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

Abstract

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.

Witnessing incompatibility of quantum channels

“…The incompatibility of quantum channels has been defined and studied in [8,22,23]. That definition has been generalized in [24] for different types of devices in general probabilistic theories, while in [25] it is extended to cover the case of two channels with arbitrary outcome algebras.…”

Convex decomposition of dimension-altering quantum channels

“…This means any convex combination of channels still leads to a valid quantum channel. The convex set of quantum channels and also other quantum objects such as states and observables have been a major focus of mathematical characterization lately [9][10][11][12][13], and exploring the convexity can benefit tasks involving quantum channels, such as quantum channel simulation [14,15].…”

Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements

“…Corollary 4 and a weaker version of Corollary 5 (restricted to m = 2 and requiring the tuple to be extremal both on P m and J m ) were already proved in Ref. [7], in the context of operator algebras. (1) , .…”

“…Joint measurement uniqueness can be further connected to the concepts of greatest and maximal lower bounds [6]. The extremality of the measurements in the tuple was also studied, and found to be related to the extremality and uniqueness of the corresponding joint measurement [7], but not equivalent. This property is also sufficient for some relations between joint measurability and coexistence [8], but the extremality of the tuple itself was never considered.…”