arXiv subjects

Igor G. Vladimirov

Publications and source records attributed to Igor G. Vladimirov.

At least 73 records · Page 4Linked to original sources

arXiv2015

A stochastic density matrix approach to approximation of probability distributions and its application to nonlinear systems

This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonormal basis functions with positive semi-definite Hermitian matrices of unit trace. Such matrices are called stochastic density matrices in order to reflect an analogy with the quantum mechanical density matrices. The SDM approximation of a PDF satisfies the normalization condition and is nonnegative everywhere in contrast to the truncated Gram-Charlier and Edgeworth expansions. For bases with an algebraic structure, such as the Hermite polynomial and Fourier bases, the SDM approximation can be chosen so as to satisfy given moment specifications and can be optimized using a quadratic proximity criterion. We apply the SDM approach to the Fokker-Planck-Kolmogorov PDF dynamics of Markov diffusion processes governed by nonlinear stochastic differential equations. This leads to an ordinary differential equation for the SDM dynamics of the approximating PDF. As an example, we consider the Smoluchowski SDE on a multidimensional torus.

math.PR↗

arXiv2015

A transverse Hamiltonian variational technique for open quantum stochastic systems and its application to coherent quantum control

This paper is concerned with variational methods for nonlinear open quantum systems with Markovian dynamics governed by Hudson-Parthasarathy quantum stochastic differential equations. The latter are driven by quantum Wiener processes of the external boson fields and are specified by the system Hamiltonian and system-field coupling operators. We consider the system response to perturbations of these energy operators and introduce a transverse Hamiltonian which encodes the propagation of the perturbations through the unitary system-field evolution. This provides a tool for the infinitesimal perturbation analysis and development of optimality conditions for coherent quantum control problems. We apply the transverse Hamiltonian variational technique to a mean square optimal coherent quantum filtering problem for a measurement-free cascade connection of quantum systems.

quant-ph↗

arXiv2015

Weyl variations and local sufficiency of linear observers in the mean square optimal coherent quantum filtering problem

This paper is concerned with the coherent quantum filtering (CQF) problem, where a quantum observer is cascaded in a measurement-free fashion with a linear quantum plant so as to minimize a mean square error of estimating the plant variables of interest. Both systems are governed by Markovian Hudson-Parthasarathy quantum stochastic differential equations driven by bosonic fields in vacuum state. These quantum dynamics are specified by the Hamiltonians and system-field coupling operators. We apply a recently proposed transverse Hamiltonian variational method to the development of first-order necessary conditions of optimality for the CQF problem in a larger class of observers. The latter is obtained by perturbing the Hamiltonian and system-field coupling operators of a linear coherent quantum observer along linear combinations of unitary Weyl operators, whose role here resembles that of the needle variations in the Pontryagin minimum principle. We show that if the observer is a stationary point of the performance functional in the class of linear observers, then it is also a stationary point with respect to the Weyl variations in the larger class of nonlinear observers.

quant-ph↗

arXiv2015

Covariance Dynamics and Entanglement in Translation Invariant Linear Quantum Stochastic Networks

This paper is concerned with a translation invariant network of identical quantum stochastic systems subjected to external quantum noise. Each node of the network is directly coupled to a finite number of its neighbours. This network is modelled as an open quantum harmonic oscillator and is governed by a set of linear quantum stochastic differential equations. The dynamic variables of the network satisfy the canonical commutation relations. Similar large-scale networks can be found, for example, in quantum metamaterials and optical lattices. Using spatial Fourier transform techniques, we obtain a sufficient condition for stability of the network in the case of finite interaction range, and consider a mean square performance index for the stable network in the thermodynamic limit. The Peres-Horodecki-Simon separability criterion is employed in order to obtain sufficient and necessary conditions for quantum entanglement of bipartite systems of nodes of the network in the Gaussian invariant state. The results on stability and entanglement are extended to the infinite chain of the linear quantum systems by letting the number of nodes go to infinity. A numerical example is provided to illustrate the results.

