arXiv subjects

Igor G. Vladimirov

Publications and source records attributed to Igor G. Vladimirov.

At least 19 recordsLinked to original sources

arXiv2026

Renyi's Relative Entropy Lyapunov Functional for Convergence to Invariant Measure in Dissipative Stochastic Hamiltonian Systems

This paper is concerned with multivariable stochastic Hamiltonian systems governed by an ordinary differential equation for the position and an Ito stochastic differential equation for the momentum. The momentum dynamics involve both conservative and nonconservative forcing including Langevin viscous damping and the Wiener process as a random force. The setting allows for rotational degrees of freedom, a non-quadratic potential energy function and position-dependent mass, diffusion and damping matrices. In addition to modelling nonlinear physical dynamics subject to random forcing, the system is motivated by a stochastic optimisation viewpoint, where the search for global minima of the potential energy (for example, using the gradient descent) is carried out through convergence to a Boltzmann equilibrium measure which favours smaller values of this function. Under certain regularity conditions, we show that the $χ^2$-divergence or, equivalently, Renyi's second-order relative entropy with respect to the invariant measure has a nonpositive time derivative and thus provides a Lyapunov functional candidate for this convergence. However, this derivative vanishes at some moments of time, which is related to the behaviour of the probability distribution of the momentum conditioned on the position and has a link to a family of auxiliary quantum harmonic oscillator Hamiltonians parameterised by the position space of the underlying classical system. This position-momentum conditioning is used in order to prove that such pre-equilibrium break times in the entropy dissipation are isolated, thus making the $χ^2$-divergence a strictly decreasing Lyapunov functional for the Fokker-Planck-Kolmogorov equation which governs the joint position-momentum distribution in such systems.

math.PR↗

arXiv2026

Pointwise and dynamic programming control synthesis for finite-level open quantum memory systems

This paper is concerned with finite-level quantum memory systems for retaining initial dynamic variables in the presence of external quantum noise. The system variables have an algebraic structure, similar to that of the Pauli matrices, and their Heisenberg picture evolution is governed by a quasilinear quantum stochastic differential equation. The latter involves a Hamiltonian whose parameters depend affinely on a classical control signal in the form of a deterministic function of time. The memory performance is quantified by a mean-square deviation of quantum system variables of interest from their initial conditions. We relate this functional to a matrix-valued state of an auxiliary classical control-affine dynamical system. This leads to a pointwise control design where the control signal minimises the time-derivative of the mean-square deviation with an additional quadratic penalty on the control. In an alternative finite-horizon setting with a terminal-integral cost functional, we apply dynamic programming and obtain a quadratically nonlinear Hamilton-Jacobi-Bellman equation, for which a solution is outlined in the form of a recursively computed asymptotic expansion.

math.OC↗

arXiv2025

Measurement-based Initial Point Smoothing and Control Approach to Quantum Memory Systems

This paper is concerned with a quantum memory system for storing quantum information in the form of its initial dynamic variables in the presence of environmental noise. In order to compensate for the deviation from the initial conditions, the classical parameters of the system Hamiltonian are affected by the actuator output of a measurement-based classical controller. The latter uses an observation process produced by a measuring apparatus from the quantum output field of the memory system. The underlying system is modelled as an open quantum harmonic oscillator whose Heisenberg evolution is governed by linear Hudson-Parthasarathy quantum stochastic differential equations. The controller is organised as a classical linear time-varying system, so that the resulting closed-loop system has quantum and classical dynamic variables. We apply linear-quadratic-Gaussian control and fixed-point smoothing at the level of the first two moments and consider controllers with a separation structure which involve a continuously updated estimate for the initial quantum variables. The initial-point smoother is used for actuator signal formation so as to minimise the sum of a mean-square deviation of the quantum memory system variables at a given time horizon from their initial values and an integral quadratic penalty on the control signal.

quant-ph↗

arXiv2025

Quantum memory optimisation using finite-horizon, decoherence time and discounted mean-square performance criteria

This paper is concerned with open quantum memory systems for approximately retaining quantum information, such as initial dynamic variables or quantum states to be stored over a bounded time interval. In the Heisenberg picture of quantum dynamics, the deviation of the system variables from their initial values lends itself to closed-form computation in terms of tractable moment dynamics for open quantum harmonic oscillators and finite-level quantum systems governed by linear or quasi-linear Hudson-Parthasarathy quantum stochastic differential equations, respectively. This tractability is used in a recently proposed optimality criterion for varying the system parameters so as to maximise the memory decoherence time when the mean-square deviation achieves a given critical threshold. The memory decoherence time maximisation approach is extended beyond the previously considered low-threshold asymptotic approximation and to Schrödinger type mean-square deviation functionals for the reduced system state governed by the Lindblad master equation. We link this approach with the minimisation of the mean-square deviation functionals at a finite time horizon and with their discounted version which quantifies the averaged performance of the quantum system as a temporary memory under a Poisson flow of storage requests.

