Symmetric and symplectic ERKN methods for oscillatory Hamiltonian systems

The ERKN methods proposed by H. Yang et al. [Comput. Phys. Comm. 180 (2009) 1777] are an important improvement of J.M. Franco?s ARKN methods for perturbed oscillators [J.M. Franco, Comput. Phys. Comm. 147 (2002) 770]. This paper focuses on the symmetry and symplecticity conditions for ERKN methods solving oscillatory Hamiltonian systems. Two examples of symmetric and symplectic ERKN (SSERKN) methods of orders two and four respectively are constructed. The phase and stability properties of the new methods are analyzed. The results of numerical experiments show the robustness and competence of the SSERKN methods compared with some well-known methods in the literature.     

Stability radii of linear discrete-time systems and symplectic pencils

In this paper, we introduce and analyse robustness measures for the stability of discrete-time system x(t + 1) = Ax(t) under parameter perturbations of the form A→A + BDC where B,C are given matrices. In particular we characterize the stability radius of the uncertain system x(t + 1) = (A + BDC)x(t), D an unknown complex perturbation matrix, via an associated symplectic pencil and present an algorithm for the computation of that radius.     

Robust stability manifolds for multilinear interval systems

The stability of a class of multilinearly perturbed families of systems is considered. It is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved. The extremal property of these subsets is established. The results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces     

Poisson schemes for Hamiltonian systems on Poisson manifolds

In this paper, we construct Poisson difference schemes of any order accuracy based on Padé approximation for linear Hamiltonian systems on Poisson manifolds with constant coefficients. For nonlinear Hamiltonian systems on Poisson manifolds, we point out that symplectic diagonal implicit Runge-Kutta methods are also Poisson schemes. The preservation of distinguished functions and quadratic first integrals of the original Hamiltonian systems of these schemes are also discussed.     

Non-local stabilization of nonlinear systems using switching manifolds

The stabilization of nonlinear systems is considered by reducing the problem to a lower dimensional switching manifold which is made globally attracting. The switching manifold is designed using the stable manifold of the unforced system. The technique is first developed in local case and then in the global situation of nonlinear vector fields on manifolds. The method generalizes the standard L yapunov approach.     

Concurrence for infinite-dimensional quantum systems

Concurrence is an important entanglement measure for states in finite-dimensional quantum systems that was explored intensively in the last decade. In this paper, we extend the concept of concurrence to infinite-dimensional bipartite systems and show that it is continuous and does not increase under local operation and classical communication.     

Local controllability of quantum systems

We give a criterion that is sufficient for controllability of multipartite quantum systems. We generalize the graph infection criterion to the quantum systems that cannot be described with the use of a graph theory. We introduce the notation of hypergraphs and reformulate the infection property in this setting. The introduced criterion has a topological nature and therefore it is not connected to any particular experimental realization of quantum information processing.     

Formal energy of symplectic scheme for Hamiltonian systems and its applications (II)

In this paper, for 4-fold and 8-fold compositions of symplectic schemes, the authors obtain the formulae for calculation of the first three terms of the power series in stepsize of their formal energies. Utilizing the special properties of revertible schemes, the authors construct higher order revertible symplectic schemes for general Hamiltonian systems only through formal energies. From any order s (even) to order s + 2, a determining conclusion is obtained. And from any order s (even) to order s + 4, an algebraic equation system, when s is evaluated whose solution gives a rise by 4 in order, is about to be set up. As examples, for the cases of s = 2 and s = 4, the numerical results are to be gained.     

Comparison of quantum discord and relative entropy in some bipartite quantum systems

The study of quantum correlations in high-dimensional bipartite systems is crucial for the development of quantum computing. We propose relative entropy as a distance measure of correlations may be measured by means of the distance from the quantum state to the closest classical–classical state. In particular, we establish relations between relative entropy and quantum discord quantifiers obtained by means of orthogonal projection measurements. We show that for symmetrical X-states density matrices the quantum discord is equal to relative entropy. At the end of paper, various examples of X-states such as two-qubit and qubit–qutrit have been demonstrated.     

