Quantum control theory and applications: A survey <s> Introduction <s> The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Introduction <s> The design of optimal final-state controllers of quantum-mechanical systems that are insensitive to errors in the molecular Hamiltonian or to errors in the initial state of the system is considered. Control arises through the interaction of the system with an external field; the goal is optimal design of these latter fields for various physical objectives in the presence of system uncertainty. Sensitivity to modeling errors and other uncertainties in the molecular Hamiltonian is minimized by considering averaged costs for a family of Hamiltonian functions ${\mathit{H}}_{0}$(\ensuremath{\alpha}) indexed by the random variable \ensuremath{\alpha} taking values on a compact set in Euclidean space. Similarly, sensitivity of the optimal control to the initial state is minimized by viewing the initial condition as a Hilbert-space-valued random variable and considering an optimization problem with a cost functional that is averaged over the class of initial conditions. A precise formulation of the control problem is given, and its well-posedness is established. Cost propagators are defined to display the dependence of the performance index on the initial conditions explicitly, which allows analytic averaging of initial conditions. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Introduction <s> We simulate a method to teach a laser pulse sequences to excite specified molecular states. We use a learning procedure to direct the production of pulses based on fitness'' information provided by a laboratory measurement device. Over a series of pulses the algorithm learns an optimal sequence. The experimental apparatus, which consists of a laser, a sample of molecules and a measurement device, acts as an analog computer that solves Schroedinger's equation n/Iexactly, in real time. We simulate an apparatus that learns to excite specified rotational states in a diatomic molecule. <s> BIB003 </s> Quantum control theory and applications: A survey <s> Introduction <s> Current experimental and theoretical progress toward the goal of controlling quantum dynamics is summarized. Two key developments have now revitalized the field. First, appropriate ultrafast laser pulse shaping capabilities have only recently become practical. Second, the introduction of engineering control concepts has put the required theoretical framework on a rigorous foundation. Extrapolations to determine what is realistically possible are presented. <s> BIB004 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB005 </s> Quantum control theory and applications: A survey <s> Introduction <s> In this paper we analyze the controllability of quantum systems arising in molecular dynamics. We model these systems as systems with finite numbers of levels, and examine their controllability. To do this we pass to their unitary generators and use results on the controllability of invariant systems on Lie groups. Examples of molecular systems, modeled as finite-dimensional control systems, are provided. A simple algorithm to detect the controllability of a molecular system is provided. Finally, we apply this algorithm to a five-level system. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB007 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB008 </s> Quantum control theory and applications: A survey <s> Introduction <s> This review puts into perspective the present state and prospects for controlling quantum phenomena in atoms and molecules. The topics considered include the nature of physical and chemical control objectives, the development of possible quantum control rules of thumb, the theoretical design of controls and their laboratory realization, quantum learning and feedback control in the laboratory, bulk media influences, and the ability to utilize coherent quantum manipulation as a means for extracting microscopic information. The preview of the field presented here suggests that important advances in the control of molecules and the capability of learning about molecular interactions may be reached through the application of emerging theoretical concepts and laboratory technologies. <s> BIB009 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB010 </s> Quantum control theory and applications: A survey <s> Introduction <s> Abstract Theoretical results are presented on the ability to arbitrarily steer about a wavefunction for a quantum system under time-dependent external field control. Criteria on the field free Hamiltonian and the field coupling term in the Hamiltonian are presented that assure full wavefunction controllability. Numerical simulations are given to illustrate the criteria. A discussion on the theoretical and practical relationship between dynamical conservation laws and controllability is also included. