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Papers by Igor G Vladimirov

Research paper thumbnail of Frequency-Domain Computation of Quadratic-Exponential Cost Functionals for Linear Quantum Stochastic Systems

Research paper thumbnail of Measurement-based Feedback Control of Linear Quantum Stochastic Systems with Quadratic-Exponential Criteria

Research paper thumbnail of Parametric Randomization, Complex Symplectic Factorizations, and Quadratic-Exponential Functionals for Gaussian Quantum States

Infinite Dimensional Analysis, Quantum Probability and Related Topics

This paper combines probabilistic and algebraic techniques for computing quantum expectations of ... more This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states. Such quadratic-exponential functionals (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive filtering and control problems for linear quantum stochastic systems. We employ a Lie-algebraic correspondence between complex symplectic matrices and quadratic-exponential functions of system variables of a quantum harmonic oscillator. The complex symplectic factorizations are used together with a parametric randomization of the quasi-characteristic or moment-generating functions according to an auxiliary classical Gaussian distribution. This reduces the QEF to an exponential moment of a quadratic form of classical Gaussian random variables with a complex symmetric matrix and is applicab...

Research paper thumbnail of Effects of Parametric Uncertainties in Cascaded Open Quantum Harmonic Oscillators and Robust Generation of Gaussian Invariant States

SIAM Journal on Control and Optimization

This paper is concerned with the generation of Gaussian invariant states in cascades of open quan... more This paper is concerned with the generation of Gaussian invariant states in cascades of open quantum harmonic oscillators governed by linear quantum stochastic differential equations. We carry out infinitesimal perturbation analysis of the covariance matrix for the invariant Gaussian state of such a system and the related purity functional subject to inaccuracies in the energy and coupling matrices of the subsystems. This leads to the problem of balancing the state-space realizations of the component oscillators through symplectic similarity transformations in order to minimize the mean square sensitivity of the purity functional to small random perturbations of the parameters. This results in a quadratic optimization problem with an effective solution in the case of cascaded one-mode oscillators, which is demonstrated by a numerical example. We also discuss a connection of the sensitivity index with classical statistical distances and outline infinitesimal perturbation analysis for translation invariant cascades of identical oscillators. The findings of the paper are applicable to robust state generation in quantum stochastic networks.

Research paper thumbnail of A numerical approach to optimal coherent quantum LQG controller design using gradient descent

Automatica

This paper is concerned with coherent quantum linear quadratic Gaussian (CQLQG) control. The prob... more This paper is concerned with coherent quantum linear quadratic Gaussian (CQLQG) control. The problem is to find a stabilizing measurement-free quantum controller for a quantum plant so as to minimize a mean square cost for the fully quantum closed-loop system. The plant and controller are open quantum systems interconnected through bosonic quantum fields. In comparison with the observation-actuation structure of classical controllers, coherent quantum feedback is less invasive to the quantum dynamics. The plant and controller 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 in the form of quadratic constraints on the state-space matrices, reflecting the CCR preservation in time. The symmetry of the problem is taken into account by introducing equivalence classes of coherent quantum controllers generated by symplectic similarity transformations. We discuss a modified gradient flow, which is concerned with norm-balanced realizations of controllers. A line-search gradient descent algorithm with adaptive stepsize selection is proposed for the numerical solution of the CQLQG control problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing coherent quantum controller, thus taking the PR constraints into account. A convergence analysis of the algorithm is presented. Numerical examples of designing locally optimal CQLQG controllers are provided in order to demonstrate the algorithm performance.

Research paper thumbnail of Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action*

2016 IEEE Conference on Norbert Wiener in the 21st Century (21CW), 2016

Research paper thumbnail of A phase-space formulation of the Belavkin-Kushner-Stratonovich filtering equation for nonlinear quantum stochastic systems*

2016 IEEE Conference on Norbert Wiener in the 21st Century (21CW), 2016

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic syst... more This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We also discuss a more specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.

