Quantum control theory and applications: A survey <s> Quantum feedback control <s> We present a quantum theory of feedback in which the homodyne photocurrent alters the dynamics of the source cavity. To the nonlinear stochastic (Ito) evolution of the conditioned system state we add a feedback term linear in the instantaneous stochastic (Stratonovich) photocurrent. Averaging over the photocurrent gives a feedback master equation which has the desired driftlike term, plus a diffusionlike term. We apply the model to phase locking a regularly pumped laser, and show that under ideal conditions the noise spectra of the output light exhibit perfect squeezing on resonance. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Quantum feedback 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> BIB002 </s> Quantum control theory and applications: A survey <s> Quantum feedback 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> Quantum feedback control <s> We consider separating the problem of designing Hamiltonian quantum feedback control algorithms into a measurement (estimation) strategy and a feedback (control) strategy, and we consider optimizing desirable properties of each under the minimal constraint that the available strength of both is limited. This motivates concepts of information extraction and disturbance that are distinct from those usually considered in quantum information theory. Using these concepts, we identify an information tradeoff in quantum feedback control. <s> BIB004 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> In the conventional picture of quantum feedback control, sensors perform measurements on the system, a classical controller processes the results of the measurements, and actuators supply semiclassical potentials to alter the behavior of the quantum system. In this picture, the sensors tend to destroy coherence in the process of making measurements, and although the controller can use the actuators to act coherently on the quantum system, it is processing and feeding back classical information. This paper proposes an alternative method for quantum feedback control, in which the sensors, controller, and actuators are quantum systems that interact coherently with the system to be controlled. In this picture, the controller gets, processes, and feeds back quantum information. Controllers that operate using such quantum feedback loops can perform tasks such as entanglement transfer that are not possible using classical feedback. Necessary and sufficient conditions are presented for Hamiltonian quantum systems to be controllable and observable using both classical and quantum feedback. <s> BIB005 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> We propose a quantum feedback scheme for producing deterministically reproducible spin squeezing. The results of a continuous nondemolition atom number measurement are fed back to control the quantum state of the sample. For large samples and strong cavity coupling, the squeezing parameter minimum scales inversely with atom number, approaching the Heisenberg limit. Furthermore, ceasing the measurement and feedback when this minimum has been reached will leave the sample in the maximally squeezed spin state. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> We compare two different approaches to the control of the dynamics of a continuously monitored open quantum system. The first is Markovian feedback, as introduced in quantum optics by Wiseman and Milburn [Phys. Rev. Lett. 70, 548 (1993)]. The second is feedback based on an estimate of the system state, developed recently by Doherty and Jacobs [Phys. Rev. A 60, 2700 (1999)]. Here we choose to call it, for brevity, Bayesian feedback. For systems with nonlinear dynamics, we expect these two methods of feedback control to give markedly different results. The simplest possible nonlinear system is a driven and damped two-level atom, so we choose this as our model system. The monitoring is taken to be homodyne detection of the atomic fluorescence, and the control is by modulating the driving. The aim of the feedback in both cases is to stabilize the internal state of the atom as close as possible to an arbitrarily chosen pure state, in the presence of inefficient detection and other forms of decoherence. Our results (obtained without recourse to stochastic simulations) prove that Bayesian feedback is never inferior, and is usually superior, to Markovian feedback. However, it would be far more difficult to implement than Markovian feedback and it loses its superiority when obvious simplifying approximations are made. It is thus not clear which form of feedback would be better in the face of inevitable experimental imperfections. <s> BIB007 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability. <s> BIB008 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> We present quantum feedback control for deterministic entanglement generation at the single-photon level. The protocol of controlling both total photon number and phase difference is based on the cascade structure of cavities placed in an optical closed loop, quantum nondemolition measurement with cross-Kerr interactions, and Lyapunov stability for feedback design. <s> BIB009 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> We consider using Hamiltonian feedback control to increase the speed at which a continuous measurement purifies (reduces) the state of a quantum system, and thus to increase the speed of the preparation of pure states. For a measurement of an observable with $N$ equispaced eigenvalues, we show that there exists a feedback algorithm which will speed up the rate of state reduction by at least a factor of $2(N+1)/3$. <s> BIB010 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> This paper reconsiders the claimed rapidity of a scheme for the purification of the quantum state of a qubit, proposed recently in Jacobs (2003 Phys. Rev. A 67 030301(R)). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs' rapid purification protocol uses Hamiltonian feedback control to maximize the average purity of the qubit for a given time, with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it minimizes the average time taken for a qubit to reach a given purity. In fact, the average time taken for the no-feedback protocol beats that for Jacobs' protocol by a factor of two. We discuss how this is compatible with Jacobs' result and the usefulness of the different approaches. