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Differential categories provide an axiomatization of the basics ofdifferentiation and categorical models of differential linear logic. Asdifferentiation is an important tool throughout quantum mechanics and quantuminformation, it makes sense to study applications of the theory of differentialcategories to categorical quantum foundations. In categorical quantumfoundations, compact closed categories (and therefore traced symmetric monoidalcategories) are one of the main objects of study, in particular the category offinite-dimensional Hilbert spaces FHilb. In this paper, we will explain why theonly differential category structure on FHilb is the trivial one. This followsfrom a sort of in-compatibility between the trace of FHilb and possibledifferential category structure. That said, there are interesting non-trivialexamples of traced/compact closed differential categories, which we alsodiscuss.  The goal of this paper is to introduce differential categories to the broadercategorical quantum foundation community and hopefully open the door to furtherwork in combining these two fields. While the main result of this paper mayseem somewhat "negative" in achieving this goal, we discuss interestingpotential applications of differential categories to categorical quantumfoundations.

Why FHilb is Not an Interesting (Co)Differential Category