Morita equivalence for quantum graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911607950409728 |
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| author | Chatzinikolaou, Alexandros Hoefer, Gage Koutsonikos-Kouloumpis, Nikolaos Paraskevas, Ioannis Apollon |
| author_facet | Chatzinikolaou, Alexandros Hoefer, Gage Koutsonikos-Kouloumpis, Nikolaos Paraskevas, Ioannis Apollon |
| contents | We introduce an operator-algebraic framework for Morita equivalence of quantum graphs based on $Δ$-equivalence of operator systems introduced by Eleftherakis, Kakariadis and Todorov. Adopting the perspective of Weaver, we view quantum graphs as quantum relations, that is, operator systems endowed with a bimodule structure over the commutant of a von Neumann algebra. Within this framework, we show that two irreducibly acting quantum graphs are Morita equivalent if and only if they are both full pullbacks of a common quantum graph. This extends a result of Eleftherakis, Kakariadis and Todorov for graph operator systems to the quantum graph setting. In passing we construct a true-twin reduction analogue for an irreducibly acting quantum graph. We further characterise the case where we have simultaneous TRO-equivalence of the quantum graphs and their associated algebras, thus giving a second, stronger notion of Morita equivalence. In the special case of noncommutative graphs, corresponding to the zero-error quantum communication setting, the two notions coincide and we obtain a characterisation in terms of strong co-homomorphisms of noncommutative graphs. Finally, we show that connectivity, the independence number, Shannon capacity, quantum complexity and subcomplexity, Haemers bound, and the Lovász number are invariant under Morita equivalence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_18065 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Morita equivalence for quantum graphs Chatzinikolaou, Alexandros Hoefer, Gage Koutsonikos-Kouloumpis, Nikolaos Paraskevas, Ioannis Apollon Operator Algebras Mathematical Physics 16D90, 46L07, 47L25, 81P45 We introduce an operator-algebraic framework for Morita equivalence of quantum graphs based on $Δ$-equivalence of operator systems introduced by Eleftherakis, Kakariadis and Todorov. Adopting the perspective of Weaver, we view quantum graphs as quantum relations, that is, operator systems endowed with a bimodule structure over the commutant of a von Neumann algebra. Within this framework, we show that two irreducibly acting quantum graphs are Morita equivalent if and only if they are both full pullbacks of a common quantum graph. This extends a result of Eleftherakis, Kakariadis and Todorov for graph operator systems to the quantum graph setting. In passing we construct a true-twin reduction analogue for an irreducibly acting quantum graph. We further characterise the case where we have simultaneous TRO-equivalence of the quantum graphs and their associated algebras, thus giving a second, stronger notion of Morita equivalence. In the special case of noncommutative graphs, corresponding to the zero-error quantum communication setting, the two notions coincide and we obtain a characterisation in terms of strong co-homomorphisms of noncommutative graphs. Finally, we show that connectivity, the independence number, Shannon capacity, quantum complexity and subcomplexity, Haemers bound, and the Lovász number are invariant under Morita equivalence. |
| title | Morita equivalence for quantum graphs |
| topic | Operator Algebras Mathematical Physics 16D90, 46L07, 47L25, 81P45 |
| url | https://arxiv.org/abs/2604.18065 |