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IEEE Conference on Decision and Control (CDC)

Nonclassical filtering problems in quantum electronics

John S. Baras, Robert O. Harger

Published Dec 1, 1975• 2 citations
Publisher

Abstract

Filtering problems arising in quantum electronics are investigated. In particular the minimum variance linear filtering problem incorporating quantum mechanical measurements is solved for both scalar and vector signal processes. The simplification of the filter under certain assumptions is demonstrated. Examples from optical communication systems illustrate the results.

DOI
10.1109/cdc.1975.270614

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Filtering problems arising in quantum electronics are investigated. In particular the minimum variance linear filtering problem incorporating quantum mechanical measurements is solved for both scalar and vector signal processes. The simplification of the filter under certain assumptions is demonstrated. Examples from optical communication systems illustrate the results.
The problem of estimating a member of a discrete-time vector process from past and present quantum mechanical measurements is considered; specifically, the minimum-variance linear estimator based on optimal present measurements is studied. Necessary and sufficient conditions that characterize the optimal processing matrix coefficients and the optimal measurements are discussed and interpreted. The optimal linear filter is compared to the optimal quantum estimator without postprocessing of past data. When the signal sequence is pairwise Gaussian and the optimal quantum measurement without postproccssing has the properties that it is linear in a specific sense and that its outcome and the corresponding element of the signal sequence are jointly Gaussian, then the optimal linear filter separates. That is, the optimal measurement can be taken to be the same as thc optimal measurement without regard to past data, and the past and present data are processed classically. Thc results are illustrated by considering the filtering of the in-phase and quadrature amplitudes of a laser field received in a single-mode cavity along with thermal noise. In this case, when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively in a very efficient manner.
The problem of estimating a member of a scalar random signal sequence with quantum-mechanical measurements is considered. The minimum variance linear estimator based on an optimal present quantum measurement and optimal linear processing of past measurements is found. When the average optimal measurement without postprocessing, for a fixed signal, is linear in the random signal and the signal sequence is pairwise Gaussian, the optimal processing separates: the optimal measurement is the same as the optimal measurement without regard to past data, and the past and present data are processed classically. The results are illustrated by considering the estimator of the real amplitude of a laser signal received in a single-mode cavity along with thermal noise; when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively. For a one-step memory signal sequence it is shown that the optimal observable generally differs from the optimal observable disregarding the past; the optimal measurement can be computed recursively.
This paper considers a version of the Wiener filtering problem for equalization of passive linear quantum systems. We demonstrate that taking into consideration the quantum nature of the signals involved leads to features typically not encountered in classical equalization problems. Most significantly, finding a mean-square optimal quantum equalizing filter amounts to solving a nonconvex constrained optimization problem. We discuss two approaches to solving this problem, both involving a relaxation of the constraint. In both cases, unlike classical equalization, there is a threshold on the variance of the noise below which an improvement of the mean-square error cannot be guaranteed.
The paper is concerned with the formulation of a continuous-time quantum-mechanical filter. In a previous paper (Ong et al., 1984), the invertibility of a quantum system coupled to a weak time-dependent classical field was studied. The physical system is modeled as an infinite-dimensional bilinear system. Necessary and sufficient conditions of invertibility were derived under the assumption that the output observable is a quantum nondemolition observable (QNDO), characterized by the classical property that its expected value is equal to its measured value. In this paper necessary and sufficient conditions are developed for an observable to qualify as a QNDO; if in addition the criteria for invertibility are met, the given observable defines a quantum nondemolition filter (QNDF). The associated filtering algorithm thus separates cleanly into the choice of output observable (a QNDO) and the choice of procedure for processing the measurement outcomes. This approach has the advantage over previous schemes that no optimization is necessary. Applications to demodulation of optical signals and to the detection and monitoring of gravitational waves are envisioned.
The paper is concerned with the formulation of a continuous-time quantum-mechanical filter. In a previous paper (Ong et al., 1984), the invertibility of a quantum system coupled to a weak time-dependent classical field was studied. The physical system is modeled as an infinite-dimensional bilinear system. Necessary and sufficient conditions of invertibility were derived under the assumption that the output observable is a quantum nondemolition observable (QNDO), characterized by the classical property that its expected value is equal to its measured value. In this paper necessary and sufficient conditions are developed for an observable to qualify as a QNDO; if in addition the criteria for invertibility are met, the given observable defines a quantum nondemolition filter (QNDF). The associated filtering algorithm thus separates cleanly into the choice of output observable (a QNDO) and the choice of procedure for processing the measurement outcomes. This approach has the advantage over previous schemes that no optimization is necessary. Applications to demodulation of optical signals and to the detection and monitoring of gravitational waves are envisioned.
Communication theory problems incorporating quantum effects for optical-frequency applications are discussed. Under suitable conditions, a unique quantum channel model corresponding to a given classical space-time varying linear random channel is established. A procedure is described by which a proper density-operator representation applicable to any receiver configuration can be constructed directly from the channel output field. Some examples illustrating the application of our methods to the development of optical quantum channel representations are given. Optimizations of communication system performance under different criteria are considered. In particular, certain necessary and sufficient conditions on the optimal detector in M-ary quantum signal detection are derived. Some examples are presented. Parameter estimation and channel capacity are discussed briefly.