Abstract
We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.
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Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425. IEEE (2004). https://doi.org/10.1109/LICS.2004.1319636. arXiv:quant-ph/0402130
Arkhipov, A.: Extending and characterizing quantum magic games (2012). arXiv:1209.3819
Atserias, Albert, Mančinska, L., Roberson, D.E., Šámal, R.,Severini, S., Varvitsiotis, A.: Quantum and non-signalling graph isomorphisms (2016). arXiv:1611.09837
Baez, J.: Lecture notes on quantum gravity. Fall (2004) Week 9. Available online at http://math.ucr.edu/home/baez/qg-fall2004/f04week09.pdf
Bahturin Y.A., Sehgal S.K., Zaicev M.V.: Group gradings on associative algebras. J. Algebra 241(2), 677–698 (2001). https://doi.org/10.1006/jabr.2000.8643
Banica T.: Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224(2), 243–280 (2005). https://doi.org/10.1016/j.jfa.2004.11.002 arXiv:math/0311402
Banica T.: Quantum automorphism groups of small metric spaces. Pac. J. Math. 219(1), 27–51 (2005) arXiv:math/0304025
Banica, T.: Higher transitive quantum groups: theory and models. arXiv:1712.04067 (2017)
Banica T., Bichon J.: Quantum automorphism groups of vertex-transitive graphs of order \({\leq}\) 11. J. Algebra. Comb. 26(1), 83–105 (2007). https://doi.org/10.1007/s10801-006-0049-9 arXiv:math/0601758
Banica T., Bichon J.: Quantum groups acting on 4 points. J. für die reine und angewandte Mathematik (Crelles J.) 626, 75–114 (2009). https://doi.org/10.1515/crelle.2009.003 arXiv:math/0703118
Banica T., Bichon J., Chenevier G.: Graphs having no quantum symmetry. Annales de l’institut Fourier 57(3), 955–971 (2007). https://doi.org/10.5802/aif.2282 arXiv:math/0605257
Banica T., Collins B.: Integration over the Pauli quantum group. J. Geom. Phys. 58(8), 942–961 (2008). https://doi.org/10.1016/j.geomphys.2008.03.002 arXiv:math/0610041
Ben David N., Ginosar Y., Meir E.: Isotropy in group cohomology. Bull. Lond. Math. Soc. 46(3), 587–599 (2014). https://doi.org/10.1112/blms/bdu018 arXiv:1309.2438
Bichon J.: Quantum automorphism groups of finite graphs. Proc. Am. Math. Soc. 131(3), 665–673 (2003). https://doi.org/10.1090/S0002-9939-02-06798-9 arXiv:math/9902029
Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.-H.: Tensor Categories and Endomorphisms of von Neumann Algebras. Springer, New York (2015). https://doi.org/10.1007/978-3-319-14301-9. arXiv:1407.4793
Borceux, F.: Handbook of Categorical Algebra, volume 1 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1994). https://doi.org/10.1017/CBO9780511525858
Brassard G.: Quantum communication complexity. Found. Phys. 33(11), 1593–1616 (2003). https://doi.org/10.1023/A:1026009100467 arXiv:math/quant-ph/0101005
Brassard G., Broadbent A., Tapp A.: Quantum pseudo-telepathy. Found. Phys. 35(11), 1877–1907 (2005). https://doi.org/10.1007/s10701-005-7353-4 arXiv:quant-ph/0407221
Carqueville N., Runkel I.: Orbifold completion of defect bicategories. Quantum Topol. 7(2), 203–279 (2016). https://doi.org/10.4171/qt/76 arXiv:1210.6363
Cleve, R., Hoyer, P., Toner, B., Watrous, J.: Consequences and limits of nonlocal strategies. In: Computational Complexity. Proceedings of 19th IEEE Annual Conference on IEEE, pp. 236–249. IEEE, (2004). https://doi.org/10.1109/CCC.2004.1313847. arXiv:quant-ph/0404076
Coecke B.: Quantum picturalism. Contemp. Phys. 51(1), 59–83 (2010). https://doi.org/10.1080/00107510903257624 arXiv:0908.1787
Coecke, B., Pavlović, D., Vicary, J.: A new description of orthogonal bases. Math. Struct. Comput. Sci. (2009). https://doi.org/10.1017/s0960129512000047. arXiv:0810.0812
Duan R., Severini S., Winter A.: Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans. Inf. Theory 59(2), 1164–1174 (2013). https://doi.org/10.1109/TIT.2012.2221677 arXiv:1002.2514
Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories. American Mathematical Society (2015). https://doi.org/10.1090/surv/205. http://www-math.mit.edu/etingof/~egnobookfinal.pdf
Heunen C., Karvonen M.: Monads on dagger categories. Theory Appl. Categ. 31(35), 1016–1043 (2016) arXiv:1602.04324
Heunen C., Vicary J., Wester L.: Mixed quantum states in higher categories. Electron. Proc. Theor. Comput. Sci. 172, 304–315 (2014). https://doi.org/10.4204/eptcs.172.22 arXiv:1405.1463