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Introduction <s> Chemistry has, throughout its history, been a science of passive control: having made the conditions as favourable as possible, the chemist is obliged to let the complex reorganization of reactant molecules into products take place unhindered. But what if one could meddle at the very heart of a reaction to affect the dynamics of molecular evolution? Welcome to the world of quantum interference control. <s> BIB012 </s> Quantum control theory and applications: A survey <s> Introduction <s> In this paper, we study the design of pulse sequences for nuclear magnetic resonance spectroscopy as a problem of time optimal control of the unitary propagator. Radio-frequency pulses are used in coherent spectroscopy to implement a unitary transfer between states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group. What is the minimum time required to steer the system from some initial point to a specified final point? In nuclear magnetic resonance (NMR) spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin system. <s> BIB013 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB014 </s> Quantum control theory and applications: A survey <s> Introduction <s> PREFACE ACKNOWLEDGEMENTS PART 1: PROBABILITY IN CLASSICAL AND QUANTUM MECHANICS 1. Classical probability theory and stochastic processes 2. Quantum Probability PART 2: DENSITY MATRIX THEORY 3. Quantum Master Equations 4. Decoherence PART 3: STOCHASTIC PROCESSES IN HILBERT SPACE 5. Probability distributions on Hilbert space 6. Stochastic dynamics in Hilbert space 7. The stochastic simulation method 8. Applications to quantum optical systems PART 4: NON-MARKOVIAN QUANTUM PROCESSES 9. Projection operator techniques 10. Non-Markovian dynamics in physical systems PART 5: RELATIVISTIC QUANTUM PROCESSES 11. Measurements in relativistic quantum mechanics 12. Open quantum electrodynamics <s> BIB015 </s> Quantum control theory and applications: A survey <s> Introduction <s> Various notions from geometric control theory are used to characterize the behavior of the Markovian master equation for N-level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states. It is shown that the system can be accessible but neither small-time controllable nor controllable in finite time. In particular, if the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time. The two level case is treated in detail. <s> BIB016 </s> Quantum control theory and applications: A survey <s> Introduction <s> We apply the techniques of control theory and of sub-Riemannian geometry to laser-induced population transfer in two- and three-level quantum systems. The aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences. Sub-Riemannian geometry and singular-Riemannian geometry provide a natural framework for this minimization, where the optimal control is expressed in terms of geodesics. We first show that in two-level systems the well-known technique of “π-pulse transfer” in the rotating wave approximation emerges naturally from this minimization. In three-level systems driven by two resonant fields, we also find the counterpart of the “π-pulse transfer.” This geometrical picture also allows one to analyze the population transfer by adiabatic passage. <s> BIB017 </s> Quantum control theory and applications: A survey <s> Introduction <s> Identifying the real and imaginary parts of wave functions with coordinates and momenta, quantum evolution may be mapped onto a classical Hamiltonian system. In addition to the symplectic form, quantum mechanics also has a positive-definite real inner product that provides a geometrical interpretation of the measurement process. Together they endow the quantum Hilbert space with the structure of a Kaeller manifold. Quantum control is discussed in this setting. Quantum time evolution corresponds to smooth Hamiltonian dynamics and measurements to jumps in the phase space. This adds additional power to quantum control, nonunitarily controllable systems becoming controllable by 'measurement plus evolution'. A picture of quantum evolution as the Hamiltonian dynamics in a classical-like phase space is the appropriate setting to carry over techniques from classical to quantum control. This is illustrated by a discussion on optimal control and sliding mode techniques. <s> BIB018 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB019 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB020 </s> Quantum control theory and applications: A survey <s> Introduction <s> We are currently in the midst of a second quantum revolution. The first quantum revolution gave us new rules that govern physical reality. The second quantum revolution will take these rules and use them to develop new technologies. In this review we discuss the principles upon which quantum technology is based and the tools required to develop it. We discuss a number of examples of research programs that could deliver quantum technologies in coming decades including; quantum information technology, quantum electromechanical systems, coherent quantum electronics, quantum optics and coherent matter technology. <s> BIB021 </s> Quantum control theory and applications: A survey <s> Introduction <s> We present controllability results for quantum systems interacting with lasers. Exact controllability for the wavefunction in these bilinear systems is proved in the finite-dimensional case under very natural hypotheses. <s> BIB022 </s> Quantum control theory and applications: A survey <s> Introduction <s> Preface.Preliminaries of the Interaction of Light with Matter.Weak-Field Photodissociation.Weak-Field Coherent Control.Optimal Control Theory.Decoherence and Loss of Control.Case Studies in Coherent Control.Coherent Control of Bimolecular Processes.Coherent Control of the Synthesis and Purification of Chiral Molecules.Coherent Control Beyond the Weak-Field Regime: Bound States and Resonances.Photodissociation Beyond the Weak-Field Regime.Coherent Control Beyond the Weak-Field Regime: The Continuum.Strong-Field Coherent Control.Case Studies in Optimal