A discrete invitation to quantum filtering and feedback control

Luc Bouten, Ramon Van Handel, Matthew R. James

Research output: Contribution to journalReview articlepeer-review

76 Scopus citations


The engineering and control of devices at the quantum mechanical level-such as those consisting of small numbers of atoms and photons-is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstr ate estimation and feedback control of quantum mechanical systems in what is essentiall y a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for ph oton counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.

Original languageEnglish (US)
Pages (from-to)239-316
Number of pages78
JournalSIAM Review
Issue number2
StatePublished - 2009
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics


  • Conditionalexpectation
  • Discrete quantum filtering
  • Dynamic programming
  • Quantum feedback control
  • Quantum probability
  • Stochastic Lyapunov functions


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