Minimal normal measurement models of quantum instruments

TL;DR

This paper investigates the structure of minimal normal measurement models of quantum instruments, revealing conditions under which the apparatus Hilbert space aligns with the minimal Stinespring dilation space and identifying special cases requiring an extra dimension.

Contribution

It characterizes when the apparatus Hilbert space matches the minimal dilation space and identifies exceptions in infinite-dimensional cases with infinite outcome multiplicities.

Findings

Apparatus Hilbert space usually unitarily isomorphic to minimal Stinespring dilation space.

In certain infinite-dimensional cases, an extra dimension is needed for the apparatus Hilbert space.

Identifies and corrects errors in previous related literature.

Abstract

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. We also point out errors in earlier papers of one of the authors (J-P.P.).