quant-ph↗

arXiv2025

Filtering and 1/3 Power Law for Optimal Time Discretisation in Numerical Integration of Stochastic Differential Equations

This paper is concerned with the numerical integration of stochastic differential equations (SDEs) which govern diffusion processes driven by a standard Wiener process. With the latter being replaced by a sequence of increments at discrete moments of time, we revisit a filtering point of view on the approximate strong solution of the SDE as an estimate of the hidden system state whose conditional probability distribution is updated using a Bayesian approach and Brownian bridges over the intermediate time intervals. For a class of multivariable linear SDEs, where the numerical solution is organised as a Kalman filter, we investigate the fine-grid asymptotic behaviour of terminal and integral mean-square error functionals when the time discretisation is specified by a sufficiently smooth monotonic transformation of a uniform grid. This leads to constrained optimisation problems over the time discretisation profile, and their solutions reveal a 1/3 power law for the asymptotically optimal grid density functions. As a one-dimensional example, the results are illustrated for the Ornstein-Uhlenbeck process.

eess.SY↗

arXiv2025

Optimization of partially isolated quantum harmonic oscillator memory systems by mean square decoherence time criteria

This paper is concerned with open quantum harmonic oscillators with position-momentum system variables, whose internal dynamics and interaction with the environment are governed by linear quantum stochastic differential equations. A recently proposed approach to such systems as Heisenberg picture quantum memories exploits their ability to approximately retain initial conditions over a decoherence horizon. Using the quantum memory decoherence time defined previously in terms of a fidelity threshold on a weighted mean-square deviation of the system variables from their initial values, we apply this approach to a partially isolated subsystem of the oscillator, which is not directly affected by the external fields. The partial isolation leads to an appropriate system decomposition and a qualitatively different short-horizon asymptotic behaviour of the deviation, which yields a longer decoherence time in the high-fidelity limit. The resulting approximate decoherence time maximization over the energy parameters for improving the quantum memory performance is discussed for a coherent feedback interconnection of such systems.

quant-ph↗

arXiv2025

Decoherence time maximization and partial isolation for open quantum harmonic oscillator memory networks

This paper considers a network of open quantum harmonic oscillators which interact with their neighbours through direct energy and field-mediated couplings and also with external quantum fields. The position-momentum dynamic variables of the network are governed by linear quantum stochastic differential equations associated with the nodes of a graph whose edges specify the interconnection of the component oscillators. Such systems can be employed as Heisenberg picture quantum memories with an engineered ability to approximately retain initial conditions over a bounded time interval. We use the quantum memory decoherence time defined previously in terms of a fidelity threshold on a weighted mean-square deviation for a subset (or linear combinations) of network variables from their initial values. This approach is applied to maximizing a high-fidelity asymptotic approximation of the decoherence time over the direct energy coupling parameters of the network. The resulting optimality condition is a set of linear equations for blocks of a sparse matrix associated with the edges of the direct energy coupling graph of the network. We also discuss a setting where the quantum network has a subset of dynamic variables which are affected by the external fields only indirectly, through a complementary ``shielding'' system. This holds under a rank condition on the network-field coupling matrix and can be achieved through an appropriate field-mediated coupling between the component oscillators. The partially isolated subnetwork has a longer decoherence time in the high-fidelity limit, thus providing a particularly relevant candidate for a quantum memory.

quant-ph↗

arXiv2024

Discrete-time Negative Imaginary Systems from ZOH Sampling

A new definition of discrete-time negative imaginary (NI) systems is provided. This definition characterizes the dissipative property of a zero-order hold sampled continuous-time NI system. Under some assumptions, asymptotic stability can be guaranteed for the closed-loop interconnection of an NI system and an output strictly negative imaginary system, with one of them having a one step advance. In the case of linear systems, we also provide necessary and sufficient frequency-domain and LMI conditions under which the definition is satisfied. Also provided is a simple DC gain condition for the stability results in the linear case.

eess.SY↗

arXiv2023

Position-momentum conditioning, relative entropy decomposition and convergence to equilibrium in stochastic Hamiltonian systems

