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Some examples of quantum graphs

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Abstract

We summarize different approaches to the theory of quantum graphs and provide several ways to construct concrete examples. First, we classify all undirected quantum graphs on the quantum space \(M_2\). Secondly, we apply the theory of 2-cocycle deformations to Cayley graphs of abelian groups. This defines a twisting procedure that produces a quantum graph, which is quantum isomorphic to the original one. For instance, we define the anticommutative hypercube graphs. Thirdly, we construct an example of a quantum graph, which is not quantum isomorphic to any classical graph.

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Notes

  1. This actually makes it quite easy to classify all such graphs, which we are going to do in Sect. 3.3.

  2. Fans of chemistry might enjoy the following parallel: The finite quantum sets X are molecules, which consist of atoms \(M_{n_i}({\mathbb {C}})\). The atoms \(M_{n_i}\) are defined by their proton number \(n_i\), they have atomic mass \(n_i^2\), and they cannot be decomposed by any chemical processes (taking quantum subsets), but can be split by nuclear fission (taking quantum set quotient).

  3. Such a concrete realization is also known as a nice unitary error basis see [16].

References

  1. Albuquerque, H., Majid, S.: Clifford algebras obtained by twisting of group algebras. J. Pure Appl. Algebra 171(2), 133–148 (2002). https://doi.org/10.1016/S0022-4049(01)00124-4

    Article  MathSciNet  MATH  Google Scholar 

  2. Babai, L.: Spectra of Cayley graphs. J. Comb. Theory Ser. B 27(2), 180–189 (1979). https://doi.org/10.1016/0095-8956(79)90079-0

    Article  MathSciNet  MATH  Google Scholar 

  3. Banica, T.: Théorie des représentations du groupe quantique compact libre \(O(n)\). Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 322, 241–244 (1996)

  4. Banica, T.: Symmetries of a generic coaction. Math. Ann. 314, 763–780 (1999). https://doi.org/10.1007/s002080050315

    Article  MathSciNet  MATH  Google Scholar 

  5. Banica, T.: Quantum groups and Fuss–Catalan algebras. Commun. Math. Phys. 226, 221–232 (2002). https://doi.org/10.1007/s002200200613

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bazlov, Y., Berenstein, A.: Cocycle twists and extensions of braided doubles. In: Berenstein, A., Retakh, V. (eds.) Noncommutative Birational Geometry, Representations and Combinatorics, volume 592 of Contemporary Mathematics, pp. 19–70. American Mathematical Society, Providence (2013). https://doi.org/10.1090/conm/592

    Chapter  MATH  Google Scholar 

  7. Banica, T., Bichon, J., Collins, B.: The hyperoctahedral quantum group. J. Ramanujan Math. Soc. 22, 345–384 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Brannan, M., Chirvasitu, A., Eifler, K., Harris, S., Paulsen, V., Xiaoyu, S., Wasilewski, M.: Bigalois extensions and the graph isomorphism game. Commun. Math. Phys. 375, 1177–1809 (2020). https://doi.org/10.1007/s00220-019-03563-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Brannan, M., Eifler, K., Voigt, C., Weber, M.: Quantum Cuntz–Krieger algebras. Trans. Am. Math. Soc. Ser. B 9, 782–826 (2022). https://doi.org/10.1090/btran/88

  10. Coecke, B., Pavlovic, D., Vicary, J.: A new description of orthogonal bases. Math. Struct. Comput. Sci. 23(3), 555–567 (2013). https://doi.org/10.1017/S0960129512000047

    Article  MathSciNet  MATH  Google Scholar 

  11. Doi, Y.: Braided bialgebras and quadratic bialgebras. Commun. Algebra 21(5), 1731–1749 (1993). https://doi.org/10.1080/00927879308824649

    Article  MathSciNet  MATH  Google Scholar 

  12. De Rijdt, A., Vander Vennet, N.: Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries. Annales de l’Institut Fourier 60(1), 169–216 (2010). https://doi.org/10.5802/aif.2520

    Article  MathSciNet  MATH  Google Scholar 

  13. Gromada, D.: Quantum symmetries of Cayley graphs of abelian groups (2021). arXiv:2106.08787

  14. Gromada, D.: Presentations of projective quantum groups. Comptes Rendus. Mathématique 360, 899–907 (2022). https://doi.org/10.5802/crmath.353

    Article  MathSciNet  MATH  Google Scholar 

  15. Gromada, D., Weber, M.: Generating linear categories of partitions. Kyoto J. Math. 62(4), 865–909 (2022). https://doi.org/10.1215/21562261-2022-0028

    Article  MathSciNet  MATH  Google Scholar 

  16. Klappenecker, A., Rötteler, M.: Unitary error bases: constructions, equivalence, and applications. In: Lu, H.F. (ed.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 139–149. Springer, Berlin (2003). https://doi.org/10.1007/3-540-44828-4_16

