Higher-order CPM Constructions

TL;DR

This paper introduces a higher-order generalization of the CPM construction using finite abelian group symmetries, providing a functorial framework and linking to probabilistic theories and interference phenomena.

Contribution

It presents a novel higher-order CPM construction, characterizes its iterative closure via monads, and connects to existing categorical probabilistic frameworks.

Findings

Construction is functorial and closed under iteration

Connects to previous CPM and probabilistic theories

Provides examples linking to interference and hyper-decoherence

Abstract

We define a higher-order generalisation of the CPM construction based on arbitrary finite abelian group symmetries of symmetric monoidal categories. We show that our new construction is functorial, and that its closure under iteration can be characterised by seeing the construction as an algebra for an appropriate monad. We provide several examples of the construction, connecting to previous work on the CPM construction and on categorical probabilistic theories, as well as upcoming work on higher-order interference and hyper-decoherence.