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Journal of the Optical Society of America

Estimation of Object Parameters by a Quantum-Limited Optical System

Carl W. Helstrom

Published Feb 1, 1970• 22 citations
Publisher

Abstract

By means of the quantum-mechanical form of the Cramér-Rao inequality, a lower bound is set to the mean-square error in an unbiased estimate of a parameter of an incoherently radiating object observed in the presence of thermal background light by an optical system collecting light through a finite aperture. Estimates of absolute radiance, frequency, and position of the object are specifically analyzed. The bounds reduce in the classical limit to those previously obtained, but are valid in the quantum limit as well. When the background vanishes, the bounds depend only weakly on the effective number of independent spatial and temporal degrees of freedom of the object light at the aperture.

DOI
10.1364/josa.60.000233

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By means of the quantum-mechanical form of the Cramér-Rao inequality, a lower bound is set to the mean-square error in an unbiased estimate of a parameter of an incoherently radiating object observed in the presence of thermal background light by an optical system collecting light through a finite aperture. Estimates of absolute radiance, frequency, and position of the object are specifically analyzed. The bounds reduce in the classical limit to those previously obtained, but are valid in the quantum limit as well. When the background vanishes, the bounds depend only weakly on the effective number of independent spatial and temporal degrees of freedom of the object light at the aperture.
The detection of an incoherent object, light from which is received in the presence of thermal radiant energy, is treated by quantum detection theory under the assumption that the product of the observation time and the bandwidth of the object light is large. The threshold detector measures a quantum-mechanical operator that is a quadratic functional of the field at the aperture of the receiver. The statistics of the outcome of this measurement are approximately Poisson, and the detectability of the object depends—as under a background limitation—on the number of effectively independent spatio-temporal modes of the field. The detectabilities of objects with a rectangular and a Lorentz spectrum are compared under the assumption that the light from each possesses first-order coherence at the aperture.
The Shannon number is shown to specify the number of degrees of freedom that can be resolved in a coherent object scrutinized by a distant optical instrument that extracts information about the object from the light passing through its aperture. The quantum-mechanical nature of the light is taken into account, and the problem of determining the object amplitude distribution is treated by means of quantum estimation theory, which defines a limitation on the accuracy of an estimate of the object distribution that cannot be surpassed by any means of processing the received light. Fundamentally, the limitation arises from the inability of more than a finite number of modes of excitation of the object to deposit a significant number of photons in the aperture.
I propose classical and quantum limits to the statistical resolution of two incoherent optical point sources from the perspective of minimax parameter estimation. Unlike earlier results based on the Cramér-Rao bound (CRB), the limits proposed here, based on the worst-case error criterion and a Bayesian version of the CRB, are valid for any biased or unbiased estimator and obey photon-number scalings that are consistent with the behaviours of actual estimators. These results prove that, from the minimax perspective, the spatial-mode demultiplexing measurement scheme recently proposed by Tsang, Nair, and Lu [Phys. Rev. X 2016, 6 031033.] remains superior to direct imaging for sufficiently high photon numbers.
The widely used quantum Cramér–Rao bound (QCRB) sets a lower bound for the mean square error of unbiased estimators in quantum parameter estimation, however, in general QCRB is only tight in the asymptotical limit. With a limited number of measurements biased estimators can have a far better performance for which QCRB cannot calibrate. Here we introduce a valid lower bound for all estimators, either biased or unbiased, which can serve as standard of merit for all quantum parameter estimations.