This paper is concerned with a class of multivariable stochastic Hamiltonian systems whose generalised position is related by an ordinary differential equation to the momentum governed by an Ito stochastic differential equation. The latter is driven by a standard Wiener process and involves both conservative and viscous damping forces. With the mass, diffusion and damping matrices being position-dependent, the resulting nonlinear model of Langevin dynamics describes dissipative mechanical systems (possibly with rotational degrees of freedom) or their electromechanical analogues subject to external random forcing. We study the time evolution of the joint position-momentum probability distribution for the system and its convergence to equilibrium by decomposing the Fokker-Planck-Kolmogorov equation (FPKE) and the Kullback-Leibler relative entropy with respect to the invariant measure into those for the position distribution and the momentum distribution conditioned on the position. This decomposition reveals a manifestation of the Barbashin-Krasovskii-LaSalle principle and higher-order dissipation inequalities for the relative entropy as a Lyapunov functional for the FPKE.

math-ph↗

arXiv2023

Decoherence time control by interconnection for finite-level quantum memory systems

This paper is concerned with open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices for finite-level systems. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics according to a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This setting includes the zero-Hamiltonian isolated system dynamics as a particular case, where the system variables are constant in time, which makes them potentially applicable as a quantum memory. In a more realistic case of nonvanishing system-field coupling, we define a memory decoherence time when a mean-square deviation of the system variables from their initial values becomes relatively significant as specified by a weighting matrix and a fidelity parameter. We consider the decoherence time maximization over the energy parameters of the system and obtain a condition under which the zero Hamiltonian provides a suboptimal solution. This optimization problem is also discussed for a direct energy coupling interconnection of such systems.

quant-ph↗

arXiv2023

Decoherence time in quantum harmonic oscillators as quantum memory systems

This paper is concerned with open quantum harmonic oscillators (OQHOs) described by linear quantum stochastic differential equations. This framework includes isolated oscillators with zero Hamiltonian, whose system variables remain unchanged (in the Heisenberg picture of quantum dynamics) over the course of time, making such systems potentially applicable as quantum memory devices. In a more realistic case of system-environment coupling, we define a memory decoherence horizon as a typical time for a mean-square deviation of the system variables from their initial values to become relatively significant as specified by a weighting matrix and a fidelity parameter. We consider the maximization of the decoherence time over the energy and coupling matrices of the OQHO as a memory system in its storage phase and obtain a condition under which the zero Hamiltonian delivers a suboptimal solution. This optimization problem is also discussed for an interconnection of OQHOs.

quant-ph↗

arXiv2023

Quadratic-exponential coherent feedback control of linear quantum stochastic systems

This paper considers a risk-sensitive optimal control problem for a field-mediated interconnection of a quantum plant with a coherent (measurement-free) quantum controller. The plant and the controller are multimode open quantum harmonic oscillators governed by linear quantum stochastic differential equations, which are coupled to each other and driven by multichannel quantum Wiener processes modelling the external bosonic fields. The control objective is to internally stabilize the closed-loop system and minimize the infinite-horizon asymptotic growth rate of a quadratic-exponential functional which penalizes the plant variables and the controller output. We obtain first-order necessary conditions of optimality for this problem by computing the partial Frechet derivatives of the cost functional with respect to the energy and coupling matrices of the controller in frequency domain and state space. An infinitesimal equivalence between the risk-sensitive and weighted coherent quantum LQG control problems is also established. In addition to variational methods, we employ spectral factorizations and infinite cascades of auxiliary classical systems. Their truncations are applicable to numerical optimization algorithms (such as the gradient descent) for coherent quantum risk-sensitive feedback synthesis.

math.OC↗

arXiv2023

Entropy bounds for invariant measure perturbations in stochastic systems with uncertain noise

This paper is concerned with stochastic systems whose state is a diffusion process governed by an Ito stochastic differential equation (SDE). In the framework of a nominal white-noise model, the SDE is driven by a standard Wiener process. For a scenario of statistical uncertainty, where the driving noise acquires a state-dependent drift and thus deviates from its idealised model, we consider the perturbation of the invariant probability density function (PDF) as a steady-state solution of the Fokker-Planck-Kolmogorov equation. We discuss an upper bound on a logarithmic Dirichlet form for the ratio of the invariant PDF to its nominal counterpart in terms of the Kullback-Leibler relative entropy rate of the actual noise distribution with respect the Wiener measure. This bound is shown to be achievable, provided the PDF ratio is preserved by the nominal steady-state probability flux. The logarithmic Dirichlet form bound is used in order to obtain an upper bound on the relative entropy of the perturbed invariant PDF in terms of quadratic-exponential moments of the noise drift in the uniform ellipticity case. These results are illustrated for perturbations of Gaussian invariant measures in linear stochastic systems involving linear noise drifts.

