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IEEE Transactions on Information Theory

Quantum-mechanical linear filtering of random signal sequences

John S. Baras, Robert O. Harger, Young Park

Published Jan 1, 1976• 11 citations
Publisher

Abstract

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.

DOI
10.1109/tit.1976.1055499

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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.
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A review of the fundamental ideas and methods of the optimal reception and processing of quantum signals is given. Estimation via an operator based on the usual and generalized measurements (e.g., quasi-measurements) are discussed. The theory of operator estimation enables one to obtain Bayes limits for the usual estimates. The change from usual measurements to quasi-measurements generally improves performance. We show that some quasi-measurements can be realized as indirect measurements, and introduce an operator measure II(db) which describes the quasi-measurements on the space of measurements. Estimation based on quasi-measurements is described by operator measures Q(du) on the space of estimates. Minimization with respect to Q(du) is minimization simultaneously for all quasi-measurements and all estimates. For the M-ary hypothesis testing problem finding the optimum reduces to finding non-negative definite Hermitian operators Q1,…, Q m satisfying Q1+…+Q m = 1 where 1 is the identity operator, which extremize. Optimal Bayes quantum estimation is discussed. In the case of Gaussian quantum signal and minimum variance estimation, finding the quasi-measurements leads to optimal linear estimation. Further suboptimal methods for finding the operator in more general cases are discussed.
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A quantum stochastic model for an open dynamical system (quantum receiver) and output multi-channel of observation with an additive nonvacuum quantum noise is given. A quantum stochastic Master equation for the corresponding instrument is derived and quantum stochastic filtering equations both for the Heisenberg operators and the reduced density matrix of the system under the nondemolition observation are found. Thus the dynamical problem of quantum filtering is generalized for a noncommutative output process, and a quantum stochastic model and optimal filtering equation for the dynamical estimation of an input Markovian process is found. The results are illustrated on an example of optimal estimation of an input Gaussian diffusion signal, an unknown gravitational force say in a quantum optical or Weber's antenna for detection and filtering a gravitational waves.
A key concept of quantum information theory is that accessing information encoded in a quantum system requires us to discriminate between several possible states the system could be in. A natural generalization of this problem, namely, quantum sequence discrimination, appears in various quantum information processing tasks, the objective being to determine the state of a finite sequence of quantum states. Since such a sequence is a composite quantum system, the fundamental question is whether an optimal measurement is local, i.e., comprising measurements on the individual members, or collective, i.e., requiring joint measurement(s). In some known instances of this problem, the optimal measurement is local, whereas in others, it is collective. But, so far, a definite prescription based solely on the problem description has been lacking. In this paper, we prove that if the members of a given sequence are secretly and independently drawn from an ensemble or even from different ensembles, the optimum success probability is achievable by fixed local measurements on the individual members of the sequence, and no collective measurement is necessary. This holds for both minimum-error and unambiguous state discrimination paradigms.