Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> A quantum stochastic model for the Markovian dynamics of an open system under the nondemolition unsharp observation which is continuous in time, is given. A stochastic equation for the posterior evolution of a quantum continuously observed system is derived and the spontaneous collapse (stochastically continuous reduction of the wave packet) is described. The quantum Langevin evolution equation is solved for the case of a quasi-free Hamiltonian in the initial CCR algebra with a linear output channel, and the posterior dynamics corresponding to an initial Gaussian state is found. It is shown for an example of the posterior dynamics of a quantum oscillator that any mixed state under a complete nondemolition measurement collapses exponentially to a pure Gaussian one. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> The history of the theory of quantum nondemolition (QND) measurements from the 1920s until today is reviewed. The definition and main principles of QND measurements are outlined. Achievements in the experimental realization of QND measurements and several new promising schemes of QND measurements are described. A list of the most important problems (from the authors{close_quote} point of view) in the area of QND measurements is presented. The problem of measurement of a quantum oscillator phase is considered. A new method of phase measurement is proposed. Examples of possible solutions of fundamental physical problems using QND methods are given. {copyright} {ital 1996 The American Physical Society.} <s> BIB002 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> I propose a quantum-trajectories approach to parametric identification of the effective Hamiltonian for a Markovian open quantum system, and discuss an application motivated by recent experiments in cavity quantum electrodynamics. This example illustrates a strategy for quantum parameter estimation that efficiently utilizes the information carried by correlations between measurements distributed in time. <s> BIB003 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> A Markovian model for a quantum automata, i.e. an open quantum dynamical discrete-time system with input and output channels and a feedback, is described. A dynamical theory of quantum discrete-time adaptive measurements and multi-stage quantum statistical decisions is developed and applied to the optimal feedback control problem for the quantum dynamical objects. Quantum analogies of Stratonovich non-stationary filtering, and Bellman quantum dynamical programming in the discrete time are derived. ::: A Gaussian Langevin model of the quantum one-dimensional linear Markovian dynamical system matched with a quantum linear transmission line as an input-output quantum noisy channel is studied. The optimal quantum multi-stage decision rule consisting of a classical linear optimal control strategy and the quantum optimal filtering of the noise is found. The latter contains the optimal quantum coherent measurement on the output of the line and the recursive processing by the Kalman filter. ::: A time-continuous limit of the above model is considered, and the quantum nondemolition measurement, time-continuous filtering and the optimal dynamical programming are found in this limit. All the results are illustrated by an example of the optimal control problem for a quantum open oscillator matched to a quantum transmission line. <s> BIB004 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> An inexpensive faucet aerator is provided by three molded parts and a thin metal cup that holds two of the molded parts in assembled relationship. The first molded part is an elongated annulus provided with upstream and downstream recesses separated by an inner ring that helps to break up the liquid flow and serves as an abutment to support a second molded, jet-forming, part in the upstream recess and is arranged to be engaged by the third, molded, part located in the downstream recess and aiding in defining air intake means to the aerator. <s> BIB005 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> Following evolution of an open quantum system one requires full knowledge of its dynamics. In this paper we consider open quantum systems for which the Hamiltonian is ``uncertain.'' In particular, we treat in detail a simple system similar to that considered by Mabuchi [Quant. Semiclass. Opt. 8, 1103 (1996)]: a radiatively damped atom driven by an unknown Rabi frequency $\ensuremath{\Omega}$ (as would occur for an atom at an unknown point in a standing light wave). By measuring the environment of the system, knowledge about the system state, and about the uncertain dynamical parameter, can be acquired. We find that these two sorts of knowledge acquisition (quantified by the posterior distribution for $\ensuremath{\Omega},$ and the conditional purity of the system, respectively) are quite distinct processes, which are not strongly correlated. Also, the quality and quantity of knowledge gain depend strongly on the type of monitoring scheme. We compare five different detection schemes (direct, adaptive, homodyne of the x quadrature, homodyne of the y quadrature, and heterodyne) using four different measures of the knowledge gain (Shannon information about $\ensuremath{\Omega},$ variance in $\ensuremath{\Omega},$ long-time system purity, and short-time system purity). <s> BIB006 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> A closed loop learning control concept is introduced for teaching lasers to manipulate quantum systems for the purpose of optimally identifying Hamiltonian information. The closed loop optimal identification algorithm operates by revealing the distribution of Hamiltonians consistent with the data. The control laser is guided to perform additional experiments, based on minimizing the dispersion of the distribution. Operation