Axiomatizing complete positivity

TL;DR

This paper extends the axiomatization of categorical models for quantum theory to include classical systems, using decoherence structures to generalize environment structures for complete positivity.

Contribution

It introduces decoherence structures to axiomatize mixed quantum theory with classical systems alongside quantum systems.

Findings

Axiomatization of mixed quantum theory with classical systems achieved

Decoherence structures generalize environment structures

Framework supports categorical models of complete positivity

Abstract

There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, whereas the other also allows classical systems on an equal footing. The former has been axiomatized using environment structures. We extend this axiomatization to the latter by introducing decoherence structures.