Jones Vaughan, F.R.: Planar algebras, I. arXiv:math/9909027 (1999)
Kelly, G., Laplaza, M.: Coherence for compact closed categories. J. Pure Appl. Algebra 19(Supplement C):193–213 (1980). https://doi.org/10.1016/0022-4049(80)90101-2
Kitaev A., Kong L.: Models for gapped boundaries and domain walls. Commun. Math. Phys. 313(2), 351–373 (2012). https://doi.org/10.1007/s00220-012-1500-5 arXiv:1104.5047
Klappenecker, A., Rötteler, M.: Unitary error bases: constructions, equivalence, and applications. In: Marc Fossorier, Tom Høholdt, Alain Poli (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 139–149. Springer, Berlin (2003). https://doi.org/10.1007/3-540-44828-4_16
Kong L., Runkel I.: Morita classes of algebras in modular tensor categories. Adv. Math. 219(5), 1548–1576 (2008). https://doi.org/10.1016/j.aim.2008.07.004 arXiv:0708.1897
Kuperberg, G., Weaver, N.: (2012) A von Neumann algebra approach to quantum metrics/quantum relations, volume 215. American Mathematical Society. https://doi.org/10.1090/S0065-9266-2011-00637-4. arXiv:1005.0353
Lack, S., Street, R.: The formal theory of monads II. J. Pure Appl. Algebra 175(1–3):243–265 (2002). http://maths.mq.edu.au/~slack/papers/ftm2.html. https://doi.org/10.1016/s0022-4049(02)00137-8
Lupini, M., Maninska,L., Roberson, D.E.: Nonlocal games and quantum permutation groups (2017). arXiv:1712.01820
Marsden, D.: Category theory using string diagrams (2014). arXiv:1401.7220
Mermin N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65(27), 3373 (1990)
Musto, B., Reutter,D., Verdon, D.: A compositional approach to quantum functions (2017). To appear. arXiv:1711.07945
Ocneanu, A.: Quantized groups, string algebras, and Galois theory for algebras. In: Evans, D.E., Takesaki, M. (eds.) Operator Algebras and Applications, pp. 119–172. Cambridge University Press (CUP), Cambridge (1989). https://doi.org/10.1017/cbo9780511662287.008
Ostrik V.: Module categories, weak Hopf algebras and modular invariants. Transf. Groups 8(2), 177–206 (2003). https://doi.org/10.1007/s00031-003-0515-6 arXiv:math/0111139
Ostrik V.: Module categories over the Drinfeld double of a finite group. IMRN 27, 1507–1520 (2003). https://doi.org/10.1155/S1073792803205079 arXiv:math/0202130
Reutter, D., Vicary, J.: Biunitary constructions in quantum information (2016). arXiv:1609.07775
Runkel, I., Fjelstad, J., Fuchs, J., Schweigert, C.: Topological and conformal field theory as Frobenius algebras (2007). https://doi.org/10.1090/conm/431/08275. arXiv:math/0512076
Schmidt, S.: The Petersen graph has no quantum symmetry (2018). arXiv:1801.02942
Selinger, P.: A survey of graphical languages for monoidal categories. In: New Structures for Physics, Lecture Notes in Physics, pp. 289–355. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12821-9_4. arXiv:0908.3347
Stahlke D.: Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans. Inf. Theory 62(1), 554–577 (2016). https://doi.org/10.1109/TIT.2015.2496377 arXiv:1405.5254
Vicary J.: Categorical formulation of finite-dimensional quantum algebras. Commun. Math. Phys. 304(3), 765–796 (2010). https://doi.org/10.1007/s00220-010-1138-0 arXiv:0805.0432
Vicary, Jamie.: Higher quantum theory. (2012). arXiv:1207.4563
Wang S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195(1), 195–211 (1998). https://doi.org/10.1007/s002200050385 arXiv:math/9807091
Weaver, N.: Quantum relations (2010). arXiv:1005.0354
Weaver, N.: Quantum graphs as quantum relations (2015). arXiv:1506.03892
Woronowicz, S.L.: Compact quantum groups. In: Symétries quantiques (Les Houches, 1995), pp. 845–884. North-Holland, Amsterdam (1998)
Acknowledgments
We are grateful to Jamie Vicary for many useful discussions and to David Roberson for sending us an early draft of [34]. We thank an anonymous referee for helpful comments on an earlier draft of this work. This work was supported by the Engineering and Physical Sciences Research Council and the Clarendon Trust.
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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
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DOI: https://doi.org/10.1007/s00220-018-3225-6