Completely positive projections and biproducts

TL;DR

This paper analyzes the CP*-construction in categorical quantum mechanics, comparing it to earlier methods like adding biproducts and splitting idempotents, highlighting its embedding properties and limitations.

Contribution

It provides a detailed comparison of the CP*-construction with previous approaches, clarifying its position and embedding relations within categorical quantum mechanics.

Findings

CP*-construction unites quantum and classical systems.

It embeds the biproduct-adding approach.

It embeds into the idempotent-splitting approach, but not equivalently.

Abstract

The recently introduced CP*-construction unites quantum channels and classical systems, subsuming the earlier CPM-construction in categorical quantum mechanics. We compare this construction to two earlier attempts at solving this problem: freely adding biproducts to CPM, and freely splitting idempotents in CPM. The CP*-construction embeds the former, and embeds into the latter, but neither embedding is an equivalence in general.