Categories of relations as models of quantum theory

Chris Heunen
(University of Oxford)
Sean Tull
(University of Oxford)

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.

In Chris Heunen, Peter Selinger and Jamie Vicary: Proceedings of the 12th International Workshop on Quantum Physics and Logic (QPL 2015), Oxford, U.K., July 15-17, 2015, Electronic Proceedings in Theoretical Computer Science 195, pp. 247–261.
Published: 4th November 2015.

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