Categories of relations as models of quantum theory
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.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 arXiv:
 arXiv:1506.05028
 Bibcode:
 2015arXiv150605028H
 Keywords:

 Mathematics  Category Theory;
 Computer Science  Logic in Computer Science;
 Quantum Physics
 EPrint:
 In Proceedings QPL 2015, arXiv:1511.01181