Complete Positivity without Positivity and Without Compactness

Abstract

Given any †-symmetric monoidal category C we construct a new category Mix(C), which, in the case that C is a †-compact category, is isomorphic to Selinger's CPM(C) [Sel]. Hence, if C is the category FdHilb of finite dimensional Hilbert spaces and linear maps we exactly obtain completely positive maps as morphisms. This means that mixedness of states and operations, within the categorical quantum axiomatics developed in [AC1, AC2, Sel, CPv, CPq], is a concept which exists independently of the quantum and classical structure. Moreover, since our construction does not require †-compactness, it can be applied to categories which have infinite dimensional Hilbert spaces as objects. Finally, in general Mix(C) is not a †-category, so does not admit a notion of positivity. This means that, in the abstract, the notion of 'complete positivity' can exist independently of a notion of 'positivity', which points at a very unfortunate terminology.