Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups

TL;DR

This paper develops a diagrammatic framework for directed nontracial quantum graphs and classifies undirected reflexive quantum graphs on 2x2 matrices, revealing quantum isomorphisms with classical graphs and monoidal equivalences.

Contribution

It introduces a diagrammatic formulation for directed nontracial quantum graphs and classifies undirected reflexive quantum graphs on M_2, connecting quantum and classical graph isomorphisms.

Findings

Classified undirected reflexive quantum graphs on M_2.

Established quantum isomorphisms with classical graphs.

Reproved monoidal equivalences between certain quantum groups.

Abstract

Motivated by string diagrammatic approach to undirected tracial quantum graphs by Musto, Reutter, Verdon (2018), in the former part of this paper we diagrammatically formulate directed nontracial quantum graphs by Brannan, Chirvasitu, Eifler, Harris, Paulsen, Su, Wasilewski (2019). In the latter part, we supply a concrete classification of undirected reflexive quantum graphs on $M_2$ and their quantum automorphism groups in both tracial and nontracial settings. We also obtain quantum isomorphisms between tracial quantum graphs on $M_2$ and certain classical graphs, which reproves the monoidal equivalences between $SO(3)$ and $S_4^+$, and $O(2)$ and $H_2^+$.