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Pretty good state transfer in qubit chains – the Heisenberg Hamiltonian. (English) Zbl 1359.81047
Summary: Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with \(n\) qubits, there is pretty good state transfer between the nodes at the \(j\)th and \((n - j + 1)\)th positions if \(n\) is a power of 2. Moreover, this condition is also necessary for \(j = 1\). We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory.{
©2017 American Institute of Physics}

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
68M10 Network design and communication in computer systems
60G50 Sums of independent random variables; random walks
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
11K60 Diophantine approximation in probabilistic number theory
Full Text: DOI arXiv
[1] Ladd, T. D.; Jelezko, F.; Laflamme, R.; Nakamura, Y.; Monroe, C.; O’Brien, J. L., Quantum computers, Nature, 464, 7285, 45-53, (2010)
[2] Bose, S., Quantum communication through an unmodulated spin chain, Phys. Rev. Lett., 91, 20, 207901, (2003)
[3] Nikolopoulos, G. M.; Jex, I., Quantum State Transfer and Network Engineering, (2014), Springer · Zbl 1303.81038
[4] Kay, A., Perfect, efficient, state transfer and its application as a constructive tool, Int. J. Quantum Inf. Sci., 8, 4, 641-676, (2010) · Zbl 1194.81046
[5] Godsil, C. D., When can perfect state transfer occur?, Electron. J. Linear Algebra, 23, 877-890, (2012) · Zbl 1253.05093
[6] Bose, S.; Casaccino, A.; Mancini, S.; Severini, S., Communication in XYZ all-to-all quantum networks with a missing link, Int. J. Quantum Inf., 7, 4, 713-723, (2009) · Zbl 1172.81004
[7] Burgarth, D.; Bose, S., Conclusive and arbitrarily perfect quantum-state transfer using parallel spin-chain channels, Phys. Rev. A, 71, 5, 052315, (2005)
[8] Vinet, L.; Zhedanov, A., Almost perfect state transfer in quantum spin chains, Phys. Rev. A, 86, 5, 052319, (2012)
[9] Godsil, C. D.; Kirkland, S.; Severini, S.; Smith, J., Number-theoretic nature of communication in quantum spin systems, Phys. Rev. Lett., 109, 5, 050502, (2012)
[10] Banchi, L., Ballistic quantum state transfer in spin chains: General theory for quasi-free models and arbitrary initial states, Eur. Phys. J. Plus, 128, 11, 1-18, (2013)
[11] Apollaro, T. J. G.; Banchi, L.; Cuccoli, A.; Vaia, R.; Verrucchi, P., 99%-fidelity ballistic quantum-state transfer through long uniform channels, Phys. Rev. A, 85, 5, 052319, (2012)
[12] Lorenzo, S.; Apollaro, T. J. G.; Sindona, A.; Plastina, F., Quantum-state transfer via resonant tunneling through local-field-induced barriers, Phys. Rev. A, 87, 4, 042313, (2013)
[13] Apollaro, T. J. G.; Lorenzo, S.; Sindona, A.; Paganelli, S.; Giorgi, G. L.; Plastina, F., Many-qubit quantum state transfer via spin chains, Physica Scripta, 014036, (2015)
[14] Omar, Y.; Sousa, R., Pretty good state transfer of entangled states through quantum spin chains, New J. Phys., 16, 12, 123003, (2014)
[15] Burgarth, D., Quantum state transfer with spin chains, (2007), University of London
[16] Campos Venuti, L.; Degli Esposti Boschi, C.; Roncaglia, M., Qubit teleportation and transfer across antiferromagnetic spin chains, Phys. Rev. Lett., 99, 060401, (2007)
[17] Campos Venuti, L.; Giampaolo, S. M.; Illuminati, F.; Zanardi, P., Long-distance entanglement and quantum teleportation in XX spin chains, Phys. Rev. A, 76, 052328, (2007)
[18] Mikeska, H.-J.; Kolezhuk, A. K., One-dimensional magnetism, Quantum Magnetism, 1-83, (2004), Springer: Springer, Berlin, Heidelberg
[19] Albanese, C.; Christandl, M.; Datta, N.; Ekert, A., Mirror inversion of quantum states in linear registers, Phys. Rev. Lett., 93, 23, 230502, (2004)
[20] Hanson, R.; Kouwenhoven, L. P.; Petta, J. R.; Tarucha, S.; Vandersypen, L. M. K., Spins in few-electron quantum dots, Rev. Mod. Phys., 79, 4, 1217, (2007)
[21] Kane, B. E., A silicon-based nuclear spin quantum computer, Nature, 393, 6681, 133-137, (1998)
[22] Fukuhara, T.; Kantian, A.; Endres, M.; Cheneau, M.; Schauß, P.; Hild, S.; Bellem, D.; Schollwöck, U.; Giamarchi, T.; Gross, C., Quantum dynamics of a mobile spin impurity, Nat. Phys., 9, 4, 235-241, (2013)
[23] Coutinho, G., Quantum state transfer in graphs, (2014), University of Waterloo
[24] Kay, A., Basics of perfect communication through quantum networks, Phys. Rev. A, 84, 2, 022337, (2011)
[25] Levitan, B. M.; Zhikov, V. V., Almost Periodic Functions and Differential Equations, (1982), CUP Archive · Zbl 0499.43005
[26] Brouwer, A. E.; Haemers, W. H., Spectra of Graphs, (2012), Universitext: Universitext, Springer, New York · Zbl 1231.05001
[27] Chr Hemmer, P.; C Maximon, L.; Wergeland, H., Recurrence time of a dynamical system, Phys. Rev., 111, 689-694, (1957) · Zbl 0083.21701
[28] Coutinho, G.; Guo, K.; van Bommel, C., Pretty good state transfer between internal nodes of paths
[29] van Bommel, C., A Complete characterization of pretty good state transfer on paths
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