On quantum symmetries of graphs
DOI:
https://doi.org/10.3842/umzh.v78i5-6.9774Keywords:
Quantum graph, Quantum automorphism, Nonlocal gameAbstract
UDC 519.17, 517.98, 519.8
Let $G$ be a simple finite graph and let $\mathcal U_G$ be the corresponding quantum graph. We study the game algebra $C(\mathrm{Qut}(\mathcal U_G))$ of the quantum automorphisms of $\mathcal U_G.$ It is shown that, for the complete graph $K_n,$ the algebra $C(\mathrm{Qut}(\mathcal U_{K_n}))$ is not commutative even for all $n\geq 3,$ unlike $C(\mathrm{Qut}(K_n))=C(S_n^+).$ Moreover, we prove that, for any graph $G$ with $|V(G)|\geq 3,$ the quantum graph $\mathcal U_G$ admits nonlocal symmetry, which means that there exists a perfect quantum no-signaling correlation for the quantum automorphism game for $\mathcal U_G,$ which is not local.
References
1. A. Atserias, L. Mančinska, D. E. Roberson, et al., Quantum and non-signalling graph isomorphisms, J. Combin. Theory, Ser. B, 136, 289–328 (2019).
2. T. Banica, Le groupe quantique compact libre ${U}(n)$, Comm. Math. Phys., 190, № 1, 143–172 (1997).
3. T. Banica, Symmetries of a generic coaction, Math. Ann., 314, № 4, 763–780 (1999).
4. T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal., 224, 243–280 (2005).
5. T. Banica, J. Bichon, B. Collins, Quantum permutation groups: a survey, Noncommutative harmonic analysis with applications to probability, Banach Center Publ., vol. 78, Polish Academy of Sciences, Institute of Mathematics, Warsaw (2007), pp. 13–34.
6. M. Brannan, S. J. Harris, I. G. Todorov, L. Turowska, Quantum no-signalling bicorrelations, Adv. Math., 449, Paper № 109732 (2024).
7. R. Duan, A. Winter, No-signalling assisted zero-error capacity of quantum channels and an information theoretic interpretation of the Lovász number, IEEE Trans. Inform. Theory, 62, № 2, 891–914 (2016).
8. M. B. Fulton, The quantum automorphism group and undirected trees, Ph.D. Thesis, Virginia Polytechnic Institute and State University (2006).
9. M. Lupini, L. Mančinska, D. E. Roberson, Nonlocal games and quantum permutation groups, J. Funct. Anal., 279, № 5, 108592 (2020).
10. L. Mančinska, V. I. Paulsen, I. G. Todorov, A. Winter, Products of synchronous games, Studia Math., 272, № 3, 299–317 (2023).
11. V. I. Paulsen, M. Rahaman, Bisynchronous games and factorizable maps, Ann. Henri Poincaré, 22, № 2, 593–614 (2021).
12. V. I. Paulsen, S. Severini, D. Stahlke, I. G. Todorov, A. Winter, Estimating quantum chromatic numbers, J. Funct. Anal., 270, № 6, 2188–2222 (2016).
13. D. E. Roberson, S. Schmidt, Quantum symmetry vs nonlocal symmetry}; arXiv:2012.13328v2 [math.QA] (2021).
14. I. G. Todorov, L. Turowska, Quantum no-signalling correlations and non-local games, Comm. Math. Phys., 405, № 6, Paper № 141 (2024).
15. C. Voigt, {On the structure of quantum automorphism groups}, J. Reine Angew. Math., 732, 255–273 (2017).
16. S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys., 195, 195–211 (1998).
17. M. Weber, Quantum permutation matrices, Complex Anal. Oper. Theory, 17, № 3, Paper № 37 (2023).
