Entropic partial orderings of quantum measurements

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 $\rho$) holds if and only if $M \equiv N$ are post-processing equivalent, which shows that the first three orderings induce identical equivalence classes.