Categorified Path Calculus

TL;DR

This paper extends path calculus to a surface diagram calculus for bicategories of matrices over bimonoidal categories, enabling new graphical proofs in categorical quantum mechanics.

Contribution

It introduces a categorified surface diagram calculus for matrices over bimonoidal categories, generalizing path calculus and applying it to quantum protocols.

Findings

Develops a surface diagram calculus for bicategories of matrices

Provides a graphical proof of the quantum teleportation protocol

Shows how additional structures like biproducts and duals enhance the calculus

Abstract

Path calculus, or graphical linear algebra, is a string diagram calculus for the category of matrices over a base ring. It is the usual string diagram calculus for a symmetric monoidal category, where the monoidal product is the direct sum of matrices. We categorify this story to develop a surface diagram calculus for the bicategory of matrices over a base bimonoidal category. This yields a surface diagram calculus for any bimonoidal category by restricting to diagrams for 1x1 matrices. We show how additional structure on the base category, such as biproducts, duals and the dagger, adds structure to the resulting calculus. Applied to categorical quantum mechanics this yields a new graphical proof of the teleportation protocol.