We study the partially ordered set of equivalence classes of quantum
measurements endowed with the post-processing partial order. The
post-processing order is fundamental as it enables to compare measurements by
their intrinsic noise and it gives grounds to define the important concept of
quantum incompatibility. Our approach is based on mapping this set into a
simpler partially ordered set using an order preserving map and investigating
the resulting image. The aim is to ignore unnecessary details while keeping the
essential structure, thereby simplifying e.g. detection of incompatibility. One
possible choice is the map based on Fisher information introduced by Huangjun
Zhu, known to be an order morphism taking values in the cone of positive
semidefinite matrices. We explore the properties of that construction and
improve Zhu’s incompatibility criterion by adding a constraint depending on the
number of measurement outcomes. We generalize this type of construction to
other ordered vector spaces and we show that this map is optimal among all
quadratic maps.

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