Toy quantum categories

TL;DR

This paper demonstrates that Rob Spekken's toy quantum theory can be understood as a specific example within a broader categorical framework, revealing that its quantum-like features are rooted in general category-theoretic principles.

Contribution

It introduces a categorical approach to quantum axiomatics, showing toy quantum models as subcategories of the dagger compact category FRel and interpreting quantum observables categorically.

Findings

Toy quantum theory is a subcategory of FRel

Quantum properties are general category-theoretic features

Complementary observables are interpretable on the two-element set

Abstract

We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general category-theoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set FRel.