A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

Vladimirov, Igor G.

doi:10.1007/s11768-017-7012-2

Cited by 5 publications

(14 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where use is also made of the relation 1 τ + 1 w = 1 ς which follows from the definition of ς in (67) and implies that A τ − 1 2w I n = A ς in view of (57). The Lyapunov inequality (72) leads to the lower bound for P 11 − P 1 in (68).…”

Section: Observer Back-action On Covariance Dynamics Of the Plantmentioning

confidence: 99%

“…The second inequality in (74) is obtained by applying (23) to (57). Therefore, Lemma 5 guarantees that the deviation P 11 − P 1 is small in comparison with P 1 (that is, r(P 11 P −1 1 − I n ) 1) and the back-action effect is negligible, if the second moments of the observer output η are small enough in the sense that the parameter κ in (73) satisfies κ max(1, r(P 1 Σ −1 1 )) 1.…”

Section: Observer Back-action On Covariance Dynamics Of the Plantmentioning

confidence: 99%

“…(159) Here, P 1 and P 2 are the second-moment matrices (57) and (58) for the uncoupled plant and observer variables given by…”

Section: Lemmamentioning

confidence: 99%

“…Fully quantum variational techniques, using perturbation analysis [45,54,55,57] beyond the class of OQHOs and symplectic geometric tools [46], suggest that the complicated sets of nonlinear equations for optimal quantum controllers and filters may appear to be more amenable to solution if they are approached using Hamiltonian structures similar to those in the underlying quantum dynamics. Such structures are particularly transparent in closed QHOs.…”

Section: Introductionmentioning

confidence: 99%

“…The initial covariance conditions in (56) for the plant and observer (prepared independently) are Σ uncertainty relation constraints Σ k + iΘ 0 0. With the effective time horizon chosen to be τ = 4.0614 τ * , the second-moment matrices of the uncoupled plant and observer variables in(57),(58) are P For the CQF problem (97), the observer back-action penalty matrix Π in (98) and the weighting matrix S 0 in the estimation error (142) are given by…”

mentioning

confidence: 99%

See 4 more Smart Citations

Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action

Vladimirov¹,

Petersen²

2016

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

Self Cite

View full text Add to dashboard Cite

This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of second-order moments of the system variables. Small-gain-theorem bounds are obtained for the back-action of the observer on the covariance dynamics of the plant in terms of the plant-observer coupling. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant-observer coupling matrix in order to mitigate the back-action effect. For the discounted mean square optimal CQF problem with penalized back-action, we establish first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic techniques, this set of equations is represented in a more explicit form for equally dimensioned plant and observer. For a class of such observers with autonomous estimation error dynamics, we obtain a solution of the CQF problem and outline a homotopy method. The computation of the performance criteria and the observer synthesis are illustrated by numerical examples.

show abstract

Section: Observer Back-action On Covariance Dynamics Of the Plantmentioning

confidence: 99%

Section: Observer Back-action On Covariance Dynamics Of the Plantmentioning

confidence: 99%

“…(159) Here, P 1 and P 2 are the second-moment matrices (57) and (58) for the uncoupled plant and observer variables given by…”

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action

Vladimirov¹,

Petersen²

2016

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

Self Cite

View full text Add to dashboard Cite

show abstract

Direct coupling coherent quantum observers with discounted mean square performance criteria and penalized back-action

Vladimirov

Petersen

2022

Math. Control Signals Syst.

View full text Add to dashboard Cite

A Transverse Hamiltonian Approach to Infinitesimal Perturbation Analysis of Quantum Stochastic Systems

Vladimirov¹

2023

Preprint

View full text Add to dashboard Cite

This paper is concerned with variational methods for open quantum systems with Markovian dynamics governed by Hudson-Parthasarathy quantum stochastic differential equations. These QSDEs are driven by quantum Wiener processes of the external bosonic fields and are specified by the system Hamiltonian and system-field coupling operators. We consider the system response to perturbations in these operators and introduce a transverse Hamiltonian which encodes the propagation of the perturbations through the unitary system-field evolution. This approach provides an infinitesimal perturbation analysis tool which can be used for the development of optimality conditions in quantum control and filtering problems. Such settings employ, as performance criteria, quadratic (or more complicated) cost functionals of the system and field variables to be minimised over the energy and coupling parameters of system interconnections. We demonstrate an application of the transverse Hamiltonian variational approach to a mean square optimal coherent quantum filtering problem for a measurement-free field-mediated cascade connection of a quantum system with a quantum observer.

show abstract

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

Cited by 5 publications

References 51 publications

Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action

Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action

Direct coupling coherent quantum observers with discounted mean square performance criteria and penalized back-action

A Transverse Hamiltonian Approach to Infinitesimal Perturbation Analysis of Quantum Stochastic Systems

Contact Info

Product

Resources

About