Processing images in entangled quantum systems

We introduce a novel method for storing and retrieving binary geometrical shapes in quantum mechanical systems. In contrast to standard procedures in classical computer science in which image reconstruction requires not only the storage of light parameters (like light frequency) but also the storage and use of additional information like correlation and pixel spatial disposition, we show that the employment of maximally entangled qubits allows to reconstruct images without using any additional information. Moreover, we provide a concrete application of our proposal in the field of image recognition and briefly explore potential experimental realizations. Our proposal could be employed to enable emergent quantum technology to be used in high-impact scientific disciplines in which extensive use of image processing is made.     

Technologies for trapped-ion quantum information systems

Scaling up from prototype systems to dense arrays of ions on chip, or vast networks of ions connected by photonic channels, will require developing entirely new technologies that combine miniaturized ion trapping systems with devices to capture, transmit, and detect light, while refining how ions are confined and controlled. Building a cohesive ion system from such diverse parts involves many challenges, including navigating materials incompatibilities and undesired coupling between elements. Here, we review our recent efforts to create scalable ion systems incorporating unconventional materials such as graphene and indium tin oxide, integrating devices like optical fibers and mirrors, and exploring alternative ion loading and trapping techniques.

     

Investigation of center manifolds of three-dimensional systems using computer algebra

For a family of three dimensional systems with center manifolds filled with closed trajectories (corresponding to periodic solutions of the system) we give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Bifurcations of critical periods of the system are studied as well. The study is performed using computer algebra systems MATHEMATICA and SINGULAR.     

Decompositions for control systems on manifolds with an affine connection

In this letter we present a decomposition for control systems whose drift vector field is the geodesic spray associated with an affine connection. With the geometric insight attained with this decomposition, we are able to easily prove some special results for this class of control systems. Examples illustrate the theory.     

Robust steering of n-level quantum systems

Robust open-loop steering of a finite-dimensional quantum system is a central problem in a growing number of applications of information engineering. In this note, we reformulate the problem in the classical control-theoretic setting, and provide a precise definition of robustness of the control strategy. We then discuss and compare some significant problems from nuclear magnetic resonance in the light of the given definition. We obtain quantitative results that are consistent with the qualitative ones available in the physics literature.     

Observer-based Hamiltonian identification for quantum systems

A symmetry-preserving observer-based parameter identification algorithm for quantum systems is proposed. Starting with a 2-level quantum system (qubit), where the unknown parameters consist of the atom-laser frequency detuning and coupling constant, we prove an exponential convergence result. The analysis is inspired by Lyapunov and adaptive control techniques and is based on averaging theory. The observer is then extended to the multi-level case where all the atom-laser coupling constants are unknown. The extension of the convergence analysis is discussed through some heuristic arguments. The relevance and the robustness with respect to various noises are tested through numerical simulations.     

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     

Analysis and design of integral sliding manifolds for systems with unmatched perturbations

The robustness properties of integral sliding-mode controllers are studied. This note shows how to select the projection matrix in such a way that the euclidean norm of the resulting perturbation is minimal. It is also shown that when the minimum is attained, the resulting perturbation is not amplified. This selection is particularly useful if integral sliding-mode control is to be combined with other methods to further robustify against unmatched perturbations. H/sub /spl infin// is taken as a special case. Simulations support the general analysis and show the effectiveness of this particular combination.     

Stabilization of relative equilibria for underactuated systems on Riemannian manifolds

This paper describes a systematic procedure to exponentially stabilize relative equilibria of mechanical systems. We review the notion of relative equilibria and their stability in a Riemannian geometry context. Potential shaping and damping control are employed to obtain full exponential stabilization of the desired trajectory. Two necessary conditions are that the effective potential be positive definite over a specified subspace and that the system be linearly controllable. Relevant applications to underwater and aerospace vehicle control are described.