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Quantum Information Processing

, Volume 11, Issue 5, pp 1219–1249 | Cite as

Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations

  • Andre Ahlbrecht
  • Christopher Cedzich
  • Robert Matjeschk
  • Volkher B. Scholz
  • Albert H. Werner
  • Reinhard F. Werner
Article

Abstract

Quantum walks subject to decoherence generically suffer the loss of their genuine quantum feature, a quadratically faster spreading compared to classical random walks. This intuitive statement has been verified analytically for certain models and is also supported by numerical studies of a variety of examples. In this paper we analyze the long-time behavior of a particular class of decoherent quantum walks, which, to the best of our knowledge, was only studied at the level of numerical simulations before. We consider a local coin operation which is randomly and independently chosen for each time step and each lattice site and prove that, under rather mild conditions, this leads to classical behavior: With the same scaling as needed for a classical diffusion the position distribution converges to a Gaussian, which is independent of the initial state. Our method is based on non-degenerate perturbation theory and yields an explicit expression for the covariance matrix of the asymptotic Gaussian in terms of the randomness parameters.

Keywords

Quantum walk Spatio-temporal coin fluctuation Asymptotic behavior Perturbation theory 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Andre Ahlbrecht
    • 1
  • Christopher Cedzich
    • 1
  • Robert Matjeschk
    • 1
  • Volkher B. Scholz
    • 1
    • 2
  • Albert H. Werner
    • 1
  • Reinhard F. Werner
    • 1
  1. 1.Inst. f. Theoretical PhysicsLeibniz Universität HannoverHannoverGermany
  2. 2.Inst. f. Theoretical PhysicsETH ZurichZurichSwitzerland

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