the discriminant of a decomposable symplectic involution a formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4 as an application an alternative proof of shapiro s pfister factor conjecture is given for tensor products of at most five quaternion algebras

orbits of quantum states and geometry of bloch vectors for n level systems physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than 2 to shed some light on the complicated structure of the set of quantum states we consider a stratification with strata given by unitary orbit manifolds which can be identified with flag manifolds the results are applied to study the geometry of the coherence vector for n level quantum systems it is shown that the unitary orbits can be naturally identified with spheres in only for n 2 in higher dimensions the coherence vector only defines a non surjective embedding into a closed ball a detailed analysis of the three level case is presented finally a refined stratification in terms of symplectic orbits is considered

a new conjugate gradient method based on quasi newton equation for unconstrained optimization abstract the spectral conjugate gradient method is an effective method for large scale unconstrained optimization problems in this paper based on quasi newton direction and quasi newton equation a new spectral conjugate gradient method is proposed this method is motivated by the three term modified polak ribiyre polyak prp method and spectral parameters the global convergence of algorithm is proved for general functions under a strong wolfe line search numerical results show that the new algorithm is superior to the three term modified prp method

topological properties of reachable sets and the control of quantum bits abstract we consider the problem of controlling the state of a two level quantum system quantum bit via an externally applied electro magnetic field the describing model is a bilinear right invariant system whose state varies on the lie group of 2 2 special unitary matrices we study the topological structure of the reachable sets if two or more independent controls are used then every state can be achieved in arbitrary time however this is no longer true if only one control is available and in this case we give an exact characterization of states reachable in arbitrary time we prove small time local controllability for any state and the existence of a critical time which is the smallest time after which every transfer of state is possible we provide upper and lower bounds for such a time the mathematical development is motivated by the problem of manipulating the state of a quantum bit every transfer of state may be interpreted as a quantum logic operation and not every logic operation can be obtained in arbitrary time the analysis we present provides information about the feasibility of a given operation as well as estimates for the speed of a quantum computer

hamiltonian control of quantum dynamical semigroups stabilization and convergence speed we consider finite dimensional markovian open quantum systems and characterize the extent to which time independent hamiltonian control may allow to stabilize a target quantum state or subspace and optimize the resulting convergence speed for a generic lindblad master equation we introduce a dissipation induced decomposition of the associated hilbert space and show how it serves both as a tool to analyze global stability properties for given control resources and as the starting point to synthesize controls that ensure rapid convergence the resulting design principles are illustrated in realistic markovian control settings motivated by quantum information processing including quantum optical systems and nitrogen vacancy centers in diamond

a method for solving exact controllability problems governed by closed quantum spin systems the liouville von neumann master equation models closed quantum spin systems that arise in nuclear magnetic resonance applications in this paper an efficient and robust computational framework to solve exact controllability problems governed by the liouville von neumann master equation is presented the proposed control framework is based on a new optimisation formulation of exact controllability quantum spin problems that allows the application of efficient computational techniques this formulation results in an optimality system with four differential equations and an optimality condition the differential equations are approximated with an appropriate modified crank nicholson scheme and the resulting discretised optimality system is solved with a matrix free krylov newton scheme combined with a cascadic nonlinear conjugate gradient initialisation results of numerical experiments demonstrate the ability of the proposed framework to solve quantum spin exact controllability control problems

practical stabilization of a quantum particle in a one dimensional infinite square potential well we consider a nonrelativistic charged particle in a one dimensional infinite square potential well this quantum system is subjected to a control which is a uniform in space time depending electric field it is represented by a complex probability amplitude solution of a schrodinger equation on a one dimensional bounded domain with dirichlet boundary conditions we prove the almost global practical stabilization of the eigenstates by explicit feedback laws

transfer function approach to quantum control part i dynamics of quantum feedback systems this paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators a quantum optical closed loop including a plant and controller is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view in order to facilitate the use of control theory in microscopic world described by quantum theory in particular an input output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed this would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics this paper is divided into two parts the first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device cavity that can be regarded as a unit of quantum dynamics of bosonic modes the formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations the input output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems in the follow up paper the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory uncertainty principle e g and some applications are presented

minimum time frictionless atom cooling in harmonic traps frictionless atom cooling in harmonic traps is formulated as a time optimal control problem and a synthesis of optimal controlled trajectories is obtained