Control.References.Author Index.Subject Index. <s> BIB023 </s> Quantum control theory and applications: A survey <s> Introduction <s> Experiments in coherent spectroscopy correspond to control of quantum mechanical ensembles guiding them from initial to final target states. The control inputs (pulse sequences) that accomplish these transformations should be designed to minimize the effects of relaxation and to optimize the sensitivity of the experiments. For example in nuclear magnetic resonance (NMR) spectroscopy, a question of fundamental importance is what is the maximum efficiency of coherence or polarization transfer between two spins in the presence of relaxation. Furthermore, what is the optimal pulse sequence which achieves this efficiency? In this paper, we give analytical answers to the above questions. Unexpected gains in sensitivity are reported for one of the most commonly used experimental building blocks in NMR spectroscopy. Surprisingly, in the case when longitudinal relaxation is small, the relaxation optimized pulse elements (ROPE) that transfer maximum polarization between coupled spins are longer than conventional sequences. <s> BIB024 </s> Quantum control theory and applications: A survey <s> Introduction <s> The shot-noise detection limit in current high-precision magnetometry [Nature (London) 422, 596 (2003)] is a manifestation of quantum fluctuations that scale as 1/sqrt[N] in an ensemble of N atoms. Here, we develop a procedure that combines continuous measurement and quantum Kalman filtering [Rep. Math. Phys. 43, 405 (1999)]] to surpass this conventional limit by exploiting conditional spin squeezing to achieve 1/N field sensitivity. Our analysis demonstrates the importance of optimal estimation for high bandwidth precision magnetometry at the Heisenberg limit and also identifies an approximate estimator based on linear regression. <s> BIB025 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB026 </s> Quantum control theory and applications: A survey <s> Introduction <s> A nonvolatile semiconductor memory including multiple memory cells. Each memory cell comprises a FET TM having a ferroelectric insulation film and two MOS transistors T1 and T2 connected in series to the ends of the source-drain path of the FET, respectively. To write data into a memory cell, an electric field is applied in a predetermined direction between the gate and the substrate of the transistor TM. The electric field polarizes the gate insulation film of the transistor TM, which is made of ferroelectric material, in the direction, thereby writing data into the memory cell. In a data read mode, if the transistor TM is on, a current flows through the transistor TM, and the potential of a bit line to which the transistor TM is coupled decreases. In contrast, if the transistor TM is off, no currents flow through this transistor TM, and the potential of the bit line to which the transistor TM is coupled remains unchanged. Thus, the data stored n the memory cell can be discriminated by detecting whether the potential of the bit line corresponds to a "1" bit or a "0" bit. <s> BIB027 </s> Quantum control theory and applications: A survey <s> Introduction <s> Many applications of magnetic resonance are limited by rapid loss of spin coherence caused by large transverse relaxation rates. In NMR of large proteins, increased relaxation losses lead to poor sensitivity of experiments and increased measurement time. In this article, we develop broadband relaxation-optimized pulse sequences that approach fundamental limits of coherence transfer efficiency in the presence of very general relaxation mechanisms that include cross-correlated relaxation. These broadband transfer schemes use techniques of chemical shift refocusing (specific trajectory adapted refocusing echoes) that are tailored to specific trajectories of coupled spin evolution. We present simulations and experimental data indicating significant enhancement in the sensitivity of multidimensional NMR experiments of large molecules through these methods. <s> BIB028 </s> Quantum control theory and applications: A survey <s> Introduction <s> Fifty years of developments in nuclear magnetic resonance (NMR) have resulted in an unrivaled degree of control of the dynamics of coupled two-level quantum systems. This coherent control of nuclear spin dynamics has recently been taken to a new level, motivated by the interest in quantum information processing. NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. This article surveys and summarizes a broad variety of pulse control and tomographic techniques which have been developed for, and used in, NMR quantum computation. Many of these will be useful in other quantum systems now being considered for the implementation of quantum information processing tasks. <s> BIB029 </s> Quantum control theory and applications: A survey <s> Introduction <s> We describe in this article some key themes that emerged during a Caltech/AFOSR Workshop on ‘Principles and Applications of Control in Quantum Systems’ (PRACQSYS), held 21–24 August 2004 at the California Institute of Technology. This workshop brought together engineers, physicists and applied mathematicians to construct an overview of new challenges that arise when applying constitutive methods of control theory to nanoscale systems whose behaviour is manifestly quantum. Its primary conclusions were that the number of experimentally accessible quantum control systems is steadily growing (with a variety of motivating applications), that appropriate formal perspectives enable straightforward application of the essential ideas of classical control to quantum systems, and that quantum control motivates extensive study of model classes that have previously received scant consideration. Copyright © 2005 John Wiley & Sons, Ltd. <s> BIB030 </s> Quantum control theory and applications: A survey <s> Introduction <s> The goal of this article is to provide a largely self-contained introduction to the modelling of controlled quantum systems under continuous observation, and to the design of feedback controls that prepare particular quantum states. We describe a bottom-up approach, where a field-theoretic model is subjected to statistical inference and is ultimately controlled. As an example, the formalism is applied to a highly idealized interaction of an atomic ensemble with an optical field. Our aim is to provide a unified outline for the modelling, from first principles, of realistic experiments in quantum control. <s> BIB031 </s> Quantum control theory and applications: A survey <s> Introduction <s> In this paper we consider the minimum time population transfer problem for the z component of the spin of a (spin 1/2) particle, driven by a magnetic field, that is constant along the z axis and controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e., after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on two-dimensional (2-D) manifolds. Let (−E,E) be the two energy levels, and ∣Ω(t)∣≤M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level −E, and we provide the explicit expression of the time optimal trajectories, steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M∕E≪1, every time optimal trajectory is bang-bang and, in particular, the corresponding control is periodic with frequency of the order of the resonance frequency ωR=2E. On the other side, for M∕E>1, the time optimal ... <s> BIB032 </s> Quantum control theory and applications: A survey <s> Introduction <s> A Lyapunov-based approach for trajectory tracking of the Schrodinger equation is proposed. In the finite dimensional case, convergence is precisely analyzed. Connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied. When the linearized system is controllable, such a feedback ensures almost global asymptotic convergence. When the linearized system is not controllable, the stability of the closed-loop system is not asymptotic. To overcome such lack of convergence, we propose, when the reference trajectory is an eigenstate, a modification based on adiabatic invariance. Simulations illustrate the simplicity and also the interest for trajectory generation. <s> BIB033 </s> Quantum control theory and applications: A survey <s> Introduction <s> Manipulation of infinite-dimensional quantum systems is important to controlling complex quantum dynamics with many practical physical and chemical backgrounds. In this paper, a general investigation is casted to the controllability problem of quantum systems evolving on infinite-dimensional manifolds. Recognizing that such problems are related with infinite-dimensional controllability algebras, we introduce an algebraic mathematical framework to describe quantum control systems possessing such controllability algebras. Then we present the concept of smooth controllability on infinite-dimensional manifolds, and draw the main result on approximate strong smooth controllability. This is a nontrivial extension of the existing controllability results based on the analysis over finite-dimensional vector spaces to analysis over infinite-dimensional manifolds. It also opens up many interesting problems for future studies. <s> BIB034 </s> Quantum control theory and applications: A survey <s> Introduction <s> This paper presents a constructive proof of complete kinematic state controllability of finite-dimensional open quantum systems whose dynamics are represented by Kraus maps. For any pair of states (pure or mixed) on the Hilbert space of the system, we explicitly show how to construct a Kraus map that transforms one state into another. Moreover, we prove by construction the existence of a Kraus map that transforms all initial states into a predefined target state (such a process may be used, for example, in quantum information dilution). Thus, in sharp contrast to unitary control, Kraus-map dynamics allows for the design of controls which are robust to variations in the initial state of the system. The capabilities of non-unitary control for population transfer between pure states illustrated for an example of a two-level system by constructing a family of non-unitary Kraus maps to transform one pure state into another. The problem of dynamic state controllability of open quantum systems (i.e., controllability of state-to-state transformations, given a set of available dynamical resources such as coherent controls, incoherent interactions with the environment, and measurements) is also discussed. <s> BIB035 </s> Quantum control theory and applications: A survey <s> Introduction <s> Abstract We review theoretical work in the field of “coherent control” (CC) of bound- and continuum-state dynamics. We explain the principles of bichromatic and polychromatic control and show that CC is made possible through the execution of a disentanglement transformation . We discuss limitations to control and show that controllability in continuum states is linked to a generalized Schwarz inequality of vector