Spekkens's toy theory as a category of processes

TL;DR

This paper presents a categorical framework for Spekkens's toy qubit theory, demonstrating its structural properties and its potential to model quantum phenomena in a discrete setting.

Contribution

It provides the first comprehensive categorical description of Spekkens's toy theory, showing its closure under composition and its alignment with categorical quantum mechanics.

Findings

The toy theory forms a subcategory of finite sets and relations.

It is closed under parallel and sequential composition.

States and effects are in bijective correspondence.

Abstract

We provide two mathematical descriptions of Spekkens's toy qubit theory, an inductively one in terms of a small set of generators, as well as an explicit closed form description. It is a subcategory MSpek of the category of finite sets, relations and the cartesian product. States of maximal knowledge form a subcategory Spek. This establishes the consistency of the toy theory, which has previously only been constructed for at most four systems. Our model also shows that the theory is closed under both parallel and sequential composition of operations (= symmetric monoidal structure), that it obeys map-state duality (= compact closure), and that states and effects are in bijective correspondence (= dagger structure). From the perspective of categorical quantum mechanics, this provides an interesting alternative model which enables us to describe many quantum phenomena in a discrete manner, and to which mathematical concepts such as basis structures, and complementarity thereof, still apply. Hence, the framework of categorical quantum mechanics has delivered on its promise to encompass theories other than quantum theory.