Covariant KSGNS construction and quantum instruments

TL;DR

This paper develops a generalized covariant KSGNS construction for positive kernels and quantum instruments, characterizing extremal points and minimal dilations in the context of group actions and $C^*$-algebras.

Contribution

It introduces a new framework for covariant positive kernels and quantum instruments, providing necessary and sufficient conditions for extremality and a covariant minimal dilation construction.

Findings

Characterization of extremal covariant positive kernels.

Determination of extreme points of normalized covariant CP maps.

Analysis of covariant quantum instruments for unimodular type-I groups.

Abstract

We study positive kernels on $X\times X$, where $X$ is a set equipped with an action of a group, and taking values in the set of $\mathcal A$-sesquilinear forms on a (not necessarily Hilbert) module over a $C^*$-algebra $\mathcal A$. These maps are assumed to be covariant with respect to the group action on $X$ and a representation of the group in the set of invertible ($\mathcal A$-linear) module maps. We find necessary and sufficient conditions for extremality of such kernels in certain convex subsets of positive covariant kernels. Our focus is mainly on a particular example of these kernels: a completely positive (CP) covariant map for which we obtain a covariant minimal dilation (or KSGNS construction). We determine the extreme points of the set of normalized covariant CP maps and, as a special case, study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. As an example, we discuss the case of instruments that are covariant with respect to a square-integrable representation.