Categories of Quantum and Classical Channels (extended 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.