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IEEE Transactions on Information Theory

Noncommuting observables in quantum detection and estimation theory

Carl W. Helstrom, Robert Kennedy

Published Jan 1, 1974• 98 citations
Publisher

Abstract

Basing decisions and estimates on simultaneous approximate measurements of noncommuting observables in a quantum receiver is shown to be equivalent to measuring commuting projection operators on a larger Hilbert space than that of the receiver itself. The quantum-mechanical Cramér-Rao inequalities derived from right logarithmic derivatives and symmetrized logarithmic derivatives of the density operator are compared, and it is shown that the latter give superior lower bounds on the error variances of individual unbiased estimates of arrival time and carrier frequency of a coherent signal. For a suitably weighted sum of the error variances of simultaneous estimates of these, the former yield the superior lower bound under some conditions.

DOI
10.1109/tit.1974.1055173

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In quantum theory, there are various possibilities of defining logarithmic-derivative operators, each giving rise to a different Likelihood operator, and also to quantum Fisher information (QFI). In each case, the Cramér–Rao (CR) bound is derived from the first principles. It is also noted that if one does not make (often tacitly made) an assumption that the density operator [Formula: see text] and its derivative commute, then they lead to different QFI. Applications to one-mode coherent states (an infinite-dimensional quantum system) and to two-level quantum systems are studied.
Multiparameter quantum metrology plays a fundamental role in uncovering and exploiting the distinctive features of quantum systems. In this work, we propose an effective and experimentally feasible scheme to significantly enhance the simultaneous quantum estimation of the photon magnon and magnon mechanical coupling strengths in a hybrid cavity magnon mechanical platform. Our approach relies on the assistance of a coherent feedback loop combined with the injection of a coherent driving field. We show that an appropriate tuning of the system and feedback parameters leads to a substantial reduction of the estimation errors associated with both coupling strengths. To quantify the metrological performance of the proposed scheme, we employ the quantum Cramer Rao bound (QCRB) as a fundamental benchmark for multiparameter estimation. We explicitly compute and compare the QCRBs derived from the symmetric logarithmic derivative (SLD) and the right logarithmic derivative (RLD) formalisms. Our results demonstrate that the RLD based QCRB is systematically lower than the SLD based bound, indicating superior estimation precision in the considered noncommutative estimation scenario. We further analyze the performance of heterodyne detection and show that, in suitable parameter regimes, the corresponding classical estimation precision closely approaches the ultimate quantum limit predicted by our scheme. Finally, we discuss the experimental feasibility of the proposed setup within currently available cavity magnon mechanical platforms. Owing to its general character, the framework developed here can be readily extended to the high precision estimation of other physical parameters in hybrid quantum systems.