Tensors, !-graphs, and non-commutative quantum structures

TL;DR

This paper introduces a new semantics for non-commutative !-graphs using an enriched tensor notation, expanding the expressive power of graphical languages in categorical quantum mechanics and quantum groups.

Contribution

It extends !-graph techniques to non-commutative structures by developing an enriched tensor notation, enabling more complex diagrammatic reasoning.

Findings

Provides a new semantics for non-commutative !-graphs

Enables diagrammatic proofs for non-commutative algebraic structures

Expands the applicability of !-box induction in quantum theories

Abstract

Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us to easily form complex compositions of (co)algebraic structures, and prove their equality via graph rewriting. One of the biggest challenges in going beyond simple rewriting-based proofs is designing a graphical language that is expressive enough to prove interesting properties (e.g. normal form results) about not just single diagrams, but entire families of diagrams. One candidate is the language of !-graphs, which consist of graphs with certain subgraphs marked with boxes (called !-boxes) that can be repeated any number of times. New !-graph equations can then be proved using a powerful technique called !-box induction. However, previously this technique only applied to commutative (or cocommutative) algebraic structures, severely limiting its applications in some parts of CQM and (especially) quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose's abstract tensor notation.