Research paper thumbnail of Robust Filtering for Uncertain Nonlinear Systems Satisfying a Sum Quadratic Constraint

ABSTRACT This paper develops a robust extended Kalman filter for nonlinear uncertain systems with... more ABSTRACT This paper develops a robust extended Kalman filter for nonlinear uncertain systems with a deterministic description of noise and uncertainty. The system state dynamics are formulated in reverse time and the uncertainties are modeled in terms of sum quadratic constraints. The robust filtering problem is formulated as a set-valued state estimation problem which is recast into an optimal control problem. The solution of the resulting Hamilton-Jacobi-Bellman equation is obtained by using a quadratic approximation. This leads to an approximate information state for the filtering problem which is computed recursively through a difference Riccati equation derived by linearizing the observation equation and using a quadratic approximation to the system dynamics. The reduction of the robust estimator to the standard Kalman filter in the uncertainty-free linear case is also discussed.

Research paper thumbnail of Minimum Relative Entropy State Transitions in Discrete Time Systems with Statistically Uncertain Noise

49Th Ieee Conference on Decision and Control, 2010

We develop a stochastic dissipativity theory for discrete-time systems driven by an uncertain ran... more We develop a stochastic dissipativity theory for discrete-time systems driven by an uncertain random noise. The deviation of the unknown probability law of the noise from a nominal white noise distribution is quantified by the conditional relative entropy given the initial state of the system. We establish a dissipation inequality and superadditivity property for the conditional relative entropy supply. The problem of minimizing the supply required to drive the system between given state distributions over a specified time horizon is considered. We obtain a dynamic programming Bellman equation for the minimum required relative entropy supply and show that the optimal noise is Markov with respect to the state of the system. For linear systems with Gaussian nominal noise and Gaussian initial and terminal state distributions, computing the minimum required supply is reduced to solving an algebraic Riccati equation.

Research paper thumbnail of Covariance dynamics and entanglement in translation invariant linear quantum stochastic networks

2015 54th IEEE Conference on Decision and Control (CDC), 2015

This paper is concerned with a translation invariant network of identical quantum stochastic syst... more 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 (QSDEs). The dynamic variables of the network satisfy the canonical commutation relations (CCRs). 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.

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

2015 IEEE Conference on Control Applications (CCA), 2015

This paper outlines an approach to the approximation of probability density functions by quadrati... more 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.

Research paper thumbnail of A transverse hamiltonian variational technique for open quantum stochastic systems and its application to coherent quantum control

2015 IEEE Conference on Control Applications (CCA), 2015

This paper is concerned with variational methods for nonlinear open quantum systems with Markovia... more 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 their propagation 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 technique to a mean square optimal filtering problem for a measurement-free cascade connection of quantum systems.

Research paper thumbnail of 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 ... more 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.

Research paper thumbnail of A Dynamic Programming Approach to Finite-horizon Coherent Quantum LQG Control

2011 Australian Control Conference, 2011

The paper is concerned with the coherent quantum Linear Quadratic Gaussian (CQLQG) control proble... more The paper is concerned with the coherent quantum Linear Quadratic Gaussian (CQLQG) control problem for time-varying quantum plants governed by linear quantum stochastic differential equations over a bounded time interval. A controller is sought among quantum linear systems satisfying physical realizability (PR) conditions. The latter describe the dynamic equivalence of the system to an open quantum harmonic oscillator and relate its state-space matrices to the free Hamiltonian, coupling and scattering operators of the oscillator. Using the Hamiltonian parameterization of PR controllers, the CQLQG problem is recast into an optimal control problem for a deterministic system governed by a differential Lyapunov equation. The state of this subsidiary system is the symmetric part of the quantum covariance matrix of the plant-controller state vector. The resulting covariance control problem is treated using dynamic programming and Pontryagin's minimum principle. The associated Hamilton-Jacobi-Bellman equation for the minimum cost function involves Frechet differentiation with respect to matrix-valued variables. The gain matrices of the CQLQG optimal controller are shown to satisfy a quasi-separation property as a weaker quantum counterpart of the filtering/control decomposition of classical LQG controllers. Keywords Quantum control • LQG cost • Physical realizability • Symplectic invariance • Dynamic programming • Pontryagin minimum principle • Frechet differentiation Mathematics Subject Classification (2000) 81Q93 • 81S25 • 93E20 • 49J50 • 58C20 The work is supported by the Australian Research Council.

