Complete classification of directed quantum graphs on $$M_2$$

Abstract In 2022, Gromada and Matsuda classified undirected quantum graphs on the matrix algebra $$M_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> (Gromada in Lett Math Phys 112:122, 2022; Matsuda in J Math Phys. 63:092201, 2022). Later, Wasilweski provided a solid theory of directed quantum graphs (Wasilewski in 29:1281–1317, 2024) 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 $$M_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . Most prominently, we observe that there is a far bigger range of directed quantum graphs than of undirected quantum graphs on $$M_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . Moreover, for quantum graphs on a nontracial quantum set $$(M_2, \psi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> we illustrate the difference between GNS- and KMS-undirected quantum graphs.