Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism 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 $H_2^{+}$.

arXiv comment44 pages; typos corrected

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Cited by in corpus (5)

• Quantum graphs: different perspectives, homomorphisms and quantum automorphisms
• Some examples of quantum graphs
• Quantum graphs, subfactors and tensor categories I
• Complete Classification of Directed Quantum Graphs on M2
• Algebraic connectedness and bipartiteness of quantum graphs