Quantum control theory and applications: A survey <s> Incoherent 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> BIB001 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> The ability to perform measurements on a quantum system, combined with the ability to feed back the measurement results via coherent control, allows one to control the system to follow any desired coherent or incoherent quantum dynamics. Such universal dynamical control can be achieved, in principle, through the repeated application of only two coherent control operations and a simple ``Yes-No'' measurement. As a consequence, a quantum computer can simulate an arbitrary open-system dynamics using just one qubit more than required to simulate closed-system dynamics. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Incoherent control <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> BIB003 </s> Quantum control theory and applications: A survey <s> Incoherent control <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> BIB004 </s> Quantum control theory and applications: A survey <s> Incoherent control <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> BIB005 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> A systematic analysis of the behavior of the quantum Markovian master equation driven by coherent control fields is proposed. Its irreversible character is formalized using control-theoretic notions and the sets of states that can be reached via coherent controls are described. The analysis suggests to what extent (and how) it is possible to counteract the effect of dissipation. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> The most basic scenario of quantum control involves the organized manipulation of pure dynamical states of the system by means of unitary transformations. Recently, Vilela Mendes and Man'ko have shown that the conditions for controllability on the state space become less restrictive if unitary control operations may be supplemented by projective measurement. The present work builds on this idea, introducing the additional element of indirect measurement to achieve a kind of remote control. The target system that is to be remotely controlled is first entangled with another identical system, called the control system. The control system is then subjected to unitary transformations plus projective measurement. As anticipated by Schroedinger, such control via entanglement is necessarily probabilistic in nature. On the other hand, under appropriate conditions the remote-control scenario offers the special advantages of robustness against decoherence and a greater repertoire of unitary transformations. Simulations carried out for a two-level system demonstrate that, with optimization of control parameters, a substantial gain in the population of reachable states can be realized. <s> BIB007 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> Recently, a new approach for the controllability of a two-dimensional quantum system S has been proposed, based on its interaction with an initially uncorrelated two-dimensional probe P whose initial state can be arbitrarily modified. Following this scheme and considering a particular model for the environment, we show that, in some specific cases, the environment-induced entanglement is rich enough to completely control the dynamics of S. Under suitable conditions on the interaction of S, P, and the environment, we prove that the state of S can be driven to an arbitrary target state by varying the initial state of P. <s> BIB008 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> We study the problem of mapping an unknown mixed quantum state onto a known pure state without the use of unitary transformations. This is achieved with the help of sequential measurements of two non-commuting observables only. We show that the overall success probability is maximized in the case of measuring two observables whose eigenstates define mutually unbiased bases. We find that for this optimal case the success probability quickly converges to unity as the number of measurement processes increases and that it is almost independent of the initial state. In particular, we show that to guarantee a success probability close to one the number of consecutive measurements must be larger than the dimension of the Hilbert space. We connect these results to quantum copying, quantum deleting and entanglement generation. <s> BIB009 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> We study a new approach to the control of a quantum system S, which uses the coupling of S with a quantum ::: probe P. The external control affects only P, and the accessibility and controllability properties describe to what ::: extent it is possible to drive the state of S by varying the initial state of P and using the interaction between ::: the two systems. In particular, we study the case of two-dimensional system and probe. Two situations are ::: considered: either the total system S + P is a closed one, or it is surrounded by a bath of decoupled harmonic ::: oscillators. We give results on the controllability and accessibility properties of this scheme, and we discuss the ::: relation of these properties with the entangling capability of the interaction between S and P. In particular, we ::: show that the SWAP operator plays a special role. <s> BIB010 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> We propose a scheme to overcome phase damping decoherence by periodically performing open loop tracking control and projective measurement. Although it is impossible to stabilize a qubit subject to Markovian dynamics only by open loop coherent control, one can attain a 'softened' control goal with the help of periodical projective measurement. The 'softened' control objective in our scheme is to keep the state of the controlled qubit to stay near a reference pure state with a high probability for a sufficiently long time. Two suboptimal control problems are given in the sense of trace distance and fidelity, respectively, and they are eventually reduced to the design of a period T. In our scheme, one can choose the period T as long as possible if the 'softened' control goal is attained. This is in contrast to the observation that quantum Zeno effect takes place only if measurements are performed in a very frequent manner, i.e., the period T must be extremely small. