Covariant KSGNS construction and quantum instruments

: Haapasalo E, Pellonpaa JP

Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD

: 2017

: Reviews in Mathematical Physics

: REVIEWS IN MATHEMATICAL PHYSICS

: REV MATH PHYS

: ARTN 1750020

: 29

: 7

: 47

: 0129-055X

: 1793-6659

DOI: https://doi.org/10.1142/S0129055X17500209

We study completely positive (CP) A-sesquilinear-form-valued maps on a unital C*-algebra B, where the sesquilinear forms operate on a module over a C*-algebra A. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.