Tensors, !-graphs, and non-commutative quantum structures (extended version)

TL;DR

This paper extends !-graphs to represent non-commutative structures in quantum algebra using an enriched tensor notation, enabling reasoning about a broader class of quantum diagrams.

Contribution

It introduces a new semantics for non-commutative !-graphs based on enriched tensor notation, overcoming previous limitations.

Findings

Enables representation of non-commutative quantum structures

Provides a formal semantics for non-commutative !-graphs

Facilitates reasoning about quantum groups and algebraic quantum information

Abstract

!-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and compositions thereof. However, they have previously been limited by an inability to express families of diagrams involving non-commutative structures which play a central role in algebraic quantum information and the theory of 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.