Robin Cockett ; Cole Comfort ; Priyaa Srinivasan - Dagger linear logic for categorical quantum mechanics

lmcs:7030 - Logical Methods in Computer Science, November 16, 2021, Volume 17, Issue 4 -
Dagger linear logic for categorical quantum mechanics

Authors: Robin Cockett ; Cole Comfort ; Priyaa Srinivasan

    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.

    Volume: Volume 17, Issue 4
    Published on: November 16, 2021
    Accepted on: September 9, 2021
    Submitted on: December 31, 2020
    Keywords: Mathematics - Category Theory,18b99, 81P10, 03G12,F.4.1


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