the lie algebra structure and controllability of spin systems in this paper we study the controllability properties and the lie algebra structure of networks of particles with spin immersed in an electro magnetic field we relate the lie algebra structure to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active for networks with different gyromagnetic ratios we provide a necessary and sufficient condition of controllability in terms of the properties of the above mentioned graph and describe the lie algebra structure in every case for these systems all the controllability notions including the possibility of driving the evolution operator and or the state are equivalent for general networks with possibly equal gyromagnetic ratios we give a sufficient condition of controllability a general form of interaction among the particles is assumed which includes both ising and heisenberg models as special cases assuming heisenberg interaction we provide an analysis of low dimensional cases number of particles less than or equal to three which includes necessary and sufficient controllability conditions as well as a study of their lie algebra structure this also provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not

finite controllability of infinite dimensional quantum systems quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates and their linear superpositions for some quantum systems such as spin systems the quantum evolution equation the schrodinger equation is finite dimensional and old results on controllability of systems defined on on lie groups and quotient spaces provide most of what is needed insofar as controllability of non dissipative systems is concerned however in an infinite dimensional setting controlling the evolution of quantum systems often presents difficulties both conceptual and technical in this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues in particular we analyze controllability for infinite dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre defined subspaces this result which we call the finite controllability theorem provides a set of sufficient conditions for controllability in an infinite dimensional setting we consider specific physical systems that are of interest for quantum computing and provide insights into the types of quantum operations gates that may be developed

small time controllability of systems on compact lie groups and spin angular momentum in this article we develop some general results on the properties of the reachable sets for right invariant bilinear systems with state varying on compact lie groups the main results consist of a characterization of the set of states reachable in arbitrary time from the identity of the group this under suitable assumptions is proved to be a lie subgroup of the underlying lie group we apply these results to the analysis of the controllability of particles with spin the results are motivated by and generalize the results in another work d d alessandro sys control lett 41 213 221 2000 where the specific model of a spin 12 particle system in an electro magnetic field was considered

brief paper discrete time controllability for feedback quantum dynamics controllability properties for discrete time markovian quantum dynamics are investigated we find that while in general the controlled system is not finite time controllable feedback control allows for arbitrary asymptotic state to state transitions under further assumptions on the form of the measurement we show that finite time controllability can be achieved in a time that is twice the dimension of the system and we provide an iterative procedure to design the unitary control actions

controllability aspects of quantum dynamics a unified approach for closed and open systems knowledge about to what extent quantum dynamical systems can be steered by coherent controls is indispensable for future developments in quantum technology the purpose of this paper is to analyze such controllability aspects for finite dimensional bilinear quantum control systems we use a unified approach based on lie algebraic methods from nonlinear control theory to revisit known and to establish new results for closed and open quantum systems in particular we provide a simplified characterization of different notions of controllability for closed quantum systems described by the liouville von neumann equation we derive new necessary and sufficient conditions for accessibility of open quantum systems modelled by the lindblad kossakowski master equation to this end we exploit a well studied topic of differential geometry namely the classification of all matrix lie groups which act transitively on the grassmann manifold or the punctured euclidean space for the special case of coupled spin 1 2 systems we obtain a remarkably simple characterization of accessibility these accessibility results correct and refine previous statements in the quantum control literature

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

growth of sobolev norms and controllability of the schrodinger equation in this paper we obtain a stabilization result for both linear and nonlinear schrodinger equations under generic assumptions on the potential then we consider the schrodinger equations with a potential which has a random time dependent amplitude we show that if the distribution of the amplitude is sufficiently non degenerate then any trajectory of the system is almost surely non bounded in sobolev spaces

controllability of quantum mechanical systems by root space decomposition of su n the controllability property of the unitary propagator of an n level quantum mechanical system subject to a single control field is described using the structure theory of semisimple lie algebras sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases

