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<p>In 2022, Gromada and Matsuda classified undirected quantum graphs on the matrix algebra <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(M_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> (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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(M_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. Most prominently, we observe that there is a far bigger range of directed quantum graphs than of undirected quantum graphs on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(M_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. Moreover, for quantum graphs on a nontracial quantum set <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((M_2, \psi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> we illustrate the difference between GNS- and KMS-undirected quantum graphs.</p>

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Complete classification of directed quantum graphs on \(M_2\)

  • 期刊论文

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  • 发表日期：

    2026年3月4日

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  • Nina Kiefer,
  • Björn Schäfer

摘要

In 2022, Gromada and Matsuda classified undirected quantum graphs on the matrix algebra \(M_2\) M 2 (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 \(M_2\) M 2 . Most prominently, we observe that there is a far bigger range of directed quantum graphs than of undirected quantum graphs on \(M_2\) M 2 . Moreover, for quantum graphs on a nontracial quantum set \((M_2, \psi )\) ( M 2 , ψ ) we illustrate the difference between GNS- and KMS-undirected quantum graphs.