Research paper thumbnail of Propagation of Mean Anisotropy of Signals in Filter Connections

Proceedings of the 17th IFAC World Congress, 2008, 2008

The anisotropy-based approach to robust control in stochastic systems occupies a unifying interme... more The anisotropy-based approach to robust control in stochastic systems occupies a unifying intermediate position between the H 2 and H ∞-optimal control theories. Initiated at the interface of Information Theory and Robust Control about fourteen years ago, the approach employs the a-anisotropic norm of a linear system as its worst-case sensitivity to input random disturbances whose mean anisotropy is bounded by a nonnegative parameter a. The latter quantifies the temporal "colouredness" and spatial "non-roundness" of the signal by its minimal relative entropy production rate with respect to Gaussian white noises with scalar covariance matrices. Revisiting the underlying definitions, the paper emphasizes the role of feedback in the construct of mean anisotropy of signals and discusses propagation of the latter through various filter connections. The results can be used to support physical and engineering intuition for a "rational" choice of the mean anisotropy level a in the design of anisotropy-based robust controllers.

Research paper thumbnail of A gradient descent approach to optimal coherent quantum LQG controller design

2015 American Control Conference (ACC), 2015

This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control probl... more 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.

Research paper thumbnail of Parameterization of stabilizing linear coherent quantum controllers

2015 10th Asian Control Conference (ASCC), 2015

This paper is concerned with application of the classical Youla-Kučera parameterization to findin... more 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 ∞ 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.

Research paper thumbnail of Decentralized Coherent Quantum Control Design for Translation Invariant Linear Quantum Stochastic Networks with Direct Coupling

Research paper thumbnail of Robust filtering for continuous-time uncertain nonlinear systems with an integral quadratic constraint

2012 American Control Conference (ACC), 2012

This paper formulates a robust state estimator for continuous-time uncertain nonlinear systems wi... more This paper formulates a robust state estimator for continuous-time uncertain nonlinear systems with an integral quadratic constraint noise/uncertainty description. The model uncertainty and exogenous disturbances enter the state dynamics and observation channel in a unified fashion that includes the case of multiplicative noise. The robust filtering problem is formulated as a set-valued state estimation problem which is recast into an optimal control problem. An approximate solution to the resulting Hamilton-Jacobi-Bellman equation is obtained by using quadratic optimization with linearization of the observation equation. The approximate information state of the robust filter is organized as a triple of scalar, vector and matrix-valued parameters governed by a differential Riccati equation.

Research paper thumbnail of On strong solutions to the Moreau process

Research paper thumbnail of Frequency-Domain Computation of Quadratic-Exponential Cost Functionals for Linear Quantum Stochastic Systems

Research paper thumbnail of Measurement-based Feedback Control of Linear Quantum Stochastic Systems with Quadratic-Exponential Criteria

Research paper thumbnail of Parametric Randomization, Complex Symplectic Factorizations, and Quadratic-Exponential Functionals for Gaussian Quantum States

Infinite Dimensional Analysis, Quantum Probability and Related Topics

This paper combines probabilistic and algebraic techniques for computing quantum expectations of ... more This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states. Such quadratic-exponential functionals (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive filtering and control problems for linear quantum stochastic systems. We employ a Lie-algebraic correspondence between complex symplectic matrices and quadratic-exponential functions of system variables of a quantum harmonic oscillator. The complex symplectic factorizations are used together with a parametric randomization of the quasi-characteristic or moment-generating functions according to an auxiliary classical Gaussian distribution. This reduces the QEF to an exponential moment of a quadratic form of classical Gaussian random variables with a complex symmetric matrix and is applicab...

Research paper thumbnail of Effects of Parametric Uncertainties in Cascaded Open Quantum Harmonic Oscillators and Robust Generation of Gaussian Invariant States

SIAM Journal on Control and Optimization

This paper is concerned with the generation of Gaussian invariant states in cascades of open quan... more This paper is concerned with the generation of Gaussian invariant states in cascades of open quantum harmonic oscillators governed by linear quantum stochastic differential equations. We carry out infinitesimal perturbation analysis of the covariance matrix for the invariant Gaussian state of such a system and the related purity functional subject to inaccuracies in the energy and coupling matrices of the subsystems. This leads to the problem of balancing the state-space realizations of the component oscillators through symplectic similarity transformations in order to minimize the mean square sensitivity of the purity functional to small random perturbations of the parameters. This results in a quadratic optimization problem with an effective solution in the case of cascaded one-mode oscillators, which is demonstrated by a numerical example. We also discuss a connection of the sensitivity index with classical statistical distances and outline infinitesimal perturbation analysis for translation invariant cascades of identical oscillators. The findings of the paper are applicable to robust state generation in quantum stochastic networks.