symmetry and subspace controllability for spin networks with a single node control we consider the relation of symmetries and subspace controllability for spin 1 2 networks with xxz couplings subject to perturbation of a single node by a local potential z control the hamiltonians for such networks decompose into excitation subspaces focusing on the single excitation subspace it is shown for single node z controls that external symmetries are characterized by eigenstates of the system hamiltonian which have zero overlap with the control node and there are no internal symmetries it is further shown that there are symmetries which persist even in the presence of random perturbations for xxz chains with uniform coupling strengths a characterization of all possible symmetries is given which shows a strong dependence on the position of the node we control we then show for heisenberg and xx chains with uniform coupling strength subject to single node z control that the lack of symmetry is both necessary and sufficient for subspace controllability finally the latter approach is generalized to establish controllability results for simple branched networks

parallel ssle algorithm for large scale constrained optimization in this paper a parallel ssle algorithm is proposed for solving large scale constrained optimization with block separable structure at each iteration the pvd sub problems are solved inexactly by the ssle algorithm which successfully overcomes the constraint inconsistency exited in most sqp type algorithm and decreases the computation amount as well without assuming the convexity of the constraints the algorithm is proved to be globally convergent to a kkt point of the original problem

almost any quantum logic gate is universal it is shown that if one can apply some hamiltonian repeatedly to a few variables at a time one can in general effect any desired unitary time evolution on an arbitrarily large number of variables as a result almost any quantum logic gate with two or more inputs is computationally universal in that copies of the gate can be wired together to effect any desired logic circuit and to perform any desired unitary transformation on a set of quantum variables

controllability properties for finite dimensional quantum markovian master equations 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

complete controllability of finite level quantum systems complete controllability is a fundamental issue in the field of control of quantum systems not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables in this paper we investigate the question of complete controllability for finite level quantum systems subject to a single control field for which the interaction is of dipole form sufficient criteria for complete controllability of a wide range of finite level quantum systems are established and the question of limits of complete controllability is addressed finally the results are applied to give a classification of complete controllability for four level systems

optimal control of quantum systems a projection approach this paper considers the optimal control of quantum systems the controlled quantum systems are described by the probability density matrix based liouville von neumann equation using projection operators the states of the quantum system are decomposed into two sub spaces namely the main state space and the remaining state space since the control energy is limited a solution for optimizing the external control force is proposed in which the main state is brought to the desired main state at a certain target time while the population of the remaining state is simultaneously suppressed in order to diminish its effects on the final population of the main state the optimization problem is formulated by maximizing a general cost functional of states and control force an efficient algorithm is developed to solve the optimization problem finally using the hydrogen fluoride hf molecular population transfer problem as an illustrative example the effectiveness of the proposed scheme for a quantum system initially in a mixed state or in a pure state is investigated through numerical simulations

indirect controllability of quantum systems a study of two interacting quantum bits a quantum mechanical system s is indirectly controlled when the control affects an ancillary system a and the evolution of s is modified through the interaction with a only a study of indirect controllability gives a description of the set of states that can be obtained for s with this scheme in this paper we study the indirect controllability of quantum systems in the finite dimensional case after discussing the relevant definitions we give a general necessary condition for controllability in lie algebraic terms we present a detailed treatment of the case where both systems s and a are two dimensional qubits in particular we characterize the dynamical lie algebra associated with s a extending previous results and prove that complete controllability of s a and an appropriate notion of indirect controllability are equivalent properties for this system we also prove several further indirect controllability properties for the system of two qubits and illustrate the role of the lie algebraic analysis in the study of reachable states

zero forcing linear and quantum controllability for systems evolving on networks we study the dynamics of systems on networks from a linear algebraic perspective the control theoretic concept of controllability describes the set of states that can be reached for these systems our main result says that controllability in the quantum sense expressed by the lie algebra rank condition and controllability in the sense of linear systems expressed by the controllability matrix rank condition are equivalent conditions we also investigate how the graph theoretic concept of a zero forcing set impacts the controllability property if a set of vertices is a zero forcing set the associated dynamical system is controllable these results open up the possibility of further exploiting the analogy between networks linear control systems theory and quantum systems lie algebraic theory this study is motivated by several quantum systems currently under study including continuous quantum walks modeling transport phenomena

non integrability of the optimal control problem for n level quantum systems we study the problem of optimal laser induced population transfer in n level quantum systems this problem can be represented as an energy optimal control problem related to a sub riemannian problem on and it is known boscain chambrion charlot gauthier that for n 2 and n 3 the hamiltonian system associated with the pontryagin maximum principle pmp is integrable we will show that this changes completely for specifically for n 4 we will prove that the adjoint equation of the pmp does not possess any meromorphic first integral independent of the hamiltonian on the levels of the casimir functions this implies that the system is not b integrable for any in proving our nonintegrability results we will use differential galois theory

