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Recurrence for Discrete Time Unitary Evolutions

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Abstract

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \({\phi}\). In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \({\phi}\). We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    F. A. Grünbaum

  2. Departamento de Matemática Aplicada and IUMA, Universidad de Zaragoza, María de Luna 3, 50018, Zaragoza, Spain

    L. Velázquez

  3. Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167, Hannover, Germany

    A. H. Werner & R. F. Werner

Authors
  1. F. A. Grünbaum
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  2. L. Velázquez
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  3. A. H. Werner
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  4. R. F. Werner
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Correspondence to A. H. Werner.

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Communicated by S. Zelditch

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Grünbaum, F.A., Velázquez, L., Werner, A.H. et al. Recurrence for Discrete Time Unitary Evolutions. Commun. Math. Phys. 320, 543–569 (2013). https://doi.org/10.1007/s00220-012-1645-2

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  • DOI: https://doi.org/10.1007/s00220-012-1645-2

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