Categories of relations as models of quantum theory

TL;DR

This paper explores how categories of relations over regular categories serve as models for quantum theory, capturing key quantum features and extending classical relations to more complex mathematical structures.

Contribution

It introduces a framework using regular logic to lift properties of relations to models of quantum theory, connecting Frobenius structures with internal groupoids and characterizing quantum features.

Findings

Relations over compact Hausdorff spaces enable continuous symmetric encryption.

Characterization of completely positive maps in regular Mal'cev categories.

Models exhibit quantum phenomena like Heisenberg uncertainty and no-broadcasting.

Abstract

Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspondence between Frobenius structures and internal groupoids. Over compact Hausdorff spaces, this lifting gives continuous symmetric encryption. Over a regular Mal'cev category, this correspondence gives a characterization of categories of completely positive maps, enabling the formulation of quantum features. These models are closer to Hilbert spaces than relations over sets in several respects: Heisenberg uncertainty, impossibility of broadcasting, and behavedness of rank one morphisms.