Categories of Quantum and Classical Channels (extended abstract)

TL;DR

This paper introduces the CP*-construction as a generalization of the CPM-construction, unifying classical and quantum channels within a categorical framework, and providing abstract notions of quantum processes.

Contribution

It extends Selinger's CPM-construction to the CP*-construction, enabling a unified categorical treatment of classical and quantum channels and states.

Findings

CP*-construction generalizes CPM-construction to include classical and quantum data

It embeds classical stochastic maps and quantum channels within a unified framework

Provides abstract categorical notions of quantum preparation and measurement

Abstract

We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-construction in such a way that the objects in the image of the embedding can be thought of as "purely quantum" state spaces. It also embeds the category of classical stochastic maps, whose image consists of "purely classical" state spaces. By allowing classical and quantum data to coexist, this provides elegant abstract notions of preparation, measurement, and more general quantum channels.