of such an apparatus is simulated for two model finite dimensional quantum systems. <s> BIB007 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> The importance of feedback control is being increasingly appreciated in quantum physics and applications. This paper describes the use of optimal control methods in the design of quantum feedback control systems, and in particular the paper formulates and solves a risk-sensitive optimal control problem. The resulting risk-sensitive optimal control is given in terms of an unnormalized conditional state, whose dynamics include the cost function used to specify the performance objective. The risk-sensitive conditional dynamic equation describes the evolution of our knowledge of the quantum system tempered by our purpose for the controlled quantum system. Robustness properties of risk-sensitive controllers are discussed and an example is provided. <s> BIB008 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> We describe the formalism for optimally estimating and controlling both the state of a spin ensemble and a scalar magnetic field with information obtained from a continuous quantum limited measurement of the spin precession due to the field. The full quantum parameter estimation model is reduced to a simplified equivalent representation to which classical estimation and control theory is applied. We consider both the tracking of static and fluctuating fields in the transient and steady-state regimes. By using feedback control, the field estimation can be made robust to uncertainty about the total spin number. <s> BIB009 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum It\^o calculus and its use in the modelling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics. <s> BIB010 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> In the theory of quantum dynamical filtering, one of the biggest issues is that the underlying system dynamics represented by a quantum stochastic differential equation must be known exactly in order that the corresponding filter provides an optimal performance; however, this assumption is generally unrealistic. Therefore, in this paper, we consider a class of linear quantum systems subjected to time-varying norm-bounded parametric uncertainties and then propose a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound. Although in the linear case much of classical control theory can be applied to quantum systems, the quantum robust observer obtained in this paper does not have a classical analog due to the system's specific structure with respect to the uncertainties. Moreover, by considering a typical quantum control problem, we show that the proposed robust observer is fairly robust against a parametric uncertainty of the system even when the other estimators--the optimal Kalman filter and risk-sensitive observer--fail in the estimation. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> This paper studies a quantum risk-sensitive estimation problem and investigates robustness properties of the filter. This is a direct extension to the quantum case of analogous classical results. All investigations are based on a discrete approximation model of the quantum system under consideration. This allows us to study the problem in a simple mathematical setting. We close the paper with some examples that demonstrate the robustness of the risk-sensitive estimator. <s> BIB012 </s> Quantum control theory and applications: A survey <s> Identification and filtering/estimation <s> 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. <s> BIB013
To design an effective quantum feedback control system, it is necessary to obtain knowledge of a system model and system states. This can be regarded as a quantum system identification or state estimation problem. The simplest example is to estimate the system state for feedback through projective measurements on a quantum ensemble. This is a simple problem in classical control which can usually be accomplished through direct measurements. However it becomes much more complex due to the quantum collapse postulate of measurement in quantum control. This problem of identifying quantum states is also called quantum state tomography in quantum information and some procedures have been developed to experimentally determine an unknown quantum state BIB005 . Some of these procedures can be used in quantum feedback control. Another example is the identification of parameters for a quantum operation (e.g., a quantum gate or the system Hamiltonian). This procedure is called as quantum process tomography and some techniques have been developed (e.g., see BIB007 , BIB005 ). To acquire information about a quantum operation, an attractive approach is quantum nondemolition measurements which do not disturb the value to be measured. However, it is not practical to design quantum nondemolition measurements for most problems (see BIB002 for details). Furthermore, it is often useful to estimate an unknown dynamical parameter through continuously monitoring an open quantum system BIB003 , BIB006 . Since continuous observations of open quantum systems are inherently noisy, quantum filtering theory is required for extracting information from a noisy signal BIB010 . Quantum filtering theory has been developed for quantum optical systems and is used as an essential component in some quantum feedback control strategies BIB004 , BIB013 , BIB001 , BIB008 . In the framework of noncommutative (or quantum) probability theory, a broad class of quantum stochastic differential equations can be obtained (for details, see, e.g., BIB010 ). In fact, the SME model (14) is a quantum filtering equation which describes the evolution of a conditional state. Recently, several robustness properties of quantum filtering and estimation have also been investigated for linear and nonlinear quantum systems BIB009 , BIB011 , BIB012 .