Complete Classification of Directed Quantum Graphs on M2

In 2022, Gromada and Matsuda classified undirected quantum graphs on the matrix algebra $M_2$. Later, Wasilweski provided a solid theory of directed quantum graphs which was formerly only established for undirected quantum graphs. Using this framework we extend the results of Matsuda and Gromada, and present a complete classification of directed quantum graphs on $M_2$. Most prominently, we observe that there is a far bigger range of directed quantum graphs than of undirected quantum graphs on $M_2$. Moreover, for quantum graphs on a nontracial quantum set $(M_2, ψ)$ we illustrate the difference between GNS- and KMS-undirected quantum graphs.