Research paper thumbnail of A numerical approach to optimal coherent quantum LQG controller design using gradient descent

Automatica

This paper is concerned with coherent quantum linear quadratic Gaussian (CQLQG) control. The prob... more This paper is concerned with coherent quantum linear quadratic Gaussian (CQLQG) control. The problem is to find a stabilizing measurement-free quantum controller for a quantum plant so as to minimize a mean square cost for the fully quantum closed-loop system. The plant and controller are open quantum systems interconnected through bosonic quantum fields. In comparison with the observation-actuation structure of classical controllers, coherent quantum feedback is less invasive to the quantum dynamics. The plant and controller 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 in the form of quadratic constraints on the state-space matrices, reflecting the CCR preservation in time. The symmetry of the problem is taken into account by introducing equivalence classes of coherent quantum controllers generated by symplectic similarity transformations. We discuss a modified gradient flow, which is concerned with norm-balanced realizations of controllers. A line-search gradient descent algorithm with adaptive stepsize selection is proposed for the numerical solution of the CQLQG control problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing coherent quantum controller, thus taking the PR constraints into account. A convergence analysis of the algorithm is presented. Numerical examples of designing locally optimal CQLQG controllers are provided in order to demonstrate the algorithm performance.

Research paper thumbnail of Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action*

2016 IEEE Conference on Norbert Wiener in the 21st Century (21CW), 2016

Research paper thumbnail of A phase-space formulation of the Belavkin-Kushner-Stratonovich filtering equation for nonlinear quantum stochastic systems*

2016 IEEE Conference on Norbert Wiener in the 21st Century (21CW), 2016

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic syst... more This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We also discuss a more specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.

Research paper thumbnail of Robust Filtering for Uncertain Nonlinear Systems Satisfying a Sum Quadratic Constraint

ABSTRACT This paper develops a robust extended Kalman filter for nonlinear uncertain systems with... more ABSTRACT This paper develops a robust extended Kalman filter for nonlinear uncertain systems with a deterministic description of noise and uncertainty. The system state dynamics are formulated in reverse time and the uncertainties are modeled in terms of sum quadratic constraints. The robust filtering problem is formulated as a set-valued state estimation problem which is recast into an optimal control problem. The solution of the resulting Hamilton-Jacobi-Bellman equation is obtained by using a quadratic approximation. This leads to an approximate information state for the filtering problem which is computed recursively through a difference Riccati equation derived by linearizing the observation equation and using a quadratic approximation to the system dynamics. The reduction of the robust estimator to the standard Kalman filter in the uncertainty-free linear case is also discussed.

Research paper thumbnail of Minimum Relative Entropy State Transitions in Discrete Time Systems with Statistically Uncertain Noise

49Th Ieee Conference on Decision and Control, 2010

We develop a stochastic dissipativity theory for discrete-time systems driven by an uncertain ran... more We develop a stochastic dissipativity theory for discrete-time systems driven by an uncertain random noise. The deviation of the unknown probability law of the noise from a nominal white noise distribution is quantified by the conditional relative entropy given the initial state of the system. We establish a dissipation inequality and superadditivity property for the conditional relative entropy supply. The problem of minimizing the supply required to drive the system between given state distributions over a specified time horizon is considered. We obtain a dynamic programming Bellman equation for the minimum required relative entropy supply and show that the optimal noise is Markov with respect to the state of the system. For linear systems with Gaussian nominal noise and Gaussian initial and terminal state distributions, computing the minimum required supply is reduced to solving an algebraic Riccati equation.

Research paper thumbnail of Covariance dynamics and entanglement in translation invariant linear quantum stochastic networks

2015 54th IEEE Conference on Decision and Control (CDC), 2015

This paper is concerned with a translation invariant network of identical quantum stochastic syst... more 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 (QSDEs). The dynamic variables of the network satisfy the canonical commutation relations (CCRs). 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.

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

2015 IEEE Conference on Control Applications (CCA), 2015

This paper outlines an approach to the approximation of probability density functions by quadrati... more 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.

Research paper thumbnail of A transverse hamiltonian variational technique for open quantum stochastic systems and its application to coherent quantum control

2015 IEEE Conference on Control Applications (CCA), 2015

This paper is concerned with variational methods for nonlinear open quantum systems with Markovia... more 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 their propagation 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 technique to a mean square optimal filtering problem for a measurement-free cascade connection of quantum systems.