quant-ph↗

arXiv2015

Parameterization of Stabilizing Linear Coherent Quantum Controllers

This paper is concerned with application of the classical Youla-Kučera parameterization to finding a set of linear coherent quantum controllers that stabilize a linear quantum plant. The plant and controller are assumed to represent open quantum harmonic oscillators modelled by linear quantum stochastic differential equations. The interconnections between the plant and the controller are assumed to be established through quantum bosonic fields. In this framework, conditions for the stabilization of a given linear quantum plant via linear coherent quantum feedback are addressed using a stable factorization approach. The class of stabilizing quantum controllers is parameterized in the frequency domain. Also, this approach is used in order to formulate coherent quantum weighted $H_2$ and $H_\infty$ control problems for linear quantum systems in the frequency domain. Finally, a projected gradient descent scheme is proposed to solve the coherent quantum weighted $H_2$ control problem.

quant-ph↗

arXiv2015

Robust Mean Square Stability of Open Quantum Stochastic Systems with Hamiltonian Perturbations in a Weyl Quantization Form

This paper is concerned with open quantum systems whose dynamic variables satisfy canonical commutation relations and are governed by quantum stochastic differential equations. The latter are driven by quantum Wiener processes which represent external boson fields. The system-field coupling operators are linear functions of the system variables. The Hamiltonian consists of a nominal quadratic function of the system variables and an uncertain perturbation which is represented in a Weyl quantization form. Assuming that the nominal linear quantum system is stable, we develop sufficient conditions on the perturbation of the Hamiltonian which guarantee robust mean square stability of the perturbed system. Examples are given to illustrate these results for a class of Hamiltonian perturbations in the form of trigonometric polynomials of the system variables.

quant-ph↗

arXiv2015

A Gradient Descent Approach to Optimal Coherent Quantum LQG Controller Design

This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system. In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information. Both the plant and the controller are open quantum systems whose dynamic variables satisfy the canonical commutation relations (CCRs) of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to such oscillators, these QSDEs must satisfy physical realizability (PR) conditions, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time. The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation. We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account. A convergence analysis of the proposed algorithm is presented. A numerical example of a locally optimal CQLQG controller design is provided to demonstrate the algorithm performance.

quant-ph↗

arXiv2015

Quantized linear systems on integer lattices: a frequency-based approach

The roundoff errors in computer simulations of continuous dynamical systems, caused by finiteness of machine arithmetic, can lead to qualitative discrepancies between phase portraits of the resulting spatially discretized systems and the original systems. These effects can be modelled on a multidimensional integer lattice by using a dynamical system obtained by composing the transition operator of the original system with a quantizer. Such models manifest pseudorandomness which can be studied using a rigorous probability theoretic approach. To this end, the lattice $\mathbb{Z}^n$ is endowed with a class of frequency measurable subsets and a spatial frequency functional as a finitely additive probability measure on them. Using a multivariate version of Weyl's equidistribution criterion, we introduce an algebra of frequency measurable quasiperiodic subsets of the lattice. This approach is applied to quantized linear systems with the transition operator $R \circ L$, where $L$ is a nonsingular matrix of the original linear system in $\mathbb{R}^n$, and the map $R$ commutes with the additive group of translations of the lattice. For almost every $L$, the events associated with the deviation of trajectories of the quantized and original systems are frequency measurable quasiperiodic subsets of the lattice whose frequencies involve geometric probabilities on finite-dimensional tori. Using the skew products of measure preserving toral automorphisms, we prove mutual independence and uniform distribution of the quantization errors and investigate statistical properties of invertibility loss for the quantized linear system, extending V.V.Voevodin's results. When $L$ is similar to an orthogonal matrix, we establish a functional central limit theorem for the deviations of trajectories of the quantized and original systems. These results are demonstrated for rounded-off planar rotations.

math.PR↗

arXiv2014

Anisotropy-based optimal filtering in linear discrete time invariant systems

This paper is concerned with a problem of robust filtering for a finite-dimensional linear discrete time invariant system with two output signals, one of which is directly observed while the other has to be estimated. The system is assumed to be driven by a random disturbance produced from the Gaussian white noise sequence by an unknown shaping filter. The worst-case performance of an estimator is quantified by the maximum ratio of the root-mean-square (RMS) value of the estimation error to that of the disturbance over stationary Gaussian disturbances whose mean anisotropy is bounded from above by a given parameter $a \ge 0$. The mean anisotropy is a combined entropy theoretic measure of temporal colouredness and spatial "nonroundness" of a signal. We construct an $a$-anisotropic estimator which minimizes the worst-case error-to-noise RMS ratio. The estimator retains the general structure of the Kalman filter, though with modified state-space matrices. Computing the latter is reduced to solving a set of two coupled algebraic Riccati equations and an equation involving the determinant of a matrix. In two limiting cases, where $a = 0$ or $a \to +\infty$, the $a$-anisotropic estimator leads to the standard steady-state Kalman filter or the $H_{\infty}$-optimal estimator, respectively.

