2006/06/30 by Luc Bouten, Ramon van Handel, Matthew R. James · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Control theory (sociology) #Discretization #Field (mathematics) #Function (biology) #Hilbert space #Lyapunov function #Master equation #Quantization (signal processing) #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #math-ph #math.MP #math.OC #math.PR #msc:34F05 #msc:81P15 #msc:81S25 #msc:93E11 #msc:93E15 #msc:93E20 #quant-ph #stochastic dynamics and bifurcation
paper · pdf · doi:10.1137/060671504
published as SIAM Review 51, pp. 239-316, 2009 · 76 pages, 12 figures. A PDF file with high resolution figures can be found at http://minty.caltech.edu/papers.php
arxiv created 2006/12/05 · openalex publication_date 2009/05/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.