Why FHilb is Not an Interesting (Co)Differential Category
Abstract
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 finitedimensional 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 incompatibility between the trace of FHilb and possible differential category structure. That said, there are interesting nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.00208
 Bibcode:
 2020arXiv200500208P
 Keywords:

 Computer Science  Logic in Computer Science;
 Mathematics  Category Theory;
 F.4.1
 EPrint:
 In Proceedings QPL 2019, arXiv:2004.14750