spaces. We illustrate the success of these techniques in controlling photo-current directionality in semiconductors using one- vs. two-photon interferences. We next turn our attention to CC using adiabatic passage (AP) processes, enabling the achievement of both selectivity and complete population transfer. We demonstrate controllability in bound state dynamics by the explicit construction of guiding fields which solve the “non-degenerate quantum control” problem. The CC + AP methodology is then extended to the control of manifolds composed entirely of degenerate bound states. We use AP techniques in the control of photoassociation and especially in the controlled recombination of ultracold atoms to form ultracold molecules. We next discuss the optical manipulation of interferences between overlapping, decaying resonances as a way of suppressing spontaneous emission, intramolecular vibrational redistribution, and internal conversion in molecules. Finally we discuss decoherence and the way material and radiative phase-instabilities diminish controllability. We show how to combat decoherence and how to use CC to construct “decoherence-free subspaces”. <s> BIB036 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB037 </s> Quantum control theory and applications: A survey <s> Introduction <s> In this paper we are concerned with the problem of stability for quantum feedback networks. We demonstrate in the context of quantum optics how stability of quantum feedback networks can be guaranteed using only simple gain inequalities for network components and algebraic relationships determined by the network. Quantum feedback networks are shown to be stable if the loop gain is less than one--this is an extension of the famous small gain theorem of classical control theory. We illustrate the simplicity and power of the small gain approach with applications to important problems of robust stability and robust stabilization. <s> BIB038 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB039 </s> Quantum control theory and applications: A survey <s> Introduction <s> According to special geometric or physical meanings, the paper summarizes three Lyapunov functions in controlling closed quantum systems and their controller designing processes. Specially, for the average value-based method, the paper gives the generalized condition of the largest invariant set in the original reference and develops the construction method of the imaginary mechanical quantity; for the error-based method, this paper gives its strict mathematical proof train of thought on the asymptotic stability and the corresponding physical meaning. Also, we study the relations among the three Lyapunov functions and give a unified form of these Lyapunov functions. Finally, we compare the control effects of three Lyapunov methods by doing some simulation experiments. <s> BIB040 </s> Quantum control theory and applications: A survey <s> Introduction <s> An incoherent control scheme for state control of locally controllable quantum systems is proposed. This scheme includes three steps: (1) amplitude amplification of the initial state by a suitable unitary transformation, (2) projective measurement on the amplified state, and (3) final optimization by a unitary controlled transformation. The first step increases the amplitudes of some desired eigenstates and the corresponding probability of observing these eigenstates, the second step projects, with high probability, the amplified state into a desired eigenstate, and the last step steers this eigenstate into the target state. Within this scheme, two control algorithms are presented for two classes of quantum systems. As an example, the incoherent control scheme is applied to the control of a hydrogen atom by an external field. The results support the suggestion that projective measurements can serve as an effective control and local controllability information can be used to design control laws for quantum systems. Thus, this scheme establishes a subtle connection between control design and controllability analysis of quantum systems and provides an effective engineering approach for controlling quantum systems with partial controllability information. <s> BIB041 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB042 </s> Quantum control theory and applications: A survey <s> Introduction <s> D. Burgarth, K. Maruyama and F. Nori, Coupling strength estimation for spin chains despite restricted access Phys. Rev. A 79, 020305(R) (2009). <s> BIB043 </s> Quantum control theory and applications: A survey <s> Introduction <s> Abstract This paper formulates and solves a variable structure control problem for the state control of quantum systems. Following the concept of wavefunction controllability, we first present a specific quantum system which is wavefunction uncontrollable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. An extension and the potential applications of the proposed control strategy are briefly discussed and the general steps of variable structure control design for this class of quantum systems are also presented. The variable structure control scheme provides an alternative quantum engineering approach for the manipulation of quantum systems. <s> BIB044 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB045 </s> Quantum control theory and applications: A survey <s> Introduction <s> An observer-based Hamiltonian identification algorithm for quantum systems is proposed. For the 2-level case an