    Chapter  MATH  Google Scholar 

  17. Klimyk, A., Schmüdgen, K.: Quantum Groups and Their Representations. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  18. Kodiyalam, V., Sunder, V.S.: Temperley–Lieb and non-crossing partition planar algebras. In: Jain, S.K. (ed.) Noncommutative Rings, Group Rings, Diagram Algebras and their Applications, volume 456 of Contemporary Mathematics, pp. 61–72. American Mathematical Society, Providence (2008). https://doi.org/10.1090/conm/456/08884

    Chapter  Google Scholar 

  19. Lupini, M., Mancinska, L., Roberson, D.E.: Nonlocal games and quantum permutation groups. J. Funct. Anal. 279(5), 108592 (2020). https://doi.org/10.1016/j.jfa.2020.108592

    Article  MathSciNet  MATH  Google Scholar 

  20. Lovász, L.: Spectra of graphs with transitive groups. Period. Math. Hung. 6, 191–195 (1975). https://doi.org/10.1007/BF02018821

    Article  MathSciNet  MATH  Google Scholar 

  21. Malacarne, S.: Woronowicz Tannaka–Krein duality and free orthogonal quantum groups. Math. Scand. 122(1), 151–160 (2018). https://doi.org/10.7146/math.scand.a-97320

    Article  MathSciNet  MATH  Google Scholar 

  22. Matsuda, J.: Classification of quantum graphs on \(M_2\) and their quantum automorphism groups. J. Math. Phys. (2021). https://aip.scitation.org/doi/10.1063/5.0081059

  23. Montgomery, S.: Hopf Algebras and their Actions on Rings. American Mathematical Society, Providence (1993)

    Book  MATH  Google Scholar 

  24. Mancinska, L., Roberson, D.E.: Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs. In: 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pp. 661–672 (2020). https://doi.org/10.1109/FOCS46700.2020.00067

  25. Musto, B., Reutter, D., Verdon, D.: A compositional approach to quantum functions. J. Math. Phys. 59(8), 081706 (2018). https://doi.org/10.1063/1.5020566

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Musto, B., Reutter, D., Verdon, D.: The Morita theory of quantum graph isomorphisms. Commun. Math. Phys. 365, 797–845 (2019). https://doi.org/10.1007/s00220-018-3225-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Schauenburg, P.: Hopf bigalois extensions. Commun. Algebra 24(12), 3797–3825 (1996). https://doi.org/10.1080/00927879608825788

    Article  MathSciNet  MATH  Google Scholar 

  28. Tambara, D.: Representations of tensor categories with fusion rules of self-duality for abelian groups. Isr. J. Math. 118, 29–60 (2000). https://doi.org/10.1007/BF02803515

    Article  MathSciNet  MATH  Google Scholar 

  29. Trautman, A.: Clifford algebras and their representations. In: Françoise, J.-P., Naber, G.L., Tsun, T.S. (eds.) Encyclopedia of Mathematical Physics, pp. 518–530. Academic Press, Oxford (2006). https://doi.org/10.1016/B0-12-512666-2/00016-X

    Chapter  Google Scholar 

  30. Vicary, J.: Categorical formulation of finite-dimensional quantum algebras. Commun. Math. Phys. 304, 765–796 (2011). https://doi.org/10.1007/s00220-010-1138-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Wang, S.: Free products of compact quantum groups. Commun. Math. Phys. 167(3), 671–692 (1995). https://doi.org/10.1007/BF02101540

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Wang, S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195(1), 195–211 (1998). https://doi.org/10.1007/s002200050385

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Weaver, N.: Quantum relations. Mem. Am Math. Soc. 215(1010), 81–140 (2012). https://doi.org/10.1090/S0065-9266-2011-00637-4

    Article  MathSciNet  MATH  Google Scholar 

  34. Weaver, N.: Quantum graphs as quantum relations. J. Geom. Anal. 31, 9090–9112 (2021). https://doi.org/10.1007/s12220-020-00578-w

    Article  MathSciNet  MATH  Google Scholar 

  35. Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111(4), 613–665 (1987). https://doi.org/10.1007/BF01219077

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. Twisted \(SU(N)\) groups. Inventiones mathematicae 93(1), 35–76 (1988). https://doi.org/10.1007/BF01393687

    Article  ADS  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27, Praha 6, Czech Republic

    Daniel Gromada

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I would like to thank Simon Schmidt, Christian Voigt, and Moritz Weber for valuable discussions about quantum graphs. I thank to Julien Bichon for discussing monoidal equivalences related to \(SO_n\). This work was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778.

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Gromada, D. Some examples of quantum graphs. Lett Math Phys 112, 122 (2022). https://doi.org/10.1007/s11005-022-01603-5

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