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

optimal control of two level quantum systems we study the manipulation of two level quantum systems this research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two level quantum system a quantum bit we consider the problem of driving the evolution operator to a desired state while minimizing an energy type cost mathematically this problem translates into an optimal control problem for systems varying on the lie group of special unitary matrices of dimension two with cost that is quadratic in the control we develop a comprehensive theory of optimal control for two level quantum systems this includes in particular a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions the impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed

on quantum state observability and measurement we consider the problem of determining the state of a quantum system given one or more readings of the expectation value of an observable the system is assumed to be a finite dimensional quantum control system for which we can influence the dynamics by generating all the unitary evolutions in a lie group we investigate to what extent by an appropriate sequence of evolutions and measurements we can obtain information on the initial state of the system we present a system theoretic viewpoint of this problem in that we study the observability of the system in this context we characterize the equivalence classes of indistinguishable states and propose algorithms for state identification

on the convergence of asynchronous parallel algorithm for large scale linearly constrained minimization problem as a synchronization parallel framework the parallel variable transformation pvt algorithm is effective to solve unconstrained optimization problems in this paper based on the idea that a constrained optimization problem is equivalent to a differentiable unconstrained optimization problem by introducing the fischer function we propose an asynchronous pvt algorithm for solving large scale linearly constrained convex minimization problems this new algorithm can terminate when some processor satisfies terminal condition without waiting for other processors meanwhile it can enhances practical efficiency for large scale optimization problem global convergence of the new algorithm is established under suitable assumptions and in particular the linear rate of convergence does not depend on the number of processors

flatness based control of a single qubit gate this work considers the open loop control problem of steering a two level quantum system from any initial to any final condition the model of this system evolves on the state space having two inputs that correspond to the complex amplitude of a resonant laser field a symmetry preserving flat output is constructed using a fully geometric construction and quaternion computations simulation results of this flatness based open loop control are provided

a question on the discriminants of involutions of central division algebras let k be a field of characteristic not equal to 2 and d a finite dimensional central simple algebra over k whose class is 2 torsion in the brauer group br k of k by classical results of albert a ch x th 19 d admits an involution which is identity on k an invariant with values in k modulo squares in k called

universality in quantum computation we show that in quantum computation almost every gate that operates on two or more bits is a universal gate we discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates

robust incoherent control of qubit systems via switching and optimisation a robust incoherent quantum control scheme via projective measurements plus unitary transformations is proposed for driving a qubit system from an unknown initial mixed state to an arbitrary target pure state this scheme consists of two main steps projective measurement on the initial mixed state and optimal control between two pure states the first step projects the initial state into an eigenstate of the qubit system by projective measurement and guarantees that the proposed scheme is robust to different initial mixed states the second step finds a set of suitable optimal controls to drive the qubit system from the conditional eigenstate to the target pure state the connection between the two steps is accomplished by a switching strategy to accomplish the second step two approaches are presented in detail these approaches are time optimal transition with unbounded control and bang bang control with minimal switches the minimal time and minimal number of switches in these approaches can be calcu

on the problem of quantum control in infinite dimensions in the framework of bilinear control of the schrodinger equation it has been proved that the reachable set has a dense complement in hence in this setting exact quantum control in infinite dimensions is not possible on the other hand it is known that there is a simple choice of operators which when applied to an arbitrary state generate dense orbits in hilbert space compatibility of these two results is established in this paper and in particular it is proved that the closure of the reachable set of bilinear control is dense in the requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group

feedback stabilization of isospectral control systems on complex flag manifolds application to quantum ensembles the convex set of density operators of an n level quantum mechanical system foliated as a complex flag manifold where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix for an isospectral bilinear control system evolving on such an orbit the state feedback stabilization problem admits a natural lyapunov based time varying feedback design a global description of the domain of attraction of the closed loop system can be provided based on a ldquoroot spacerdquo like structure of the cone of density operators the converging conditions are time independent but depend on the topology of the flag manifold it is shown that the closed loop must have a number of equilibria at least equal to the euler characteristic of the manifold thus imposing topological obstructions to global stabilizability