exponential convergence result based on averaging arguments and some relevant transformations is provided. The convergence for multi-level cases is discussed using some heuristic arguments and the relevance of the method is tested via simulations. Finally, the robustness issue with respect to non-negligible uncertainties and experimental noises is also addressed on simulations. <s> BIB046 </s> Quantum control theory and applications: A survey <s> Introduction <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> BIB047 </s> Quantum control theory and applications: A survey <s> Introduction <s> This article focuses on wavefunction controllability of a class of quantum systems with switching control. First, we present a specific quantum system which is not wavefunction controllable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. Then we present a sufficient condition for wavefunction controllability of a class of quantum systems with switching control and briefly discuss the problem of control design for such quantum control systems. Finally, two illustrative examples are presented to demonstrate a possible control design process for the proposed method. <s> BIB048
Quantum control theory is a rapidly evolving research area, which has developed over the last three decades  - BIB009 . Controlling quantum phenomena has been an implicit goal of much quantum physics and chemistry research since the establishment of quantum mechanics BIB004 ,  . One of the main goals in quantum control theory is to establish a firm theoretical footing and develop a series of systematic methods for the active manipulation and control of quantum systems BIB030 . This goal is nontrivial since microscopic quantum systems have many unique characteristics (e.g., entanglement and coherence) which do not occur in classical mechanical systems and the dynamics of quantum systems must be described by quantum theory. Quantum control theory has already attained significant successes in physical chemistry BIB009 - BIB027 , atomic and molecular physics  ,  and quantum optics BIB005 , BIB031 , and has also contributed to the understanding of fundamental aspects of quantum mechanics BIB010 . In recent years, the development of the general principles of quantum control theory has been recognized as an essential requirement for the future application of quantum technologies BIB021 . Quantum control theory is drawing wide attention from different communities in the areas of physics, chemistry, control theory, applied mathematics and quantum information, and has become a regular topic in international conferences such as the IEEE Conference on Decision and Control. An international workshop on "Principles and Applications for Control of Quantum Systems (PRACQSYS)" has also been sponsored since 2004 to discuss recent developments in this area BIB030 . This paper will present the basic concepts and main developments of quantum control theory from a control systems perspective. In much of quantum control theory, the controllability of quantum systems is a fundamental issue. This issue concerns whether one can drive a quantum system to a desired state  . This problem has practical importance since it has a close connection with the universality of quantum computation  and the possibility of achieving atomic or molecular scale transformations BIB006 , BIB034 . Different notions of controllability, such as pure state controllability, complete controllability, eigenstate controllability and kinematic controllability, have been proposed BIB006 - BIB035 . A common research focus is on finite dimensional (finite level) quantum systems for which the controllability criteria may be expressed in terms of the structure and rank of corresponding Lie groups and Lie algebras  . This method allows for an easy mathematical treatment of the problem for the case of closed quantum systems (i.e., isolated systems considered without external influences) and in some cases, methods from classical (non-quantum) control theory can be directly applied. However, the relevant criteria may be computationally difficult when the dimension of the quantum system to be controlled is large. Therefore, a controllability method based on graph theory has been proposed, and the corresponding controllability criterion is easier to verify BIB011 , BIB022 . Compared with finite dimensional closed quantum systems, the controllability of infinite dimensional quantum systems or open quantum systems (i.e., systems considered interacting with the environment; e.g., see BIB015 ) is more difficult and only a few results have been obtained BIB034 , BIB035 , BIB016 . For a controllable quantum system, it is desirable to develop a good control strategy to accomplish the required control tasks on the quantum system BIB009 . The coherent control strategy is a widely used method where one manipulates the states of a quantum system by applying semiclassical potentials in a fashion that preserves quantum coherence BIB010 (a wave-like property of quantum systems allowing for constructive and destructive interference, e.g., see BIB004 ). An early paradigm of quantum control was openloop coherent control BIB004 , which has successfully been used in the quantum control of chemical reactions BIB012 , BIB036 . Rigorous optimal control theory has been successfully applied to the design of open-loop coherent control strategies in order to find the best way of achieving given quantum control objectives in physical chemistry