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A graph with fractional revival. (English) Zbl 1387.81221
Summary: An example of a graph that admits balanced fractional revival between antipodes is presented. It is obtained by establishing the correspondence between the quantum walk on a hypercube where the opposite vertices across the diagonals of each face are connected and, the coherent transport of single excitations in the extension of the Krawtchouk spin chain with next-to-nearest neighbour interactions.

MSC:
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
82C22 Interacting particle systems in time-dependent statistical mechanics
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[1] Robinett, R. W., Quantum wave packet revivals, Phys. Rep., 392, 1-119, (2004)
[2] Banchi, L.; Compagno, E.; Bose, S., Perfect wave-packet splitting and reconstruction in a one-dimensional lattice, Phys. Rev. A, 91, (2015)
[3] Genest, V.; Vinet, L.; Zhedanov, A., Quantum spin chains with fractional revival, Ann. Phys., 371, 348-367, (2016) · Zbl 1380.82030
[4] Bose, S., Quantum communication through spin chain dynamics: an introductory review, Contemp. Phys., 48, 13-30, (2007)
[5] Kay, A., A review of perfect state transfer and its applications as a constructive tool, Int. J. Quantum Inf., 8, 641-676, (2010) · Zbl 1194.81046
[6] Nikolopoulos, G. M.; Jex, I., Quantum state transfer and network engineering, (2014), Springer · Zbl 1303.81038
[7] Perez-Leija, A.; Keil, R.; Kay, A.; Moya-Cessa, H.; Nolte, S.; Kwek, L.-C.; Rodriguez-Lara, B. M.; Szameit, A.; Christodoulides, D. N., Coherent transport in photonic lattices, Phys. Rev. A, 87, (2013)
[8] Chapman, R. J.; Santandrea, M.; Huang, Z.; Corrielli, G.; Crespi, A.; Yung, M.-H.; Osellame, R.; Peruzzo, A., Experimental perfect state transfer of an entangled photonic qubit, Nat. Commun., 7, (2016)
[9] Christandl, M.; Datta, N.; Dorlas, T. C.; Ekert, A.; Kay, A.; Landahl, A. J., Perfect transfer of arbitrary states in quantum spin networks, Phys. Rev. A, 71, (2005)
[10] Albanese, C.; Christandl, M.; Datta, N.; Ekert, A., Mirror inversion of quantum states in linear registers, Phys. Rev. Lett., 93, (2004)
[11] Godsil, C., State transfer on graphs, Discrete Math., 312, 123-147, (2012) · Zbl 1232.05123
[12] Kendon, V.; Tamon, C., Perfect state transfer in quantum walks on graphs, J. Comput. Theor. Nanosci., 8, 422-433, (2011)
[13] Childs, A.; Farhi, E.; Gutmann, S., An example of the difference between quantum and classical random walks, Quantum Inf. Process., 1, 35-43, (2002) · Zbl 1329.82006
[14] Christandl, M.; Vinet, L.; Zhedanov, A., Analytic next-to-nearest neighbor XX models with perfect state transfer and fractional revival, Phys. Rev. A, 96, 3, (2017)
[15] Vinet, L.; Zhedanov, A., How to construct spin chains with perfect state transfer, Phys. Rev. A, 85, (2012)
[16] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-regular graphs, (1989), Springer · Zbl 0747.05073
[17] Bannai, E.; Ito, T., Algebraic combinatorics, (1984), Benjamin/Cummings · Zbl 0555.05019
[18] Bannai, E., Orthogonal polynomials in coding theory and algebraic combinatorics, (Nevai, P., Orthogonal Polynomials: Theory and Practice, NATO ASI Series, vol. 294, (1990), Springer), 25-53
[19] Stanton, D., Orthogonal polynomials and combinatorics, (Boustoz, J.; Ismail, M. E.H.; Suslov, S., Special Functions 2000: Current Perspective and Future Directions, NATO Science Series, vol. 30, (2001), Springer), 389-409 · Zbl 0993.05150
[20] Grünbaum, F. A.; Vinet, L.; Zhedanov, A., Birth and death processes and quantum spin chains, J. Math. Phys., 54, (2013) · Zbl 1285.82028
[21] Grünbaum, F. A., Block tridiagonal matrices and a beefed-up version of the Ehrenfest urn model, (Adamyan, V. M.; etal., Modern Analysis and Applications, Operator Theory: Advances and Applications, vol. 190, (2009), Springer), 267-277 · Zbl 1178.33008
[22] A. Chan, G. Coutinho, C. Tamon, L. Vinet, H. Zhan, in preparation.
[23] Štefaňák, M.; Jex, I.; Kiss, T., Recurrence and Pólya number of quantum walks, Phys. Rev. Lett., 100, (2008)
[24] Štefaňák, M.; Kiss, T.; Jex, I., Recurrence properties of unbiased coined quantum walks on infinite d-dimensional lattices, Phys. Rev. A, 78, (2008)
[25] Chandrashekar, C. M., Fractional recurrence in discrete-time quantum walk, Cent. Eur. J. Phys., 8, 6, 979-988, (2010)
[26] Grünbaum, F. A.; Velázquez, L.; Werner, A. H.; Werner, R. F., Recurrence for discrete time unitary evolutions, Commun. Math. Phys., 320, 543-569, (2013) · Zbl 1276.81087
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