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> In this paper we show how weak measurement and local feedback can be used to control entanglement generation between two qubits. To do this, we make use of a decoherence free subspace. Weak measurement and feedback can be used to drive the system into this subspace rapidly. Once within the subspace, feedback can generate entanglement rapidly, or turn off entanglement generation dynamically. We also consider, in the context of weak measurement, some of the differences between purification and generating entanglement. <s> BIB012 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> We apply quantum filtering and control to a particle in a harmonic trap under continuous position measurement, and show that a simple static feedback law can be used to cool the system. The final steady state is Gaussian and dependent on the feedback strength and coupling between the system and probe. In the limit of weak coupling, this final state becomes the ground state. An earlier model by Haine et al. [Phys. Rev. A 69, 13605 (2004)] without measurement backaction showed dark states: states that did not display error signals, thus remaining unaffected by the control. This paper shows that for a realistic measurement process this is not true, which indicates that a Bose-Einstein condensate may be driven toward the ground state from any arbitrary initial state. <s> BIB013 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> In this paper, we consider a linear quantum network composed of two distantly separated cavities that are connected via a one-way optical field. When one of the cavities is damped and the other undamped, the overall cavity state obtains a large amount of entanglement in its quadratures. This entanglement, however, immediately decays and vanishes in a finite time. That is, entanglement sudden death occurs. We show that the direct measurement feedback method proposed by Wiseman can avoid this entanglement sudden death, and, further, enhance the entanglement. It is also shown that the entangled state under feedback control is robust against signal loss in a realistic detector, indicating the reliability of the proposed direct feedback method in practical situations. <s> BIB014 </s> Quantum control theory and applications: A survey <s> Quantum feedback 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> BIB015 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> The control of individual quantum systems promises a new technology for the 21st century – quantum technology. This book is the first comprehensive treatment of modern quantum measurement and measurement-based quantum control, which are vital elements for realizing quantum technology. Readers are introduced to key experiments and technologies through dozens of recent experiments in cavity QED, quantum optics, mesoscopic electronics, and trapped particles several of which are analyzed in detail. Nearly 300 exercises help build understanding, and prepare readers for research in these exciting areas. This important book will interest graduate students and researchers in quantum information, quantum metrology, quantum control and related fields. Novel topics covered include adaptive measurement; realistic detector models; mesoscopic current detection; Markovian, state-based and optimal feedback; and applications to quantum information processing. <s> BIB016 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> We study cooling and squeezing the fluctuations of a nanomechanical beam using quantum feedback control. In our model, the nanomechanical beam is coupled to a transmission line resonator via a superconducting quantum interference device (SQUID). The leakage of the electromagnetic field from the transmission line resonator is measured using homodyne detection. This measured signal is then used to design a quantum-feedback-control signal to drive the electromagnetic field in the transmission line resonator. Although the control is imposed on the transmission line resonator, this quantum-feedback-control signal indirectly affects the thermal motion of the nanomechanical beam via the inductive beam-resonator coupling, making it possible to cool and squeeze the fluctuations of the beam, allowing it to approach the standard quantum limit. <s> BIB017 </s> Quantum control theory and applications: A survey <s> Quantum feedback control <s> In this paper, we put forward a fundamental question concerning feedback control of quantum systems: Is measurement-based feedback control still better than open-loop control? In contrast to the classical control theory, the answer is far from obvious. This is because measurement-based feedback needs measurement to reduce the system uncertainty, whereas the measurement on a quantum system will inevitably increase the system uncertainty in turn. In fact, there is a complicated tradeoff between the uncertainty introduced and the information gained by the measurement on a quantum system. To investigate this fundamental problem, we will only focus on a typical model of coherent control mode with and without the decoherence term in the paper. By establishing some fundamental limits on the performances of both the open-loop and measurement-based feedback controls, we will demonstrate via simulation that the measurement-based feedback control of quantum systems is still superior to the open-loop control in some sense for the typical model under consideration. <s> BIB018
As we know, feedback is an effective strategy in classical control theory and the aim of feedback is to compensate for the effects of unpredictable disturbances on a system under control, or to make automatic control possible when the initial state of the system is unknown. In classical control, many results have shown that feedback control is superior to open-loop control. In feedback control, it is usually necessary to obtain the information about the state of system through measurement. However, measurements of a quantum system will unavoidably disturb the state of measured quantum system, which makes the situation more complex when applying feedback to quantum systems. In spite of this difficulty, important progress has been made and feedback has been used to improve the control performance for squeezed states BIB006 ,  , quantum entanglement BIB009 , BIB014 , BIB012 , and quantum state reduction BIB008 - BIB010 , BIB011 -  in many areas such as quantum optics BIB001 , BIB016 , superconducting quantum systems  , Bose-Einstein condensate BIB013 and nanomechanical systems BIB017 . Quantum feedback control has also been compared with open-loop control (e.g., see BIB018 ). It has been proven for a specific quantum system that quantum feedback is superior to open-loop control in dealing with uncertainties in initial states and it has also been demonstrated via simulation that feedback control is still better than open-loop control for dealing with decoherence BIB018 . In quantum feedback control, the two main approaches to information acquisition are projective measurement and continuous weak measurement. The system to be controlled is a quantum system, however, the controller may be quantum, classical or a quantum/classical hybrid. Several paradigms of quantum feedback have been proposed, such as Markovian quantum feedback BIB001 ,  , Bayesian quantum feedback BIB002 , BIB003 , BIB004 , and coherent quantum feedback BIB005 , BIB015 . In Markovian quantum feedback, any time delay is ignored and a memoryless controller is assumed. That is, the measurement record is immediately fed back onto the system to alter the system dynamics and may then be forgotten BIB007 . Hence, the equation describing the resulting evolution is a Markovian master equation. In Bayesian quantum feedback, the process is divided into two steps involving state estimation and feedback control. The best estimates of the dynamical variables are obtained continuously from the measurement record, and fed back to control the system dynamics BIB002 . The dynamical equation describing the resulting evolution is non-Markovian. In coherent quantum feedback, the feedback controller itself is a quantum system, and it processes quantum information. This is greatly different from Markovian and Bayesian quantum feedback where the feedback information from measurement results is classical information and the feedback controller is a classical controller. In the following, we will present several important aspects of quantum feedback control including quantum filtering, feedback stabilization, optimal feedback control and robust control.