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 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 incompatibility 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.

Monoidal Reverse Differential Categories

“…and where (0, ∅) is the zero object. Unfortunately, however, as explained in [23], FVEC B k does not usually have a (non-trivial) differential category structure. This problem is solved when we consider k = Z 2 , as was done in [19,23].…”

“…Unfortunately, however, as explained in [23], FVEC B k does not usually have a (non-trivial) differential category structure. This problem is solved when we consider k = Z 2 , as was done in [19,23]. FVEC B Z2 is then a differential category where the coalgebra modality is induced by the exterior algebra, which is defined as follows:…”

Monoidal reverse differential categories

“…and where is the zero object. Unfortunately, however, as explained in Lemay (2019), does not usually have a (non-trivial) differential category structure. This problem is solved when we consider , as was done in Hyland and Schalk (2003), Lemay (2019).…”

“…Unfortunately, however, as explained in Lemay (2019), does not usually have a (non-trivial) differential category structure. This problem is solved when we consider , as was done in Hyland and Schalk (2003), Lemay (2019). is then a differential category where the coalgebra modality is induced by the exterior algebra, which is defined as follows:…”

Graded Differential Categories and Graded Differential Linear Logic

“…Graded differential categories also provide a solution to the problem presented by the first author in [18]. Indeed, in said paper, it is explained why the category of finite dimensional Hilbert spaces (FHILB), the main model of interest in categorical quantum mechanics, has no non-trivial differential category structure.…”