Tensors, !-graphs, and Non-commutative Quantum Structures

Aleks Kissinger,David Quick

doi:10.1007/s00354-016-0203-4

Aleks Kissinger, David Quick

Open Access

https://doi.org/10.1007/s00354-016-0203-4

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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.

Highlights

David QuickCategorical 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
Diagrammatic theories give us a way to study a wide variety of algebraic and coalgebraic structures in monoidal categories. They consist of two parts: a signature Σ and a set of diagram equations E
The signature consists of a set of objects {A, B, . . .} along with a set of generating morphisms with input and output arities formed from combining objects with ⊗ and I

Summary

David Quick

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 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. New !-graph equations can be proved using a powerful technique called !-box induction. Previously this technique only applied to commutative (or cocommutative) algebraic structures, severely limiting its applications in some parts of CQM and (especially) quantum groups. We fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose’s abstract tensor notation

Introduction

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