Classical structures of CP maps are all canonical

TL;DR

This paper proves that all classical structures in the category of completely positive maps are canonical, showing they originate from the doubling of structures in finite-dimensional Hilbert spaces, using a purity principle.

Contribution

It establishes that all isometric comonoids in CPM(fHilb) are pure and that all special dagger Frobenius algebras are canonical, answering an open question in quantum information theory.

Findings

All isometric comonoids in CPM(fHilb) are pure.

All special dagger Frobenius algebras in CPM(fHilb) are canonical.

Classical structures in CPM(fHilb) arise from doubling structures in fHilb.

Abstract

We use purity, a principle borrowed from the foundations of quantum information, to show that all isometric comonoids in the category $\operatorname{CPM}\left(\operatorname{fHilb}\right)$ are necessarily pure. As a corollary, we answer an open question about special dagger Frobenius algebras (and classical structures in particular) in $\operatorname{CPM}\left(\operatorname{fHilb}\right)$: we show that they are all canonical, i.e. that they all arise by doubling of special dagger Frobenius algebras from the category $\operatorname{fHilb}$.