Quantum control theory and applications: A survey <s> Controllability <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> BIB001 </s> Quantum control theory and applications: A survey <s> Controllability <s> Sufficient conditions for complete controllability of N-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Controllability of these model systems is of special interest since they have many applications in physics, e.g., Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Controllability <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> BIB003 </s> Quantum control theory and applications: A survey <s> Controllability <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> BIB004 </s> Quantum control theory and applications: A survey <s> Controllability <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> BIB005 </s> Quantum control theory and applications: A survey <s> Controllability <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> BIB006 </s> Quantum control theory and applications: A survey <s> Controllability <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> BIB007 </s> Quantum control theory and applications: A survey <s> Controllability <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> BIB008 </s> Quantum control theory and applications: A survey <s> Controllability <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> BIB009
The controllability of quantum systems is a fundamental theoretical notion in quantum control as well as having practical importance because of its close connection with the universality of quantum computation  and the possibility of attaining atomic or molecular scale transformations BIB001 , BIB007 . Different notions of controllability for quantum systems have been proposed and some notions have been studied in depth  . In this section, we will briefly present some aspects of controllability for quantum systems. A common research focus is on BLM of quantum systems for which the controllability criteria may be expressed in terms of the structure and rank of the corresponding Lie groups and Lie algebras  . We denote the N -dimensional complex unit sphere by S N C and the Lie algebra generated by the operators {−iH 0 , −iH 1 , . . . , −iH K } as L 0 . Also, we let U (N ) (SU (N )) represent the N -dimensional unitary group (special unitary group) and u(N ) (su(N )) be the corresponding Lie algebra. Furthermore, we let R(|ψ ) denote the reachable set of all states that are reachable from |ψ . Now, we give several definitions of controllability and their corresponding testing criteria. Definition 1 (Pure State Controllability)  The quantum system (5) is pure state controllable if for every pair of initial and final states, |ψ 0 and |ψ f in S N −1 C there exist control functions {u k (t)} and a time t > 0 such that the corresponding solution of (5) at time t, with initial condition |ψ 0 , is |ψ(t) = |ψ f . Definition 2 (Operator Controllability)  The quantum system (7) is operator controllable if there exist control functions {u k (t)} which drive the unitary operator U in (7) from I to U f , for any U f ∈ U (N ) (or SU (N )). Definition 3 (Eigenstate Controllability)  Suppose |φ 1 , . . . , |φ N are the N eigenstates of the free Hamiltonian H 0 . The quantum system (5) is eigenstate controllable if The quantum system (5) is pure state controllable if and only if the corresponding dynamical Lie algebra L 0 satisfies one of the following conditions: Theorem 5  The quantum system (7) is operator controllable if and only if From the above definitions and theorems, we can see that the notion of operator controllability is strongest of these quantum controllability notions and eigenstate controllability is weakest. If a quantum system is operator controllable, it is easy to show that it is also pure state controllable using  or by a comparison of Theorems 4 and 5. If a quantum system is pure state controllable, it follows from the definitions that it must also be eigenstate controllable. Note that pure state controllability is also called wavefunction controllability in some references (e.g., BIB003 , BIB006 , BIB009 ). Operator controllability, in the unitary case, is also called complete controllability BIB002 ,  . Some algorithms have been developed for testing the controllability of specific quantum systems BIB001 , BIB002 ,  ,  . The Lie algebra method allows for a straightforward mathematical treatment of closed quantum systems. However, the relevant criteria may be computationally difficult when the dimension of the controlled system is large. Hence, another method based on graph theory has been developed for pure state controllability for which the controllability criterion becomes easy to verify. For simplicity, we consider the quantum system (9) with only a single control input u(t); i.e., where the matrices A and B correspond to the free Hamiltonian and interaction Hamiltonian, respectively. Now, we associate the system with a non-oriented connectivity graph G = (V, E), where the set of vertices V corresponds to the set of the eigenstates |φ i and the set of edges E corresponds to the set of all pairs of eigenstates directly coupled by the matrix B = (B ij ), Let . . , K be connected components of this graph. We denote by λ i (i = 1, . . . , N ) the eigenvalues of the matrix A and let The following theorem provides a sufficient condition for pure state controllability in terms of the connectivity graph BIB003 , BIB006 : Theorem 6 The system (22) The proof of Theorem 6 can be found in BIB006 . The three conditions in Theorem 6 provide a sufficient but not a necessary condition for pure state controllability. In some circumstances, the assumptions (II) and (III) can be slightly relaxed BIB003 . Moreover, it is clear that if we add one more eigenstate into the set D, the pure state controllability criterion is easy to check for the resultant new system BIB003 . All of the above results consider finite dimensional quantum systems, which are a reasonable approximation to the quantum systems arising in many applications. Many practical quantum systems, especially those with continuous spectra, are essentially infinite dimensional; i.e., |ψ(t) in (5) is a unit vector in an infinite dimensional Hilbert space H. The controllability of such infinite dimensional quantum systems has been studied in  . For an infinite dimensional system (5), the existence of a dense analytic domain (in the sense of Nelson), together with some conditions on the corresponding Lie algebra, can guarantee its analytic controllability. The controllability of infinite dimensional quantum control systems has also been extensively studied in BIB007 ,  . Another interesting focus is on the controllability of open quantum systems. It has been proven that a finite dimensional open quantum system with Markovian dynamics (i.e., MME (13)) is not controllable when using only coherent control BIB005 . However, finite dimensional open quantum systems with Kraus-map dynamics are complete kinematic state controllable BIB008 and a specific Kraus map can be constructed for transformation from an arbitrary initial state to a predefined target state if some incoherent resources are available as control tools  , BIB004 . A control approach involving generating Kraus type dynamics can be potentially robust to variations of the initial system state BIB008 .