Sensitivity limits are usually determined using the Cramér-Rao bound. Recently, this approach has been used to obtain the ultimate resolution limit for the estimation of the separation between two incoherent point sources. However, methods that saturate these resolution limits usually require the full measurement statistics, which can be challenging to access. In this work we introduce a simple superresolution protocol to estimate the separation between two thermal sources which relies only on the average value of a single accessible observable. We show how optimal observables for this technique may be constructed for arbitrary thermal sources and we study their sensitivities when one has access to spatially resolved intensity measurements (direct imaging) and photon counting after spatial-mode demultiplexing. For demultiplexing, our method is optimal, i.e., it saturates the quantum Cramér-Rao bound. We also investigate the impact of noise on the optimal observables, their measurement sensitivity, and the scaling with the number of detected photons of the smallest resolvable separation. For low signals in the image plane, we demonstrate that our method saturates the Cramér-Rao bound even in the presence of noise.

中文翻译：

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基于矩的超分辨率：形式主义和应用

灵敏度限值通常使用 Cramér-Rao 界限确定。最近，这种方法已被用于获得估计两个非相干点源之间的间隔的最终分辨率极限。然而，达到这些分辨率极限的方法通常需要完整的测量统计数据，这可能很难访问。在这项工作中，我们引入了一个简单的超分辨率协议来估计两个热源之间的分离，该协议仅依赖于单个可访问观测值的平均值。我们展示了如何为任意热源构建该技术的最佳观测值，并且我们研究了在空间模式解复用后可以获得空间分辨强度测量（直接成像）和光子计数时的灵敏度。对于解复用，我们的方法是最优的，即它使量子 Cramér-Rao 界饱和。我们还研究了噪声对最佳观测值、它们的测量灵敏度以及最小可分辨分离的检测到的光子数量的缩放的影响。对于图像平面中的低信号，我们证明了即使在存在噪声的情况下，我们的方法也会使 Cramér-Rao 边界饱和。