Quantum Theory is a Quasi-stochastic Process Theory

John van de Wetering
(Radboud University Nijmegen, Netherlands)

There is a long history of representing a quantum state using a quasi-probability distribution: a distribution allowing negative values. In this paper we extend such representations to deal with quantum channels. The result is a convex, strongly monoidal, functorial embedding of the category of trace preserving completely positive maps into the category of quasi-stochastic matrices. This establishes quantum theory as a subcategory of quasi-stochastic processes. Such an embedding is induced by a choice of minimal informationally complete POVM's. We show that any two such embeddings are naturally isomorphic. The embedding preserves the dagger structure of the categories if and only if the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of foundational interest in the QBism community. We also study general convex embeddings of quantum theory and prove a dichotomy that such an embedding is either trivial or faithful.

In Bob Coecke and Aleks Kissinger: Proceedings 14th International Conference on Quantum Physics and Logic (QPL 2017), Nijmegen, The Netherlands, 3-7 July 2017, Electronic Proceedings in Theoretical Computer Science 266, pp. 179–196.
Published: 27th February 2018.

ArXived at: http://dx.doi.org/10.4204/EPTCS.266.12 bibtex PDF
References in reconstructed bibtex, XML and HTML format (approximated).
Comments and questions to: eptcs@eptcs.org
For website issues: webmaster@eptcs.org