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arXiv:2011.07897 · Physical Review A

Generalization of Rayleigh's criterion on parameter estimation with incoherent sources

Lijun Peng, Xiao-Ming Lu

Published Apr 1, 2021• 9 citations
arXivHTMLPDF

Abstract

The basic idea behind Rayleigh's criterion on resolving two incoherent optical point sources is that the overlap between the spatial modes from different sources would reduce the estimation precision for the locations of the sources, dubbed Rayleigh's curse. We generalize the concept of Rayleigh's curse to the abstract problems of quantum parameter estimation with incoherent sources. To manifest the effect of Rayleigh's curse on quantum parameter estimation, we define the curse matrix in terms of quantum Fisher information and introduce the global and local immunity to the curse accordingly. We further derive the expression for the curse matrix and give the necessary and sufficient condition on the immunity to Rayleigh's curse. For estimating the one-dimensional location parameters with a common initial state, we demonstrate that the global immunity to the curse on quantum Fisher information is impossible for more than two sources.

DOI
10.1103/physreva.103.042601

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The basic idea behind Rayleigh's criterion on resolving two incoherent optical point sources is that the overlap between the spatial modes from different sources would reduce the estimation precision for the locations of the sources, dubbed Rayleigh's curse. We generalize the concept of Rayleigh's curse to the abstract problems of quantum parameter estimation with incoherent sources. To manifest the effect of Rayleigh's curse on quantum parameter estimation, we define the curse matrix in terms of quantum Fisher information and introduce the global and local immunity to the curse accordingly. We further derive the expression for the curse matrix and give the necessary and sufficient condition on the immunity to Rayleigh's curse. For estimating the one-dimensional location parameters with a common initial state, we demonstrate that the global immunity to the curse on quantum Fisher information is impossible for more than two sources.
The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.
The resolution limit of standard imaging techniques is expressed by the Rayleigh criterion, which states that two point-like sources are difficult to resolve if their transverse separation is smaller that the Rayleigh length. While the criterion is useful in the case of direct detection imaging, other measurement techniques may not be subject to this limitation. Here we consider the use of imaging to estimate the distance between two incoherent point sources. In this framework, the Rayleigh criterion manifests itself through the so-called Rayleigh curse. The latter expresses the fact that the statistical error in the estimation increases for sub-wavelength separation. Here we show that, in the regime of weak signals, a structured measurement obtained by concatenating a linear interferometer with on-off photo-detection is immune to the Rayleigh curse. In this way we clarify the relation between imaging and interferometry, and establish the optimality of linear interferometry for an arbitrary number of incoherent sources.
We devise a systematic method to determine the Fisher information required for resolving two incoherent point sources with a diffraction-limited linear imaging device. The resulting Cramér-Rao bound gives the lowest variance achievable for an unbiased estimator. When only intensity in the image plane is recorded, this bound diverges as the separation between the sources tends to zero, an effect which has been dubbed as Rayleigh's curse. However, this curse can be lifted using suitable coherent measurements. In particular, we determine a class of measurements that can be easily implemented as projections on the orthogonalised derivatives of the point spread function of the system.
Superresolution has been demonstrated to overcome the limitation of Rayleigh's criterion and achieve significant precision improvement in resolving the separation of two incoherent optical point sources. However, in recent years, it was found that, if the photon numbers of two incoherent optical sources are unknown, the precision of superresolution vanishes when the photon numbers are actually different. In this work, we first analyze in detail the estimation precision of the separation for two incoherent optical sources with the same point-spread functions and show that, when the two photon numbers are different but sufficiently close, the superresolution can still be realized but with different precisions. We find the condition on how close the photon numbers need to be to realize the superresolution and derive the precision of superresolution in different regimes of the photon number difference. We further consider the superresolution for two incoherent optical sources with different point-spread functions and show that the competition between the difference of photon numbers, the difference of point-spread functions, and the separation of source locations determines the precision of superresolution. The results exhibit precision limits distinct from that of two identical point-spread functions with equal photon numbers and extend the realizable regimes of quantum superresolution. Finally, the results are illustrated by Gaussian point-spread functions. Published by the American Physical Society 2024
The quantum Fisher information (FI), when applied to the estimation of the separation of two point sources, has been shown to be non-zero in cases where the coherence between the sources is known. Although it has been claimed that ignorance of the coherence causes the quantum FI to vanish (a resurgence of Rayleigh’s curse), a more complete analysis including both the magnitude and phase of the coherence parameter is given here. Partial ignorance of the coherence is shown to potentially break Rayleigh’s curse, whereas complete ignorance guarantees its resurgence.
In spatial imaging, it is well known that two incoherent sources with a separation less than the diffraction limit cannot be resolved, a phenomenon dubbed as Rayleigh’s curse. A recent information theoretic approach suggests that this curse can be avoided using optimal (here, mode-sensitive) measurements instead of the usual intensity-only measurements, which define the classical limit [1] . This quantum enhanced resolution has been realized in both the spatial [2] and temporal domain [3] . Nevertheless, the role of optical coherence on the resolution limit still remains elusive [4] , [5] , [6] . Here, we experimentally resolve this debate using time-domain measurements.