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Control Theory and Technology

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

Igor G. Vladimirov

Published Aug 1, 2017• 2 citations
Publisher
DOI
10.1007/s11768-017-7012-2

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This paper is concerned with a filtering problem for a\nclass of nonlinear quantum stochastic systems with multichannel nondemolition\nmeasurements. The system-observation dynamics are governed\nby a Markovian Hudson-Parthasarathy quantum stochastic differential\nequation driven by quantum Wiener processes of bosonic fields in vacuum\nstate. The Hamiltonian and system-field coupling operators, as functions\nof the system variables, are represented in a Weyl quantization form.\nUsing the Wigner-Moyal phase-space framework, we obtain a stochastic\nintegro-differential equation for the posterior quasi-characteristic function\n(QCF) of the system conditioned on the measurements. This equation\nis a spatial Fourier domain representation of the Belavkin-Kushner-\nStratonovich stochastic master equation driven by the innovation process\nassociated with the measurements. We also discuss a more specific form\nof the posterior QCF dynamics in the case of linear system-field coupling\nand outline a Gaussian approximation of the posterior quantum state.
Simulation of nonlinear stochastic master equations generally requires trajectory-level evolution of large Hilbert-space density matrices, making strongly nonlinear continuously monitored systems computationally challenging. Here we identify an exactly solvable class centered on polynomial Hamiltonians of commuting quadratures with linear damping and homodyne measurement of those same quadratures. For initial states with a Gaussian measured-quadrature marginal and polynomial conditional-momentum sections, we derive an exact finite stochastic representation through any fixed momentum order. The result extends to interacting multimode systems, arbitrary measurement efficiency, and finite thermal occupation. Its computational cost is linear in trajectory length and independent of Hilbert-space dimension, requiring no Fock-space truncation, Gaussian approximation, or approximate moment closure. As an application, we derive an exact causal filter for monitored cubic-phase gates that uses the homodyne record to determine the variance-minimizing momentum displacement. Numerical simulations validate the representation against direct stochastic-master-equation evolution, demonstrate substantial speedups over converged Fock-space propagation, and show that this record-conditioned recentering can efficiently recover nonlinear squeezing otherwise lost through trajectory averaging.
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.