the gradient flow for control of closed quantum systems the gradient flow is commonly used to numerically solve the problem of maximizing quantum observables for finite dimensional closed quantum systems in this note we analyze the asymptotic behavior of the gradient flow we show that under a regularity assumption on the controls the flow almost always converges to a solution of the maximization problem

implicit lyapunov control of finite dimensional schrodinger equations an implicit lyapunov based approach is proposed for generating trajectories of a finite dimensional controlled quantum system the main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable the controlled lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem the convergence analysis is done using lasalle invariance principle closed loop simulations illustrate the performance of such feedback laws for the open loop control of a test case considered by chemists

identification of dynamical lie algebras for finite level quantum control systems the problem of identifying the dynamical lie algebras of finite level quantum systems subject to external control is considered with special emphasis on systems that are not completely controllable in particular it is shown that the dynamical lie algebra for an n level system with symmetrically coupled transitions such as a system with equally spaced energy levels and uniform transition dipole moments is a subalgebra of so n if n 2 1 and a subalgebra of sp if n 2 general criteria for obtaining either so 2 1 or sp are established

newton methods for the optimal control of closed quantum spin systems an efficient and robust computational framework for solving closed quantum spin optimal control and exact controllability problems with control constraints is presented closed spin systems are of

controlabilite des systemes bilineaires this paper deals with bilinear systems a complementary result to the well known jurdjevic kupka criterion for controllability is obtained

notions of controllability for bilinear multilevel quantum systems in this note we define four different notions of controllability of physical interest for multilevel quantum mechanical systems these notions involve the possibility of driving the evolution operator as well as the state of the system we establish the connections among these different notions as well as methods to verify controllability

stabilization of stochastic quantum dynamics via open and closed loop control in this paper we investigate parametrization free solutions of the problem of quantum pure state preparation and subspace stabilization by means of hamiltonian control continuous measurement and quantum feedback in the presence of a markovian environment in particular we show that whenever suitable dissipative effects are induced either by the unmonitored environment or by non hermitian measurements there is no need for feedback as open loop time invariant control is sufficient to achieve stabilization of the target set in probability constructive necessary and sufficient conditions on the form of the control hamiltonian can be provided in this case when time invariant control is not sufficient state stabilization can be attained by the addition of filtering based feedback control

analysis of lyapunov method for control of quantum states the natural trajectory tracking problem is studied for generic quantum states represented by density operators a control design based on the hilbert schmidt distance as a lyapunov function is considered the control dynamics is redefined on an extended space where the lasalle invariance principle can be correctly applied even for non stationary target states lasalle s invariance principle is used to derive a general characterization of the invariant set which is shown to always contain the critical points of the lyapunov function critical point analysis of the latter is used to show that for generic states it is a morse function with n isolated critical points including one global minimum one global maximum and n 2 saddles it is also shown however that the actual dynamics of the system is not a gradient flow and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points this analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular and in fact convergence to the target state trajectory in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space on the other hand if either of these sufficient conditions is not satisfied the target state ceases to be asymptotically stable a center manifold emerges around the target state and the control design ceases to be effective

on the controllability of distributed systems to control a system is to make it behave hopefully according to our wishes in a way compatible with safety and ethics at the least possible cost the systems considered here are distributed i e governed modeled by partial differential equations pdes of evolution our wish is to drive the system in a given time by an adequate choice of the controls from a given initial state to a final given state which is the target if this can be achieved respectively if we can reach any neighborhood of the target the system with the controls at our disposal is exactly respectively approximately controllable a very general and fuzzy idea is that the more a system is unstable chaotic turbulent the simplest or the cheapest it is to achieve exact or approximate controllability when the pdes are the navier stokes equations it leads to conjectures which are presented and explained recent results reported in this expository paper essentially prove the conjectures in two space dimensions in three space dimensions a large number of new questions arise some new results support without proving the conjectures such as generic controllability and cases of decrease of cost of control when the instability increases short comments are made on models arising in climatology thermoelasticity non newtonian fluids and molecular chemistry the introduction of the paper and the first part of all sections are not technical many open questions are mentioned in the text

lyapunov control methods of closed quantum systems 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