Research paper thumbnail of 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 ... more 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.

Research paper thumbnail of A Dynamic Programming Approach to Finite-horizon Coherent Quantum LQG Control

2011 Australian Control Conference, 2011

The paper is concerned with the coherent quantum Linear Quadratic Gaussian (CQLQG) control proble... more The paper is concerned with the coherent quantum Linear Quadratic Gaussian (CQLQG) control problem for time-varying quantum plants governed by linear quantum stochastic differential equations over a bounded time interval. A controller is sought among quantum linear systems satisfying physical realizability (PR) conditions. The latter describe the dynamic equivalence of the system to an open quantum harmonic oscillator and relate its state-space matrices to the free Hamiltonian, coupling and scattering operators of the oscillator. Using the Hamiltonian parameterization of PR controllers, the CQLQG problem is recast into an optimal control problem for a deterministic system governed by a differential Lyapunov equation. The state of this subsidiary system is the symmetric part of the quantum covariance matrix of the plant-controller state vector. The resulting covariance control problem is treated using dynamic programming and Pontryagin's minimum principle. The associated Hamilton-Jacobi-Bellman equation for the minimum cost function involves Frechet differentiation with respect to matrix-valued variables. The gain matrices of the CQLQG optimal controller are shown to satisfy a quasi-separation property as a weaker quantum counterpart of the filtering/control decomposition of classical LQG controllers. Keywords Quantum control • LQG cost • Physical realizability • Symplectic invariance • Dynamic programming • Pontryagin minimum principle • Frechet differentiation Mathematics Subject Classification (2000) 81Q93 • 81S25 • 93E20 • 49J50 • 58C20 The work is supported by the Australian Research Council.

Research paper thumbnail of Propagation of Mean Anisotropy of Signals in Filter Connections

Proceedings of the 17th IFAC World Congress, 2008, 2008

The anisotropy-based approach to robust control in stochastic systems occupies a unifying interme... more The anisotropy-based approach to robust control in stochastic systems occupies a unifying intermediate position between the H 2 and H ∞-optimal control theories. Initiated at the interface of Information Theory and Robust Control about fourteen years ago, the approach employs the a-anisotropic norm of a linear system as its worst-case sensitivity to input random disturbances whose mean anisotropy is bounded by a nonnegative parameter a. The latter quantifies the temporal "colouredness" and spatial "non-roundness" of the signal by its minimal relative entropy production rate with respect to Gaussian white noises with scalar covariance matrices. Revisiting the underlying definitions, the paper emphasizes the role of feedback in the construct of mean anisotropy of signals and discusses propagation of the latter through various filter connections. The results can be used to support physical and engineering intuition for a "rational" choice of the mean anisotropy level a in the design of anisotropy-based robust controllers.

Research paper thumbnail of A gradient descent approach to optimal coherent quantum LQG controller design

2015 American Control Conference (ACC), 2015

This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control probl... more 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.

Research paper thumbnail of Parameterization of stabilizing linear coherent quantum controllers

2015 10th Asian Control Conference (ASCC), 2015

This paper is concerned with application of the classical Youla-Kučera parameterization to findin... more 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 ∞ 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.

Research paper thumbnail of Decentralized Coherent Quantum Control Design for Translation Invariant Linear Quantum Stochastic Networks with Direct Coupling

Research paper thumbnail of Robust filtering for continuous-time uncertain nonlinear systems with an integral quadratic constraint

2012 American Control Conference (ACC), 2012

This paper formulates a robust state estimator for continuous-time uncertain nonlinear systems wi... more This paper formulates a robust state estimator for continuous-time uncertain nonlinear systems with an integral quadratic constraint noise/uncertainty description. The model uncertainty and exogenous disturbances enter the state dynamics and observation channel in a unified fashion that includes the case of multiplicative noise. The robust filtering problem is formulated as a set-valued state estimation problem which is recast into an optimal control problem. An approximate solution to the resulting Hamilton-Jacobi-Bellman equation is obtained by using quadratic optimization with linearization of the observation equation. The approximate information state of the robust filter is organized as a triple of scalar, vector and matrix-valued parameters governed by a differential Riccati equation.

Research paper thumbnail of On strong solutions to the Moreau process