processing relation implies all the others. We show that this hierarchy is strict for general POVMs, with examples showing that all four orderings are strictly inequivalent. Restricted to projective measurements, all are equivalent. Finally we show that observational entropy equality $S_M = S_N$ (for all $ρ$) holds if and only if $M \equiv N$ are post-processing equivalent, which shows that the first three orderings induce identical equivalence classes.