Decoherence-Free Linear Quantum Subsystems

Linear quantum systems: A tutorial

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics are symmetric and uniformly elliptical and by using the properties of the Mittag–Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.

In this paper, the synthesis of robust memory modes for linear quantum passive systems in the presence of unknown inputs has been studied, aimed at facilitating secure storage and communication of quantum information. In particular, we can switch on decoherence-free (DF) modes in the storage stage by placing the poles on the imaginary axis via a coherent feedback control scheme, and these memory modes can further be simultaneously made robust against perturbations to the system parameters by minimizing the condition number associated with imaginary poles. The DF modes can also be switched off by tuning the controller parameters to place the poles in the left half of the complex plane in the writing/reading stage. We develop explicit algebraic conditions guiding the design of such a coherent quantum controller, which involves employing an augmented system model to counter the influence of unknown inputs. Examples are provided to illustrate the procedure of synthesizing robust memory modes for linear optical quantum systems.

On Controllability of Infinite Dimensional Linear Stochastic Systems

In this paper we attempt to describe a circle of ideas which makes possible a unified exposition of a large part of linear algezbra, operator theory and both finite and infinite dimensional multivariable linear systems. As the title suggests the unifying conzcepts will be those of functional models, module theory and various factorizations of polynomial and analytic matrix valued functions. The full exposition of the ideas presented here will be the theme of a forthcoming monograph [20].

We consider the infinite dimensional linear control system described by the population dynamics model of Lotka-McKendrick with spatial diffusion. Considering control functions localized with respect to the spatial variable but active for all ages, we prove that the whole population can be steered to zero in any positive time. The main novelty we bring is that, unlike the existing results in the literature, we can also control the population of ages very close to 0. Another novelty brought in is the employed methodology: as far as we know, the present work is the first one remarking that the null controllability of the considered system can be obtained by using the Lebeau-Robbiano strategy, originally developed for the null-controllability of the heat equation.

The question of power and asymptotic stability of infinite dimensional discrete-time state space systems is investigated. It is shown that every balanced realization is asymptotically stable. Conditions are given for balanced, input normal, or output normal realization to be asymptotically and/or power stable. >

Well-posedness of infinite-dimensional linear systems with nonlinear feedback

PurposeThe purpose of this paper is to present a procedure for approximating DC operating points of nonlinear circuits. The presented approach can also be applied in case of multiple DC operating points.Design/methodology/approachA generalized Carleman linearization is used, which transforms an algebraic nonlinear equation into an equivalent infinite-dimensional linear system. In general, no close-form solution can be given for the infinite-dimensional linear system. Hence, the infinite-dimensional linear system is approximated by a finite one over a predefined interval using a self-consistent technique. The presented procedure allows to approximate all possible DC operating points within a predefined interval. To isolate all DC operating points, the initial interval is gradually divided into subintervals.FindingsIt is shown that the presented approach is not restricted to the polynomial case and allows to approximate all DC operating points. The presented approach can be applied in case of multiple DC operating points and does not depend on the domain of attraction of the DC operating points.Originality/valueA new procedure for the approximation of DC operating points of nonlinear circuits based on a generalized Carleman linearization is presented. This approach can be applied in case of multiple DC operating points and is independent of the domain of attraction. Further, this generalized approach is not restricted to the polynomial case and can be applied to a variety of circuits.

We consider the problem of minimizing the supplied energy of infinite-dimensional linear port-Hamiltonian systems and prove that optimal trajectories exhibit the turnpike phenomenon towards certain subspaces induced by the dissipation of the dynamics. The theoretical foundations are illustrated by means of numerical examples concerning a Timoshenko beam and the heat equation.

This paper proposes a simple and rigorous Fourier-matching method to study transverse-magnetic-polarized electro-magnetic resonances by a perfectly conducting slab with a finite number of subwavelength slits of width $h\ll 1$. Since variable separation is applicable in the region outside the slits, by Fourier transforming its governing equation, a magnetic field can be represented in terms of its derivative on the aperture. Next, inside each slit where variable separation is still available, the field can be represented as a Fourier series in terms of a countable set of basis functions with unknown Fourier coefficients. Finally, by matching the two subdomain representations on the aperture, we establish a linear system of an infinite number of equations governing the countable Fourier coefficients; the unknowns are further rescaled to be in the standard $\ell^2$ space. By the asymptotic expansion of each entry of the coefficient matrix, we rigorously show that its certain principal submatrix is invertible so that the infinite-dimensional linear system can be reduced to a finite-dimensional linear system. Resonance frequencies are exactly those frequencies making the linear system rank-deficient. This in turn leads to an asymptotic formula of accuracy ${\cal O}(h^3\log h)$ for computing the resonance frequencies. We emphasize that the new formula is more accurate than all existing results and is the first formula for slits of number more than two to the best of our knowledge. Numerical experiments are carried out finally to validate the proposed formula and demonstrate its accuracy.

A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The error of the dynamic iteration is convergent to 0 and subject to a effective decreasing bound. No stability condition is required and the method is in particular applicable to port-Hamiltonian formulations arising from domain decompositions.

In this paper, we propose a measurement nonlinear feedback control scheme to generate Wigner-function negativity in an optical cavity having dynamics described as a linear quantum system. In general, linear optical quantum systems can be easily constructed with reliable devices; therefore, the idea of constructing the entire system with such an optical system and nonlinear feedback is reasonable for generating Wigner-function negativity. However, existing studies have insufficiently examined the realizability or actual implementation of feedback control, which essentially requires fast responses from the sensors and actuators. In order to solve this problem, we consider the realizable feedback control of the optical phase of a pumping beam supplied to a cavity by using electro-optical modulation, which can be utilized as a fast control actuator. Then, we introduce mathematical models of the feedback-controlled system and evaluate its effect on the generation of the Wigner-function negativity by using numerical simulation. Through various numerical simulations, we show that the proposed feedback control can effectively generate the negativity of the Wigner function.

This paper studies finite- and infinite-dimensional linear control systems of the form ${{df} / {dt}} = Af + Bu$, where A is the infinitesimal generator of a $C_0 $-semigroup that preserves a cone C, and where $Bu$ takes values in C. Since the reachable states are all in C, the system is not controllable in the usual sense. Of concern is “positive controllability,” which means that the entire cone C can be (approximately) reached. It turns out that positive controllability is rather difficult to achieve but that for stable systems an important subclass of states can be reached. Different examples are provided.

We establish left and right canonical factorizations of Hilbert-space operator-valued functions G ( z ) that are analytic on neighborhoods of the complex unit circle $${\mathbb {T}}$$ T and the origin 0 and that have the form $$G(z)=I+F(z)$$ G ( z ) = I + F ( z ) with F ( z ) taking strictly contractive values on $${\mathbb {T}}$$ T . Such functions can be realized as transfer functions of infinite dimensional dichotomous discrete-time linear systems, and we employ the strict bounded real lemma for this class of operators, together with associated Kreĭn space theory, to derive explicit formulas for the left and right canonical factorizations.