## Programming Research Group
Research Report RR-07-05

*Complete Positivity without Positivity and Without Compactness
*
Bob Coecke

September 2007, 8pp.

### 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.

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