Minimal normal measurement models of quantum instruments

In this work we study the minimal normal measurement models of quantum instruments. We show that usually the apparatus' Hilbert space in such a model is unitarily isomorphic to the minimal Stinespring dilation space of the instrument. However, if the Hilbert space of the system is infinite-dimensional and the multiplicities of the outcomes of the associated observable (POVM) are all infinite then this may not be the case. In these pathological cases the minimal apparatus' Hilbert space is shown to be unitarily isomorphic to the instrument's minimal dilation space augmented by one extra dimension.