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A Discrete Invitation to Quantum Filtering and Feedback Control

Bouten, Luc and van Handel, Ramon and James, Matthew R. (2009) A Discrete Invitation to Quantum Filtering and Feedback Control. SIAM Review, 51 (2). pp. 239-316. ISSN 0036-1445.

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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 demonstrate estimation and feedback control of quantum mechanical systems in what is essentially 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 photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.

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Additional Information:© 2009 Society for Industrial and Applied Mathematics. Received October 4, 2006; accepted November 28, 2007; published May 4, 2009. Dedicated to Slava Belavkin in the year of his 60th birthday. The work of these authors was partially supported by the Army Research Office under grants DAAD19-03-1-0073 and W911NF-06-1-0378, and by the National Science Foundation under grants CCF-0323542 and PHY-0456720. The work of this author was supported by the Australian Research Council. The authors would like to thank John Stockton for providing a figure. L.B. thanks Hans Maassen for introducing him to discrete systems.
Funding AgencyGrant Number
Army Research OfficeDAAD19-03-1-0073
Army Research OfficeW911NF-06-1-0378
Australian Research CouncilUNSPECIFIED
Subject Keywords:discrete quantum filtering; quantum feedback control; quantum probability; conditional expectation; dynamic programming; stochastic Lyapunov functions. AMS Subject Classifications: 93E11, 93E15, 93E20, 81P15, 81S25, 34F05
Issue or Number:2
Record Number:CaltechAUTHORS:20090811-111434670
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14953
Deposited By: Jason Perez
Deposited On:11 Aug 2009 22:45
Last Modified:03 Oct 2019 00:53

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