一句话：推广范畴量子力学至无限维过程，建立乘法 Dagger 线性逻辑的范畴语义并定义 mixed unitary category。
范畴量子力学利用有限维 Hilbert 空间的 Dagger 紧闭结构与 string diagram 图形演算进行推理，但许多量子物理涉及无限维过程，而所有 Hilbert 空间的范畴并非紧闭，因此该框架不能直接处理无限维过程。本文启动这一推广方向：以 *-autonomous 范畴及更一般的 linearly distributive 范畴作为可容纳无限维空间的范畴基础，同时发展乘法 Dagger 线性逻辑的范畴语义，重点刻画 Dagger 在这些更一般结构中的行为。文末定义 mixed unitary category，为后续扩展范畴量子力学的核心特征提供结构基础。
Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.
@article{paperbot1266,
title = {Dagger linear logic for categorical quantum mechanics},
author = {Robin Cockett and Cole Comfort and Priyaa Srinivasan},
journal = {Logical Methods in Computer Science},
volume = {Volume 17, Issue 4},
year = {2021},
doi = {10.46298/lmcs-17(4:8)2021}
}