Title: Dagger linear logic for categorical quantum mechanics

Abstract: Categorical quantum mechanics is largely based on dagger compact closed categories. A well-known limitation of this approach is that, while it supports finite dimensional processes, it does not generalize well to infinite dimensional processes. A natural categorical generalization of compact closed categories, in which to seek a description of categorical quantum mechanics without this limitation, is in ∗-autonomous categories or, more generally, linearly distributive categories (LDCs).

Last winter, Robin, Cole and myself started development of this direction of generalization. An important first step is to establish the behavior of the dagger in this more general setting. Thus, we simultaneously developed the categorical semantics of dagger linear logic. The goal of my talk is to give the definition of Mixed Unitary Categories [1] which is a key structure in the development of the quantum mechanics in the setting of LDCs. I will also show a running example of this category during my talk.

Reference:

[1] Robin Cockett, Cole Comfort, and Priyaa Srinivasan “Dagger linear logic for categorical quantum mechanics”, arXiv:1809.00275 [math.CT], Sep. 2018.

This week we:-

– recap the definition of dagger-LDC, dagger mix and dagger isomix categories

– define unitary structure for dagger isomix categories

– show that compact LDCs are linearly equivalent to monoidal categories

– define a mixed unitary category

– (If time permits) Finiteness spaces, an example of a Mixed Unitary Category