eess.SY↗

arXiv2014

A quantum mechanical version of Price's theorem for Gaussian states

This paper is concerned with integro-differential identities which are known in statistical signal processing as Price's theorem for expectations of nonlinear functions of jointly Gaussian random variables. We revisit these relations for classical variables by using the Frechet differentiation with respect to covariance matrices, and then show that Price's theorem carries over to a quantum mechanical setting. The quantum counterpart of the theorem is established for Gaussian quantum states in the framework of the Weyl functional calculus for quantum variables satisfying the Heisenberg canonical commutation relations. The quantum mechanical version of Price's theorem relates the Frechet derivative of the generalized moment of such variables with respect to the real part of their quantum covariance matrix with other moments. As an illustrative example, we consider these relations for quadratic-exponential moments which are relevant to risk-sensitive quantum control.

quant-ph↗

arXiv2014

Physical Realizability and Mean Square Performance of Translation Invariant Networks of Interacting Linear Quantum Stochastic Systems

This paper is concerned with translation invariant networks of linear quantum stochastic systems with nearest neighbour interaction mediated by boson fields. The systems are associated with sites of a one-dimensional chain or a multidimensional lattice and are governed by coupled linear quantum stochastic differential equations (QSDEs). Such interconnections of open quantum systems are relevant, for example, to the phonon theory of crystalline solids, atom trapping in optical lattices and quantum metamaterials. In order to represent a large-scale open quantum harmonic oscillator, the coefficients of the coupled QSDEs must satisfy certain physical realizability conditions. These are established in the form of matrix algebraic equations for the parameters of an individual building block of the network and its interaction with the neighbours and external fields. We also discuss the computation of mean square performance functionals with block Toeplitz weighting matrices for such systems in the thermodynamic limit per site for unboundedly increasing fragments of the lattice.

eess.SY↗

arXiv2013

Modified Associate Formalism without Entropy Paradox: Part I. Model Description

A Modified Associate Formalism is proposed for thermodynamic modelling of solution phases. The approach is free from the entropy paradox described by Lück et al. (Z. Metallkd. 80 (1989) pp. 270--275). The model is considered in its general form for an arbitrary number of solution components and an arbitrary size of associates. Asymptotic behaviour of chemical activities of solution components in binary dilute solutions is also investigated.

cond-mat.other↗

arXiv2013

Coherent Quantum Filtering for Physically Realizable Linear Quantum Plants

The paper is concerned with a problem of coherent (measurement-free) filtering for physically realizable (PR) linear quantum plants. The state variables of such systems satisfy canonical commutation relations and are governed by linear quantum stochastic differential equations, dynamically equivalent to those of an open quantum harmonic oscillator. The problem is to design another PR quantum system, connected unilaterally to the output of the plant and playing the role of a quantum filter, so as to minimize a mean square discrepancy between the dynamic variables of the plant and the output of the filter. This coherent quantum filtering (CQF) formulation is a simplified feedback-free version of the coherent quantum LQG control problem which remains open despite recent studies. The CQF problem is transformed into a constrained covariance control problem which is treated by using the Frechet differentiation of an appropriate Lagrange function with respect to the matrices of the filter.

quant-ph↗

arXiv2012

Minimum Relative Entropy State Transitions in Linear Stochastic Systems: the Continuous Time Case

This paper is concerned with a dissipativity theory for dynamical systems governed by linear Ito stochastic differential equations driven by random noise with an uncertain drift. The deviation of the noise from a standard Wiener process in the nominal model is quantified by relative entropy. We discuss a dissipation inequality for the noise relative entropy supply. The problem of minimizing the supply required to drive the system between given Gaussian state distributions over a specified time horizon is considered. This problem, known in the literature as the Schroedinger bridge, was treated previously in the context of reciprocal processes. A closed-form smooth solution is obtained for a Hamilton-Jacobi equation for the minimum required relative entropy supply by using nonlinear algebraic techniques.

