Quantum control theory and applications: A survey <s> LQG control <s> We present a formulation of feedback in quantum systems in which the best estimates of the dynamical variables are obtained continuously from the measurement record, and fed back to control the system. We apply this method to the problem of cooling and confining a single quantum degree of freedom, and compare it to current schemes in which the measurement signal is fed back directly in the manner usually considered in existing treatments of quantum feedback. Direct feedback may be combined with feedback by estimation, and the resulting combination, performed on a linear system, is closely analogous to classical LQG control theory with residual feedback. <s> BIB001 </s> Quantum control theory and applications: A survey <s> LQG control <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> BIB002 </s> Quantum control theory and applications: A survey <s> LQG control <s> We introduce and discuss the problem of quantum feedback control in the context of established formulations of classical control theory, examining conceptual analogies and essential differences. We describe the application of state-observer-based control laws, familiar in classical control theory, to quantum systems and apply our methods to the particular case of switching the state of a particle in a double-well potential. <s> BIB003 </s> Quantum control theory and applications: A survey <s> LQG control <s> For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case: Given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state-based) control problems (the stationary linear-quadratic-Gaussian class), this question is a semidefinite program. Moreover, the answer also applies to Markovian (current-based) feedback. <s> BIB004 </s> Quantum control theory and applications: A survey <s> LQG control <s> It has recently been shown that finding the optimal measurement on the environment for stationary linear quadratic Gaussian control problems is a semidefinite program. We apply this technique to the control of the Einstein-Podolsky-Rosen correlations between two bosonic modes interacting via a parametric Hamiltonian at steady state. The optimal measurement turns out to be nonlocal homodyne measurement\char22{}the outputs of the two modes must be combined before measurement. We also find the optimal local measurement and control technique. This gives the same degree of entanglement but a higher degree of purity than the local technique previously considered [S. Mancini, Phys. Rev. A 73, 010304(R) (2006)]. <s> BIB005 </s> Quantum control theory and applications: A survey <s> LQG control <s> Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as"coherent feedback control.'' It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones. <s> BIB006 </s> Quantum control theory and applications: A survey <s> LQG control <s> This article investigates the effects of input noise on a magnetometer with quantum feedback. Previous research has shown that feedback makes the measurement robust to an unknown parameter, the number of atoms involved, with the assumption that the feedback is noise free. To evaluate the effects of the feedback noise, we extend the original model by an input noise term. We then analyze the steady state performance of the Kalman filter for both the closed-loop and open-loop cases and retrieve the second moment of the estimation error. The results are compared and criteria for evaluating the effects of input noise are obtained. Robust and optimal designs are also discussed. Computations and simulations show how quantitatively the input noise increases the estimation error and changes the region where the closed-loop case behaves better than the open-loop case. <s> BIB007 </s> Quantum control theory and applications: A survey <s> LQG control <s> In this paper, we formulate and solve a guaranteed cost control problem for a class of uncertain linear stochastic quantum systems. For these quantum systems, a connection with an associated classical (non-quantum) system is first established. Using this connection, the desired guaranteed cost results are established. The theory presented is illustrated using an example from quantum optics. <s> BIB008 </s> Quantum control theory and applications: A survey <s> LQG control <s> I present an experimental realization of a coherent-feedback control system that was recently proposed for testing basic principles of linear quantum stochastic control theory [M. R. James, H. I. Nurdin and I. R. Petersen, to appear in IEEE Transactions on Automatic Control (2008), arXiv:quant-ph/0703150v2]. For a dynamical plant consisting of an optical ring-resonator, I demonstrate ~ 7 dB broadband disturbance rejection of injected laser signals via all-optical feedback with a tailored dynamic compensator. Comparison of the results with a transfer function model pinpoints critical parameters that determine the coherent-feedback control system's performance. <s> BIB009 </s> Quantum control theory and applications: A survey <s> LQG control <s> The purpose of this paper is to formulate and solve a H∞ controller synthesis problem for a class of non-commutative linear stochastic systems which includes many examples of interest in quantum technology. The paper includes results on the class of such systems for which the quantum commutation relations are preserved (such a requirement must be satisfied in a physical quantum system). A quantum version of standard (classical) dissipativity results are presented and from this a quantum version of the Strict Bounded Real Lemma is derived. This enables a quantum version of the two Riccati solution to the H∞ control problem to be presented. This result leads to controllers which may be realized using purely quantum, purely classical or a mixture of quantum and classical elements. This issue of physical realizability of the controller is examined in detail, and necessary and sufficient conditions are given. Our results are constructive in the sense that we provide explicit formulas for the Hamiltonian function and coupling operator corresponding to the controller. <s> BIB010 </s> Quantum control theory and applications: A survey <s> LQG control <s> The purpose of this paper is to develop a synthesis theory for linear dynamical quantum stochastic systems that are encountered in linear quantum optics and in phenomenological models of linear quantum circuits. In particular, such a theory will enable the systematic realization of coherent/fully quantum linear stochastic controllers for quantum control, amongst other potential applications. We show how general linear dynamical quantum stochastic systems can be constructed by assembling an appropriate interconnection of one degree of freedom open quantum harmonic oscillators and, in the quantum optics setting, discuss how such a network of oscillators can be approximately synthesized or implemented in a systematic way from some linear and nonlinear quantum optical elements. An example is also provided to illustrate the theory. <s> BIB011
Under appropriate assumptions, some quantum optical systems can be approximately modeled by LQSDEs driven by quantum Wiener processes  , BIB004 . This simplification provides an opportunity to develop quantum linear-quadratic-Gaussian (LQG) control to obtain an optimal feedback strategy  , BIB005 , BIB002 , BIB001 , BIB003 , BIB004 , BIB007 , BIB008 . In LQG control, the goal is to find an optimal feedback control law for a stochastic linear system by optimizing a quadratic cost functional. In some cases, results from classical LQG control can be applied to quantum systems  . An important result is the separation theorem which applies when the system is linear, the cost functional is quadratic in the system variables and the noises are Gaussian. In this situation, the values of optimal estimates are fed back. Then, when calculating the feedback required for the optimal control, we may assume that the dynamical variables are known exactly BIB001 . In the quantum LQG control problem, the optimal control is also a linear feedback control law. The controller may be a classical controller BIB008 or a quantum controller BIB009 , BIB006 . For classical controllers, it is straightforward to apply results in classical LQG control to quantum LQG problems. In fact, some results on quantum LQG control with classical controllers have been developed in BIB005 , BIB001 , BIB003 , BIB004 , BIB008 . For controllers which are themselves quantum systems, we must add some additional constraints on coefficient matrices to ensure that the controller is physically realizable. For example, BIB010 has developed a notion of physical realizability based on the concept of an open quantum harmonic oscillator as the basic components of a physically realizable quantum system. Under several reasonable assumptions on a system of the form (18) (e.g., n y is even, n ω ≥ n y , F ω = I + idiag(J, . . . , J), etc.; see BIB010 for details), we have the following results on physical realizability BIB010 : The system (18) is physically realizable if and only if: where N y = ny 2 and Hence, the problem of quantum LQG control with a quantum controller is converted into the following problem: Given a canonical commutation relation, find a feedback controller, whose coefficient matrices satisfy the conditions of physical realizability in Theorem 11, that minimizes a quadratic performance index. The additional constraints on the coefficients for controllers make the problem more complex. Indeed, new algorithms are required to be developed in order to solve this problem. In BIB006 , a numerical procedure based on an alternating projection algorithm has been developed to solve this control problem. Furthermore, several results on synthesizing coherent quantum controllers have also been presented (e.g., see BIB011 ).