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> We demonstrate a scheme for controlling a large quantum system by acting on a small subsystem only. The local control is mediated to the larger system by some fixed coupling Hamiltonian. The scheme allows us to transfer arbitrary and unknown quantum states from a memory to the large system (``upload access'') as well as the inverse (``download access''). We study the sufficient conditions of the coupling Hamiltonian and give lower bounds on the fidelities for downloading and uploading. <s> BIB012 </s> Quantum control theory and applications: A survey <s> Incoherent control <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> BIB013 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> In this paper, an incoherent control scheme for accomplishing the state control of a class of quantum systems which have wavefunction-controllable subspaces is proposed. This scheme includes the following two steps: projective measurement on the initial state and learning control in the wavefunction-controllable subspace. The first step probabilistically projects the initial state into the wavefunction-controllable subspace. The probability of success is sensitive to the initial state; however, it can be greatly improved through multiple experiments on several identical initial states even in the case with a small probability of success for an individual measurement. The second step finds a local optimal control sequence via quantum reinforcement learning and drives the controlled system to the objective state through a set of suitable controls. In this strategy, the initial states can be unknown identical states, the quantum measurement is used as an effective control, and the controlled system is not necessarily unitarily controllable. This incoherent control scheme provides an alternative quantum engineering strategy for locally controllable quantum systems. <s> BIB014 </s> Quantum control theory and applications: A survey <s> Incoherent control <s> We give a sufficient criterion that guarantees that a many-body quantum system can be controlled by properly manipulating the (local) Hamiltonian of one of its subsystems. The method can be applied to a wide range of systems: it does not depend on the details of the couplings but only on their associated topology. As a special case, we prove that Heisenberg and Affleck-Kennedy-Lieb-Tasaki chains can be controlled by operating on one of the spins at their ends. In principle, arbitrary quantum algorithms can be performed on such chains by acting on a single qubit. <s> BIB015
The above discussions on open-loop quantum control focus on coherent control where we can manipulate the state of a quantum system by applying semiclassical potentials in a fashion that preserves quantum coherence BIB001 . The control inputs commonly occur as tunable parameters in the Hamiltonian of the system and can directly affect the coherent part of the dynamics. There exist some physical situations where it is not possible or very difficult to control the state of a quantum system directly with coherent resources (unitary transformations) BIB008 . For example, it has been proven that a finite dimensional open quantum system with Markovian dynamics is not controllable using only coherent control BIB003 , BIB006 . A natural extension of these quantum control methods is to introduce incoherent resources (called "incoherent control") to enhance our capability of controlling quantum systems or to help in control design in order to accomplish specified tasks. The main paradigms for introducing incoherent resources into open-loop quantum control include incoherent control fields BIB005 , the use of auxiliary systems (environments) and quantum measurements BIB004 - BIB013 , BIB009 - BIB011 . Quantum measurements are regarded as deleterious in accomplishing coherent control tasks since they usually destroy the coherent characteristics of the quantum system. However, recent results show that quantum measurements can improve the controllability of quantum systems in some situations. Indeed, sometimes a quantum system which is not unitarily controllable can be controlled by the joint action of projective measurement and unitary evolution BIB004 , BIB007 . Let us consider the model (5), and let G(A) denote the Lie group generated by the operators {−iH 0 , −iH 1 , . . . , −iH K }. Also, let O(N ) be the N -dimensional orthogonal group. We then have the following theorem BIB004 : Theorem 9 Given any goal state |ψ f , there is a family of observables M (|ψ f ) such that measurement of one of these observables on any |ψ 0 plus unitary evolution The proof of this theorem can be found in BIB004 . If G(A) is equal to either U (N ) or sp( N 2 ), the corresponding system is pure state controllable and it might still be more efficient to use the measurement plus evolution strategy BIB004 . Several incoherent control schemes based on quantum measurements have been proposed. For example, a control scheme for mapping an unknown mixed quantum state onto a known pure state has been proposed with the help of sequential measurements of two noncommutative observables and without the use of unitary transformations BIB009 . A probabilistic quantum control strategy using indirect measurement is presented for the remote control of quantum systems BIB007 . Projective measurements on several identical initial states are used to help complete the control design of quantum systems with pure state controllable subspaces BIB014 . Projective measurements are also combined with amplitude amplification for the state control of locally controllable quantum systems BIB013 . In these incoherent control schemes, we emphasize the role of measurement as a control tool. Although we may use the measurement result, we present these methods as open-loop control strategies since no real closed-loop is constructed in the control process. Quantum measurements can be used as an effective control tool in quantum control. In practical applications, the realization of a quantum measurement on a quantum system is usually accomplished by entangling the quantum system with an auxiliary system (a probe). Hence, one can also control a quantum system by controlling an auxiliary system (e.g., a probe) entangled with the quantum system. Under suitable conditions on the interaction of the system to be controlled, the auxiliary system, and the environment, the system dynamics is completely controllable by varying the initial state of the auxiliary system BIB008 , BIB010 or manipulating the local Hamiltonian of the auxiliary system BIB012 , which has potential applications to quantum networks BIB015 . Furthermore, if all kinds of coherent and incoherent resources are available, in principle, we can control a (closed or open) quantum system from an arbitrary initial state to a predefined target state  , BIB002 .