When do pieces determine the whole? Extremal marginals of a completely positive map

TL;DR

This paper investigates the conditions under which extremal marginals of completely positive maps uniquely determine the entire map, providing new insights into quantum instruments and their observables.

Contribution

It proves that extremal marginals uniquely determine the map and that extremal marginals imply the extremality of the whole map, unifying several known results.

Findings

Extremal marginal maps uniquely determine the entire map.

Both marginals extremal implies the entire map is extremal.

Provides new insights into quantum instruments and marginal observables.

Abstract

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extremal point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extremal, then the whole map is extremal. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.