Why FHilb is Not an Interesting (Co)Differential Category

Jean-Simon Pacaud Lemay
(University of OXford)

Differential categories provide an axiomatization of the basics of differentiation and categorical models of differential linear logic. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes sense to study applications of the theory of differential categories to categorical quantum foundations. In categorical quantum foundations, compact closed categories (and therefore traced symmetric monoidal categories) are one of the main objects of study, in particular the category of finite-dimensional Hilbert spaces FHilb. In this paper, we will explain why the only differential category structure on FHilb is the trivial one. This follows from a sort of in-compatibility between the trace of FHilb and possible differential category structure. That said, there are interesting non-trivial examples of traced/compact closed differential categories, which we also discuss.

The goal of this paper is to introduce differential categories to the broader categorical quantum foundation community and hopefully open the door to further work in combining these two fields. While the main result of this paper may seem somewhat "negative" in achieving this goal, we discuss interesting potential applications of differential categories to categorical quantum foundations.

In Bob Coecke and Matthew Leifer: Proceedings 16th International Conference on Quantum Physics and Logic (QPL 2019), Chapman University, Orange, CA, USA., 10-14 June 2019, Electronic Proceedings in Theoretical Computer Science 318, pp. 13–26.
Published: 1st May 2020.

ArXived at: http://dx.doi.org/10.4204/EPTCS.318.2 bibtex PDF
References in reconstructed bibtex, XML and HTML format (approximated).
Comments and questions to: eptcs@eptcs.org
For website issues: webmaster@eptcs.org