math.OC↗

arXiv2023

Nonlinear Negative Imaginary Systems with Switching

In this paper, we extend nonlinear negative imaginary (NI) systems theory to switched systems. Switched nonlinear NI systems and switched nonlinear output strictly negative imaginary (OSNI) systems are defined. We show that the interconnection of two switched nonlinear NI systems is still switched nonlinear NI. The interconnection of a switched nonlinear NI system and a switched nonlinear OSNI system is asymptotically stable under some assumptions. This stability result is then illustrated using a numerical example.

eess.SY↗

arXiv2022

Probabilistic bounds with quadratic-exponential moments for quantum stochastic systems

This paper is concerned with quadratic-exponential moments (QEMs) for dynamic variables of quantum stochastic systems with position-momentum type canonical commutation relations. The QEMs play an important role for statistical ``localisation'' of the quantum dynamics in the form of upper bounds on the tail probability distribution for a positive definite quadratic function of the system variables. We employ a randomised representation of the QEMs in terms of the moment-generating function (MGF) of the system variables, which is averaged over its parameters using an auxiliary classical Gaussian random vector. This representation is combined with a family of weighted $L^2$-norms of the MGF, leading to upper bounds for the QEMs of the system variables. These bounds are demonstrated for open quantum harmonic oscillators with vacuum input fields and non-Gaussian initial states.

quant-ph↗

arXiv2022

Coherent quantum LQG controllers with Luenberger dynamics

This paper is concerned with the coherent quantum linear-quadratic-Gaussian control problem of minimising an infinite-horizon mean square cost for a measurement-free field-mediated interconnection of a quantum plant with a stabilising quantum controller. The plant and the controller are multimode open quantum harmonic oscillators, governed by linear quantum stochastic differential equations and coupled to each other and the external multichannel bosonic fields in the vacuum state. We discuss an interplay between the quantum physical realizability conditions and the Luenberger structure associated with the classical separation principle. This leads to a quadratic constraint on the controller gain matrices, which is formulated in the framework of a swapping transformation for the conjugate positions and momenta in the canonical representation of the controller variables. For the class of coherent quantum controllers with the Luenberger dynamics, we obtain first-order necessary conditions of optimality in the form of algebraic equations, involving a matrix-valued Lagrange multiplier.

quant-ph↗

arXiv2022

Measuring decoherence by commutation relations decay for quasilinear quantum stochastic systems

This paper considers a class of open quantum systems with an algebraic structure of dynamic variables, including the Pauli matrices for finite-level systems as a particular case. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics in the form of a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This quasilinearity leads to a tractable evolution of the two-point commutator matrix of the system variables (and their multi-point mixed moments in the case of vacuum input fields) involving time-ordered operator exponentials. The resulting exponential decay in the two-point commutation relations is a manifestation of quantum decoherence, caused by the dissipative system-field interaction and making the system lose specific unitary dynamics features which it would have in isolation from the environment. We quantify the decoherence in terms of the rate of the commutation relations decay and apply system theoretic and matrix analytic techniques, such as algebraic Lyapunov inequalities and spectrum perturbation results, to the study of the asymptotic behaviour of the related Lyapunov exponents in the presence of a small scaling parameter in the system-field coupling. These findings are illustrated for finite-level quantum systems (and their interconnections through a direct energy coupling) with multichannel external fields and the Pauli matrices as internal variables.

quant-ph↗

arXiv2022

Making Nonlinear Systems Negative Imaginary via State Feedback

This paper provides a state feedback stabilization approach for nonlinear systems of relative degree less than or equal to two by rendering them nonlinear negative imaginary (NI) systems. Conditions are provided under which a nonlinear system can be made a nonlinear NI system or a nonlinear output strictly negative imaginary (OSNI) system. Roughly speaking, an affine nonlinear system that has a normal form with relative degree less than or equal to two, after possible output transformation, can be rendered nonlinear NI and nonlinear OSNI. In addition, if the internal dynamics of the normal form are input-to-state stable, then there exists a state feedback input that stabilizes the system. This stabilization result is then extended to achieve stability for systems with a nonlinear NI uncertainty.

eess.SY↗