BIB023 , BIB001 , BIB002 . Recently, time-optimal control problems for spin systems have been solved to achieve specified control objectives in minimum time BIB013 , BIB017 , BIB032 . Optimal control techniques have also been successfully applied to multidimensional nuclear magnetic resonance (NMR) experiments to improve the sensitivity of these systems in the presence of relaxation  , BIB024 , BIB028 . Another useful tool in open-loop quantum control is the Lyapunov-based control design approach BIB033 , BIB040 . In classical control, the Lyapunov method determines the control input as a function of the system state. Hence, it is a feedback control design method. In quantum control, it is difficult to acquire information about quantum states without destroying them. Hence, the Lyapunov-based control methodology is used to first construct an artificial closed-loop controller and then an open-loop control law is obtained by simulation of the artificial closed-loop system. In coherent control, the control operations consist of unitary transformations. However, some quantum systems may not be controllable using only coherent controls. For such uncontrollable quantum systems, it may be possible to enhance the capabilities of quantum control by introducing new control strategies such as variable structure control and incoherent control (i.e., one is allowed to destroy coherence of the quantum systems during the control process). For example, a variable structure control method has been introduced in BIB044 , BIB048 to enhance the capabilities of quantum control for some specific systems which are not controllable. Incoherent control enables the effects of quantum measurements and incoherent environments to be combined with unitary transformations to complete quantum control tasks and enhance the performance of quantum control systems BIB018 - BIB041 . Incoherent control strategies have also been recognized as important techniques to assist coherent control strategies in quantum control systems. Although open-loop strategies using coherent or incoherent control approaches, optimal control techniques and Lyapunov methods, have achieved important successes, their applications have many limitations. In classical control, closed-loop methods have many advantages over open-loop methods, such as robustness and reliability. For open quantum systems, coupling with uncontrollable environments makes the introduction of noises and uncertainties unavoidable  . A requirement for robustness in the presence of uncertainties and noises has been recognized as one of the key aspects in developing practical quantum technologies BIB030 , BIB042 . A natural solution to this problem is to develop closed-loop quantum control approaches. Two paradigms for closed-loop control have been proposed: closed-loop learning control and quantum feedback control BIB009 . Closedloop learning control involves a closed-loop operation where each cycle of the loop is executed with a new sample, and this approach has achieved great success in controlling quantum phenomena in chemical reactions BIB009 , BIB003 . Feedback is an essential concept in classical control theory, where information about the state variables obtained from direct measurements or state estimation is fed back to the system through a controller to achieve desired performance. Quantum feedback control has been used to improve the system performance in different tasks, including the control of squeezed states (a typical class of nonclassical states, e.g., see  ) and entangled states (see Section 2.1), state reduction and quantum error correction BIB005 , BIB019 - BIB039 . The success of feedback control is usually dependent on the acquisition of suitable feedback information. This problem becomes more complex and subtle in quantum control since quantum measurements unavoidably affect the states of measured systems. Hence, research aimed at establishing a general framework for quantum feedback is accelerating the development of some related areas including quantum filtering theory BIB008 , BIB037 , BIB045 , quantum parameter estimation BIB007 , BIB025 , BIB026 , quantum system identification BIB020 , BIB046 , BIB043 and quantum robust control BIB042 , BIB038 . This paper will survey the development of open-loop and closed-loop methods in quantum control theory from a control systems perspective. We will also briefly discuss some applications of quantum control theory to quantum information technology BIB014 . Many successful applications of quantum control in chemistry and physics are beyond the scope of this paper and, for these topics, readers can refer to several excellent books and review articles in the areas of chemical reactions BIB004 , BIB009 , [11] , BIB023 , BIB027 ,  , BIB012 , BIB036 , atomic physics  , NMR BIB030 , BIB029 and quantum optics BIB047 . The remainder of this paper is organized as follows. Section 2 briefly introduces quantum states, quantum measurements and several quantum control models. In Section 3, the controllability of quantum systems and several open-loop control strategies, including optimal control, Lyapunov methods, incoherent control and variable structure control, are reviewed. Section 4 presents some results on closed-loop control of quantum systems. In particular, in this section, closed-loop learning control is briefly mentioned and some specific aspects of quantum feedback control are presented. Conclusions and perspectives are given in Section 5.