math.OC↗

arXiv2012

Anisotropic Norm Bounded Real Lemma for Linear Discrete Time Varying Systems

We consider a finite horizon linear discrete time varying system whose input is a random noise with an imprecisely known probability law. The statistical uncertainty is described by a nonnegative parameter a which constrains the anisotropy of the noise as an entropy theoretic measure of deviation of the actual noise distribution from Gaussian white noise laws with scalar covariance matrices. The worst-case disturbance attenuation capabilities of the system with respect to the statistically uncertain random inputs are quantified by the a-anisotropic norm which is an appropriately constrained operator norm of the system. We establish an anisotropic norm bounded real lemma which provides a state-space criterion for the a-anisotropic norm of the system not to exceed a given threshold. The criterion is organized as an inequality on the determinants of matrices associated with a difference Riccati equation and extends the Bounded Real Lemma of the H-infinity-control theory. We also provide a necessary background on the anisotropy-based robust performance analysis.

eess.SY↗

arXiv2012

State distributions and minimum relative entropy noise sequences in uncertain stochastic systems: the discrete time case

The paper is concerned with a dissipativity theory and robust performance analysis of discrete-time stochastic systems driven by a statistically uncertain random noise. The uncertainty is quantified by the conditional relative entropy of the actual probability law of the noise with respect to a nominal product measure corresponding to a white noise sequence. We discuss a balance equation, dissipation inequality and superadditivity property for the corresponding conditional relative entropy supply as a function of time. The problem of minimizing the supply required to drive the system between given state distributions over a specified time horizon is considered. Such variational problems, involving entropy and probabilistic boundary conditions, are known in the literature as Schroedinger bridge problems. In application to control systems, this minimum required conditional relative entropy supply characterizes the robustness of the system with respect to an uncertain noise. We obtain a dynamic programming Bellman equation for the minimum required conditional relative entropy supply and establish a Markov property of the worst-case noise with respect to the state of the system. For multivariable linear systems with a Gaussian white noise sequence as the nominal noise model and Gaussian initial and terminal state distributions, the minimum required supply is obtained using an algebraic Riccati equation which admits a closed-form solution. We propose a computable robustness index for such systems in the framework of an entropy theoretic formulation of uncertainty and provide an example to illustrate this approach.

eess.SY↗

arXiv2012

Hardy-Schatten Norms of Systems, Output Energy Cumulants and Linear Quadro-Quartic Gaussian Control

This paper is concerned with linear stochastic control systems in state space. The integral of the squared norm of the system output over a bounded time interval is interpreted as energy. The cumulants of the output energy in the infinite-horizon limit are related to Schatten norms of the system in the Hardy space of transfer functions and the risk-sensitive performance index. We employ a novel performance criterion which seeks to minimize a combination of the average value and the variance of the output energy of the system per unit time. The resulting linear quadro-quartic Gaussian control problem involves the H2 and H4-norms of the closed-loop system. We obtain equations for the optimal controller and outline a homotopy method which reduces the solution of the problem to the numerical integration of a differential equation initialized by the standard linear quadratic Gaussian controller.

eess.SY↗

arXiv2012

Risk-sensitive Dissipativity of Linear Quantum Stochastic Systems under Lur'e Type Perturbations of Hamiltonians

This paper is concerned with a stochastic dissipativity theory using quadratic-exponential storage functions for open quantum systems with canonically commuting dynamic variables governed by quantum stochastic differential equations. The system is linearly coupled to external boson fields and has a quadratic Hamiltonian which is perturbed by nonquadratic functions of linear combinations of system variables. Such perturbations are similar to those in the classical Lur'e systems and make the quantum dynamics nonlinear. We study their effect on the quantum expectation of the exponential of a positive definite quadratic form of the system variables. This allows conditions to be established for the risk-sensitive stochastic storage function of the quantum system to remain bounded, thus securing boundedness for the moments of system variables of arbitrary order. These results employ a noncommutative analogue of the Doleans-Dade exponential and a multivariate partial differential version of the Gronwall-Bellman lemma.

quant-ph↗