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

controllability and simultaneous controllability of isospectral bilinear control systems on complex flag manifolds for isospectral bilinear control systems evolving on the so called complex flag manifolds i e on the orbits of the hermitian matrices under unitary conjugation action it is shown that controllability is almost always verified easy and generic sufficient conditions are provided the result applies to the problem of density operator controllability of finite dimensional quantum mechanical systems in addition we show that systems having different drifts corresponding for example to different larmor frequencies are simultaneously controllable by the same control field

criteria for reachability of quantum states we address the question of which quantum states can be inter converted under the action of a time dependent hamiltonian in particular we consider the problem as applied to mixed states and investigate the difference between pure and mixed state controllabilities introduced in previous work we provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group we also address the problem of which states can be prepared if the dynamical lie group is not sufficiently large to allow the system to be controllable

optimal population transfers in a quantum system for large transfer time transferring a quantum system to a final state with given populations is an important problem with applications to quantum chemistry and atomic physics in this paper we consider such transfers that minimize l2 the norm of the control this problem is challenging both analytically and numerically with the exception of the simplest cases there is no general understanding of the nature of optimal controls and trajectories we find that by examining the limit of large transfer times we can uncover such general properties in particular for transfer times large with respect to the time scale of the free dynamics of the quantum system the optimal control is a sum of terms each being a bohr frequency sinusoid modulated by a slow amplitude i e a profile that changes considerably only on the scale of the transfer time moreover we show that the optimal trajectory follows a ldquomeanrdquo evolution modulated by the fast free dynamics of the system the calculation of the ldquomeanrdquo optimal trajectory and the slow control profiles is done via an ldquoaveragedrdquo two point boundary value problem that we derive and which is much easier to solve than the one expressing the necessary conditions for optimality of the original optimal transfer problem

controllability for distributed bilinear systems this paper studies controllability of systems of the form dw dt mathcal a w p t mathcal b w where mathcal a is the infinitesimal generator of a c 0 semigroup of bounded linear operators e mathcal a t on a banach space x mathcal b x to x is a c 1 map and p in l 1 0 t mathbb r is a control the paper i gives conditions for elements of x to be accessible from a given initial state w 0 and ii shows that controllability to a full neighborhood in x of w 0 is impossible for dim x infty examples of hyperbolic partial differential equations are provided

wavefunction controllability for finite dimensional bilinear quantum systems 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

subspace controllability of bipartite symmetric spin networks abstract we consider a class of spin networks where each spin in a certain set interacts via ising coupling with a set of central spins and the control acts simultaneously on all the spins due to the permutation symmetries of the network the system is not globally controllable but it displays invariant subspaces of the underlying hilbert space the system is said to be subspace controllable if it is controllable on each of these subspaces we characterize the given invariant subspaces and the dynamical lie algebra of this class of systems and prove subspace controllability in every case

on the convergence of constrained parallel variable distribution algorithms we consider the parallel variable distribution pvd approach proposed by ferris and mangasarian siam j optim 4 1994 pp 815 832 for solving optimization problems the problem variables are distributed among p processors with each processor having the primary responsibility for updating its block of variables while allowing the remaining secondary variables to change in a restricted fashion along some easily computable directions for constrained nonlinear programs convergence in m c ferris and o l mangasarian siam j optim 4 1994 pp 815 832 was established in the special case of convex block separable constraints for general inseparable constraints it was suggested that a dual differentiable exact penalty function reformulation of the problem be used we propose to apply the pvd approach to problems with general convex constraints directly and show that the algorithm converges provided certain conditions are imposed on the change of secondary variables these conditions are both natural and practically implementable we also show that the original requirement of exact global solution of the parallel subproblems can be replaced by a less stringent sufficient descent condition the first rate of convergence result for the class of constrained pvd algorithms is also given

optimal control of one qubit gates we consider the problem of carrying an initial bloch vector to a final bloch vector in a specified amount of time under the action of three control fields a vector control field we show that this control problem is solvable and therefore it is possible to optimize the control we choose the physically motivated criteria of minimum energy expended in the control minimum magnitude of the rate of change of the control and a combination of both we find exact analytical solutions determine the fields for a general one qubit gate and use the y gate as an example we argue that in the case of less than three controls only the physical intuition does not provide a straight reasonable solution and solve the problem in the case of a unique control minimizing the energy consumption

