# Quantum Theory is a Quasi-stochastic Process Theory

## TL;DR

This paper demonstrates that quantum theory can be represented as a subcategory of quasi-stochastic processes through a functorial embedding using informationally complete POVMs, linking quantum states and channels to quasi-probability distributions.

## Contribution

It extends quasi-probability representations to quantum channels, establishing a functorial embedding of quantum operations into quasi-stochastic matrices, and explores conditions for preserving quantum structures.

## Key findings

- Quantum states and channels can be embedded into quasi-stochastic matrices.
- The embedding preserves the dagger structure if and only if symmetric informationally complete POVMs are used.
- Any two such embeddings are naturally isomorphic, showing a canonical form.

## Abstract

There is a long history of representing a quantum state using a quasi-probability distribution: a distribution allowing negative values. In this paper we extend such representations to deal with quantum channels. The result is a convex, strongly monoidal, functorial embedding of the category of trace preserving completely positive maps into the category of quasi-stochastic matrices. This establishes quantum theory as a subcategory of quasi-stochastic processes. Such an embedding is induced by a choice of minimal informationally complete POVM's. We show that any two such embeddings are naturally isomorphic. The embedding preserves the dagger structure of the categories if and only if the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of foundational interest in the QBism community. We also study general convex embeddings of quantum theory and prove a dichotomy that such an embedding is either trivial or faithful.