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arXiv:1602.06498 · IEEE Conference on Norbert Wiener in the 21st Century (21CW)

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

Igor G. Vladimirov, Ian R. Petersen

Published Jul 1, 2016• 8 citations
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Abstract

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. 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 of the observer on the covariance dynamics of the plant. 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 related Liealgebraic techniques, we represent this set of equations in a more explicit form in the case of equally dimensioned plant and observer.

DOI
10.1109/norbert.2016.7547464

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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.
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