Infinite-dimensional Categorical Quantum Mechanics

Stefano Gogioso
(Quantum Group, University of Oxford)
Fabrizio Genovese
(Quantum Group, University of Oxford)

We use non-standard analysis to define a category *Hilb suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We show the existence of unital special commutative dagger-Frobenius algebras, and we conclude *Hilb to be compact closed, with partial traces and a Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the textbook case of 1-dimensional wavefunctions with periodic boundary conditions: we show the momentum and position observables to be well defined, and to give rise to a strongly complementary pair of unital commutative dagger-Frobenius algebras.

In Ross Duncan and Chris Heunen: Proceedings 13th International Conference on Quantum Physics and Logic (QPL 2016), Glasgow, Scotland, 6-10 June 2016, Electronic Proceedings in Theoretical Computer Science 236, pp. 51–69.
Published: 1st January 2017.

ArXived at: http://dx.doi.org/10.4204/EPTCS.236.4 bibtex PDF
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