Pretty good state transfer on circulant graphs.

*(English)*Zbl 1364.05027Summary: Let \(G\) be a graph with adjacency matrix \(A\). The transition matrix of \(G\) relative to \(A\) is defined by \(H(t):=\exp(-itA)\), where \(t\in\mathbb R\). The graph \(G\) is said to admit pretty good state transfer between a pair of vertices \(u\) and \(v\) if there exists a sequence of real numbers \(\{t_k\}\) and a complex number \(\gamma\) of unit modulus such that \(\lim_{k\to\infty} H(t_k)e_u=\gamma e_v.\) We find that the cycle \(C_n\) as well as its complement \(\overline{C}_n\) admit pretty good state transfer if and only if \(n\) is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on \(2^k\) \((k\geq 3)\) vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.

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\textit{H. Pal} and \textit{B. Bhattacharjya}, Electron. J. Comb. 24, No. 2, Research Paper P2.23, 13 p. (2017; Zbl 1364.05027)

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##### References:

[1] | E. Ackelsberg, Z. Brehm, A. Chan, J. Mundinger, and C. Tamon. Laplacian State Transfer in Coronas. Linear Algebra and its Applications, 506:154-167, 2016. · Zbl 1346.05158 |

[2] | E. Ackelsberg, Z. Brehm, A. Chan, J. Mundinger, and C. Tamon. Quantum State Transfer in Coronas.arXiv:1605.05260. · Zbl 1364.05044 |

[3] | T. M. Apostol. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. New York: Springer-Verlag, 1997. |

[4] | M. Baˇsi´c. Characterization of Circulant Networks having Perfect State Transfer. Quantum Information Processing, 12(1):345-364, 2013. |

[5] | C. H. Bennett and G. Brassard. Quantum Cryptography: Public Key Distribution and Coin Tossing. Proc. IEEE Int. Conf. Computers Systems and Signal Processing, Bangalore, India, 175-179, 1984. · Zbl 1306.81030 |

[6] | A. Bernasconi, C. Godsil, and S. Severini. Quantum Networks on Cubelike Graphs. Physical Review A, 78(5):052320, 2008. |

[7] | S. Bose. Quantum Communication through an Unmodulated Spin Chain. Physical Review Letters, 91(20):207901, 2003. |

[8] | R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M. H. Yung, R. Osellame, and A. Peruzzo. Experimental Perfect State Transfer of an Entangled Photonic Qubit. Nature Communications, 7:11339, 2016. |

[9] | W. Cheung and C. Godsil. Perfect State Transfer in Cubelike Graphs. Linear Algebra and its Applications, 435(10):2468-2474, 2011. · Zbl 1222.05150 |

[10] | M. Christandl, N. Datta, T. Dorlas, A Ekert, A. Kay, and A. J. Landahl. Perfect Transfer of Arbitrary States in Quantum Spin Networks. Physical Review A, 71(3):032312, 2005. |

[11] | M. Christandl, N. Datta, A. Ekert, and A. J. Landahl. Perfect State Transfer in Quantum Spin Networks. Physical Review Letters, 92(18):187902, 2004. |

[12] | G. Coutinho and C. Godsil. Perfect State Transfer in Products and Covers of Graphs. Linear and Multilinear Algebra, 64 (2):235-246, 2016. · Zbl 1331.05141 |

[13] | G. Coutinho and C. Godsil. Perfect State Transfer is Poly-time.arXiv:1606.02264. · Zbl 1331.05141 |

[14] | G. Coutinho, C. Godsil, K. Guo, and F. Vanhove. Perfect State Transfer on Distanceregular Graphs and Association Schemes.Linear Algebra and its Applications, 478:108-130, 2015. · Zbl 1312.05147 |

[15] | A. K. Ekert. Quantum Cryptography based on Bell’s Theorem. Physical Review Letters, 67(6):661, 1991. · Zbl 0990.94509 |

[16] | J. P. Escofier. Galois Theory. 1st ed. New York: Springer-Verlag, 2001. · Zbl 0967.12001 |

[17] | X. Fan and C. Godsil. Pretty Good State Transfer on Double Stars. Linear Algebra and its Applications, 438(5):2346-2358, 2013. the electronic journal of combinatorics 24(2) (2017), #P2.2312 · Zbl 1258.05069 |

[18] | C. Godsil. Periodic Graphs. The Electronic Journal of Combinatorics, 18(1): #P23, 2011. |

[19] | C. Godsil. State Transfer on Graphs. Discrete Mathematics, 312(1):129-147, 2012. · Zbl 1232.05123 |

[20] | C. Godsil. When can Perfect State Transfer occur?Electronic Journal of Linear Algebra, 23:877-890, 2012. · Zbl 1253.05093 |

[21] | C. Godsil, S. Kirkland, S. Severini, and J. Smith. Number-theoretic Nature of Communication in Quantum Spin Systems. Physical Review Letters, 109(5): 050502, 2012. |

[22] | S. Kirkland. Sensitivity Analysis of Perfect State Transfer in Quantum Spin Networks. Linear Algebra and its Applications, 472:1-30, 2015. · Zbl 1307.05140 |

[23] | W. Klotz and T. Sander. Integral Cayley Graphs over Abelian Groups. The Electronic Journal of Combinatorics, 17: #R81, 2010. · Zbl 1189.05074 |

[24] | G. Malajovich. An Effective Version of Kronecker’s Theorem on Simultaneous Diophantine Approximation. Instituto de Matem´atica da Universidade Federal do Rio de Janeiro, Brasil, 2001. |

[25] | H. Pal and B. Bhattacharjya. A class of gcd-graphs having Perfect State Transfer. Electronic Notes in Discrete Mathematics, 53:319-329, 2016. · Zbl 1347.05092 |

[26] | H. Pal and B. Bhattacharjya. Perfect State Transfer on gcd-graphs. Linear and Multilinear Algebra,doi:10.1080/03081087.2016.1267105. · Zbl 1387.05070 |

[27] | H. Pal and B. Bhattacharjya. Perfect State Transfer on NEPS of the Path on Three Vertices. Discrete Mathematics, 339(2):831-838, 2016. · Zbl 1327.05325 |

[28] | H. Pal and B. Bhattacharjya. Pretty Good State Transfer on some NEPS. Discrete Mathematics, 340(4):746-752, 2017. · Zbl 1355.05159 |

[29] | W. So. Integral Circulant Graphs. Discrete Mathematics, 306(1):153-158, 2006. · Zbl 1084.05045 |

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