non stationary quantum walks on the cycle we consider quantum walks on the cycle in the non stationary case where the coin operation is allowed to change at each time step we characterize in algebraic terms the set of possible state transfers and prove that as opposed to the stationary case the associate probability distribution may converge to a uniform distribution among the nodes of the associated graph

parallel variable distribution an approach is presented here for solving optimization problems in which the variables are distributed among p processors each processor has primary responsibility for updating its own block of variables in parallel while allowing the remaining variables to change in a restricted fashion e g along a steepest descent quasi newton or any arbitrary direction this forget me not approach is a distinctive feature of the algorithm that has not been analyzed before the parallelization step is followed by a fast synchronization step wherein the affine hull of the points computed by the parallel processors and the current point is searched for an optimal point convergence to a stationary point under continuous differentiability is established for the unconstrained case as well as a linear convergence rate under the additional assumption of a lipschitzian gradient and strong convexity for problems constrained to lie in the cartesian product of closed convex sets convergence is established to a point sa

control landscapes are almost always trap free a geometric assessment a proof is presented that almost all closed finite dimensional quantum systems have trap free i e free from local optima landscapes for a large and physically general class of circumstances which includes qubit evolutions in quantum computing this result offers an explanation for why gradient based methods succeed so frequently in quantum control the role of singular controls is analyzed using geometric tools in the case of the control of the propagator and thus in the case of observables as well singular controls have been implicated as a source of landscape traps the conditions under which singular controls can introduce traps and thus interrupt the progress of a control optimization are discussed and a geometrical characterization of the issue is presented it is shown that a control being singular is not sufficient to cause control optimization progress to halt and sufficient conditions for a trap free landscape are presented it is further shown that the local surjectivity full rank assumption of landscape analysis can be refined to the condition that the end point map is transverse to each of the level sets of the fidelity function this mild condition is shown to be sufficient for a quantum system s landscape to be trap free the control landscape is shown to be trap free for all but a null set of hamiltonians using a geometric technique based on the parametric transversality theorem numerical evidence confirming this analysis is also presented this new result is the analogue of the work of altifini wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation these collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps

controllability of open quantum systems with kraus map dynamics 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

rapid lyapunov control of finite dimensional quantum systems abstract rapid state control of quantum systems is significant in reducing the influence of relaxation or decoherence caused by the environment and enhancing the capability in dealing with uncertainties in the model and control process bang bang lyapunov control can speed up the control process but cannot guarantee convergence to a target state this paper proposes two classes of new lyapunov control methods that can achieve rapidly convergent control for quantum states one class is switching lyapunov control where the control law is designed by switching between bang bang lyapunov control and standard lyapunov control the other class is approximate bang bang lyapunov control where we propose two special control functions which are continuously differentiable and yet have a bang bang type property related stability results are given and a construction method for the degrees of freedom in the lyapunov function is presented to guarantee rapid convergence to a target eigenstate being isolated in the invariant set several numerical examples demonstrate that the proposed methods can achieve improved performance for rapid state control of quantum systems

a parallel algorithm for a class of convex programs a parallel algorithm is proposed in this paper for solving the problem min q x x in c 1 cap cdots cap c m where q is an uniformly convex function and c i are closed convex sets in r n in each iteration of the method we solve in parallel m independent subproblems each minimizing a definite quadratic function over an individual set c i the method has attractive convergence properties and can be implemented as parallel algorithms for tackling definite quadratic programs linear programs systems of linear equations and systems of generalized nonlinear inequalities

model identification for spin networks abstract we consider the problem of determining the unknown parameters of the hamiltonian of a network of spin 1 2 particles in particular we study experiments in which the system is driven by an externally applied electro magnetic field and the expectation value of the total magnetization is measured under appropriate system theoretic assumptions we prove that if it is possible to prepare the system in a known initial state the above experiment allows to identify the parameters of the hamiltonian in the case where the initial state is itself an unknown parameter we characterize all the pairs hamiltonian initial state which give the same value of the magnetization for every form of the driving electro magnetic field the analysis is motivated by recent results on the isospectrality of hamiltonians describing magnetic molecules

lyapunov control of bilinear schrodinger equations 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

degrees of controllability for quantum systems and application to atomic systems precise definitions for different degrees of controllability for quantum systems are given and necessary and sufficient conditions for each type of controllability are discussed the results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies