Dagger linear logic for categorical quantum mechanics

TL;DR

This paper extends categorical quantum mechanics to infinite dimensional spaces by developing dagger linear logic within *-autonomous categories, enabling reasoning about a broader class of quantum processes.

Contribution

It introduces the categorical semantics of multiplicative dagger linear logic and defines mixed unitary categories to generalize quantum mechanics frameworks.

Findings

Established the behavior of the dagger in *-autonomous categories

Developed the categorical semantics of multiplicative dagger linear logic

Defined the structure of mixed unitary categories

Abstract

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.