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Repeated Quantum Non-Demolition Measurements: Convergence and Continuous Time Limit

Annales Henri Poincaré volume 14, pages 639–679 (2013)Cite this article

Abstract

We analyze general enough models of repeated indirect measurements in which a quantum system interacts repeatedly with randomly chosen probes on which von Neumann direct measurements are performed. We prove, under suitable hypotheses, that the system state probability distribution converges after a large number of repeated indirect measurements, in a way compatible with quantum wave function collapse. We extend this result to mixed states and we prove similar results for the system density matrix. We show that the convergence is exponential with a rate given by some relevant mean relative entropies. We also prove that, under appropriate rescaling of the system and probe interactions, the state probability distribution and the system density matrix are solutions of stochastic differential equations modeling continuous-time quantum measurements. We analyze the large time behavior of these continuous time processes and prove convergence.

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Authors and Affiliations

  1. Institut de Physique Théorique, CEA/DSM/IPhT, Unité de recherche associée au CNRS, Point Courrier 136, CEA-Saclay, 91191, Gif sur Yvette Cedex, France

    Michel Bauer

  2. Laboratoire de Physique Théorique de l’ENS, CNRS & École Normale Supérieure de Paris, 24, rue Lhomond, 75005, Paris, France

    Tristan Benoist & Denis Bernard

Authors
  1. Michel Bauer
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  2. Tristan Benoist
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  3. Denis Bernard
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Corresponding author

Correspondence to Denis Bernard.

Additional information

Communicated by Krzysztof Gawedzki.

D. Bernard: Member of CNRS.

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Bauer, M., Benoist, T. & Bernard, D. Repeated Quantum Non-Demolition Measurements: Convergence and Continuous Time Limit. Ann. Henri Poincaré 14, 639–679 (2013). https://doi.org/10.1007/s00023-012-0204-x

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  • DOI: https://doi.org/10.1007/s00023-012-0204-x

Keywords

  • Radon
  • Density Matrix
  • Pointer State
  • Recursion Relation
  • Relative Entropy