Order preserving maps on quantum measurements

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