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Large Time Behavior and Convergence Rate for Quantum Filters Under Standard Non Demolition Conditions

Communications in Mathematical Physics volume 331, pages 703–723 (2014)Cite this article

Abstract

A quantum system \({\mathcal S}\) undergoing continuous time measurement is usually described by a jump-diffusion stochastic differential equation. Such an equation is called a quantum filtering equation (or quantum stochastic master equation) and its solution is called a quantum filter (or quantum trajectory). This solution describes the evolution of the state of \({\mathcal S}\). In the context of quantum non demolition measurement, we investigate the large time behavior of this solution. It is rigorously shown that, for large time, this solution behaves as if a direct Von Neumann measurement has been performed at time 0. In particular the solution converges to a random pure state which can be directly linked to the wave packet reduction postulate. Using the theory of Girsanov transformation, we obtain the explicit rate of convergence towards this random state. The problem of state estimation (used in experiment) is also investigated.

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Authors and Affiliations

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

    Tristan Benoist

  2. Institut de Mathématiques de Toulouse, Equipe de Statistique et de Probabilité, Université Paul Sabatier, 31062, Toulouse Cedex 9, France

    Clément Pellegrini

Authors
  1. Tristan Benoist
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  2. Clément Pellegrini
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Correspondence to Clément Pellegrini.

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Communicated by H. Spohn

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Benoist, T., Pellegrini, C. Large Time Behavior and Convergence Rate for Quantum Filters Under Standard Non Demolition Conditions. Commun. Math. Phys. 331, 703–723 (2014). https://doi.org/10.1007/s00220-014-2029-6

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  • DOI: https://doi.org/10.1007/s00220-014-2029-6

Keywords

  • Large Time Behavior
  • Quantum Trajectory
  • True Quantum
  • Martingale Property
  • Wave Function Collapse