# Quantum Nondemolition Filtering

## Abstract

A continuous-time filter is formulated for a physical system modeled as an infinite-dimensional bilinear system. Efforts focus on a quantum system with Hamiltonian operator of the form *H* _{0} + *u*(*t*) *H* _{1} where *H* _{0} is the Hamiltonian of the undisturbed system, *H* _{1} couples the system to an external classical field, and *u*(*t*) represents the time-varying signal carried by this field. An important problem is to determine when and how the signal *u*(*t*) can be extracted from the time development of the measured value of a suitable system observable C. There exist certain observables, called quantum nondemolition observables (QNDO), which have the property that their expected and measured values coincide. The invertibility problem has been posed and solved for these quasiclassical observables. Since one is addressing an *infinite-dimensional* bilinear system, the domain issue for the operators *H* _{0}, *H* _{1} and *C* becomes nontrivial. An additional complication is that the input observable *C* is in general time dependent. Having derived conditions for invertibility, necessary and sufficient conditions are developed for an observable to qualify as a QNDO. If an observable meets both sets of criteria, it is said to constitute a quantum nondemolition filter (QNDF). By construction, the associated filtering algorithm separates cleanly into the choice of output observable (a QNDO) and the choice of procedure for processing measurement outcomes. This approach has the advantage over previous schemes that no optimization is necessary. QNDF’s may see practical application in the demodulation of optical signals and the detection and monitoring of gravitational waves.

## Keywords

Gravitational Wave Analytic Domain Bilinear System Inverse System Heisenberg Picture## Preview

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