论文标题
为什么Fhilb不是一个有趣的（CO）差异类别
Why FHilb is Not an Interesting (Co)Differential Category
论文作者
论文摘要
差分类别提供了分化基础和差分线性逻辑模型的基础知识。由于分化是整个量子力学和量子信息的重要工具，因此研究差异类别理论对分类量子基础的应用是有意义的。在分类量子基础中，紧凑的封闭类别（因此是跟踪的对称单体类别）是研究的主要对象之一，尤其是有限维的希尔伯特空间fhilb的类别。在本文中，我们将解释为什么Fhilb上唯一的差异类别结构是微不足道的。这来自Fhilb的痕迹与可能的差异类别结构之间的一种兼容性。也就是说，我们还讨论了有趣的非平凡示例，包括追踪/紧凑的封闭差异类别。 本文的目的是将差异类别引入更广泛的分类量子基础社区，并希望为将这两个领域的进一步努力打开大门。尽管本文的主要结果似乎在实现这一目标时似乎有些“负”，但我们讨论了差异类别在分类量子基础上的有趣潜在应用。
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.
