Distances for Operator-valued Information Channels

TL;DR

This paper develops new metrics for quantum probability measures and operator-valued information channels, extending classical topological structures to quantum settings and characterizing these metrics via completely bounded norms.

Contribution

It introduces three metrics on quantum probability measures, extends topological structures to operator-valued channels, and characterizes these using completely bounded norms.

Findings

Defined three metrics on quantum probability measures.

Extended topological structures to operator-valued channels.

Characterized metrics via completely bounded norms.

Abstract

We introduce three metrics on the set of quantum probability measures over a compact Hausdorff space and characterize them in terms of the completely bounded norm of the corresponding unital completely positive maps. We extend the existing topological structures between scalar-valued information channels to operator-valued ones and associate them with topologies on the set of unital completely positive maps between a commutative C*-algebra and the C*-algebra of bounded weakly measurable operator-valued functions over a compact Hausdorff space. Given a measure on the input alphabet space, we introduce the notion of an almost everywhere defined operator-valued information channel and provide a characterization result.