Covariant KSGNS construction and quantum instruments

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.