Complete Classification of Directed Quantum Graphs on M2

TL;DR

This paper provides a complete classification of directed quantum graphs on the matrix algebra M_2, revealing a broader variety than undirected graphs and exploring differences in quantum graph types on nontracial quantum sets.

Contribution

It extends existing classifications to directed quantum graphs on M_2, incorporating the framework for directed graphs and analyzing their properties.

Findings

Directed quantum graphs on M_2 are more diverse than undirected ones.

Complete classification of directed quantum graphs on M_2 achieved.

Differences between GNS- and KMS-undirected quantum graphs on nontracial sets illustrated.

Abstract

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 $(M_2, \psi)$ we illustrate the difference between GNS- and KMS-undirected quantum graphs.