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arXiv:2308.04317 · Physical Review A

Quantum limit to subdiffraction incoherent optical imaging. III. Numerical analysis

Xiao-Jie Tan, Mankei Tsang

Published Nov 16, 2023• 4 citations
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Abstract

To investigate the fundamental limit to far-field incoherent imaging, the prequels to this work [M. Tsang, Phys. Rev. A 99, 012305 (2019); Phys. Rev. A 104, 052411 (2021)]. have studied a quantum lower bound on the error of estimating an object moment and proved a scaling law for the bound with respect to the object size. As the scaling law was proved only in the asymptotic limit of vanishing object size, this work performs a numerical analysis of the quantum bound to verify that the law works well for nonzero object sizes in reality. We also use the numerical bounds to study the optimality of a measurement called spatial-mode demultiplexing (SPADE), showing that SPADE not only follows the scaling but is also numerically close to being optimal, at least for low-order moments.

DOI
10.1103/physreva.108.052416

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To investigate the fundamental limit to far-field incoherent imaging, the prequels to this work [M. Tsang, Phys. Rev. A 99, 012305 (2019); Phys. Rev. A 104, 052411 (2021)]. have studied a quantum lower bound on the error of estimating an object moment and proved a scaling law for the bound with respect to the object size. As the scaling law was proved only in the asymptotic limit of vanishing object size, this work performs a numerical analysis of the quantum bound to verify that the law works well for nonzero object sizes in reality. We also use the numerical bounds to study the optimality of a measurement called spatial-mode demultiplexing (SPADE), showing that SPADE not only follows the scaling but is also numerically close to being optimal, at least for low-order moments.
I derive classical and quantum lower error bounds for moment estimation in subdiffraction incoherent optical imaging. While the end results are similar to earlier ones [Tsang, Phys. Rev. Res. 1, 033006 (2019); Phys. Rev. A 99, 012305 (2019)], the present work avoids the flawed mathematical assumptions in the earlier works by adopting a parametric-submodel approach from semiparametric statistics. With the rigorous bounds at hand and redefining the parameter of interest, I show that an alternative measurement called spatial-mode demultiplexing (SPADE) is superior to ideal direct imaging for a subdiffraction object, even if SPADE can measure only a finite number of modes.
I propose a spatial-mode demultiplexing (SPADE) measurement scheme for the far-field imaging of spatially incoherent optical sources. For any object too small to be resolved by direct imaging under the diffraction limit, I show that SPADE can estimate its second or higher moments much more precisely than direct imaging can fundamentally do in the presence of photon shot noise. I also prove that SPADE can approach the optimal precision allowed by quantum mechanics in estimating the location and scale parameters of a subdiffraction object. Realizable with far-field linear optics and photon counting, SPADE is expected to find applications in both fluorescence microscopy and astronomy.
In a previous paper [M. Tsang, Phys. Rev. A 99, 012305 (2019)], I proposed a quantum limit to the estimation of object moments in subdiffraction incoherent optical imaging. In this sequel, I prove the quantum limit rigorously by infinite-dimensional analysis. A key to the proof is the choice of an unfavorable parametric submodel to give a bound for the semiparametric problem. By generalizing the quantum limit for a larger class of moments, I also prove that the measurement method of spatial-mode demultiplexing with just one or two modes is able to achieve the quantum limit. For comparison, I derive a classical bound for direct imaging using the parametric-submodel approach, which suggests that direct imaging is substantially inferior.
I present a semiclassical analysis of a spatial-mode demultiplexing (SPADE) measurement scheme for far-field incoherent optical imaging under the effects of diffraction and photon shot noise. Building on previous results that assume two point sources or the Gaussian point-spread function, I generalize SPADE for a larger class of point-spread functions and evaluate its errors in estimating the moments of an arbitrary subdiffraction object. Compared with the limits to direct imaging set by the Cram\'er-Rao bounds, the results show that SPADE can offer far superior accuracy in estimating second- and higher-order moments.
Resolving extended incoherent objects below the diffraction limit poses an application-rich imaging challenge whose solution may enable a new generation of observational instruments and capabilities. In this work, we invoke a practical model for general imaging by approximating an arbitrary extended incoherent object as a finite grid of thermal point emitters parameterized by their brightnesses. We derive the quantum Fisher information matrix (QFIM) for simultaneous brightness estimation and show that the symmetric logarithmic derivatives weakly commute, indicating that the Helstrom bound furnishes the ultimate quantum limit on the estimation error for incoherent imaging. Furthermore, for deeply sub-diffraction scenes, we find numerical evidence of a gap between the Nagaoka-Hayashi (NH) bound and the Helstrom bound. This gap reveals that separable measurements, though more experimentally accessible, are insufficient to reach the quantum limit, and points to the prospective advantage of joint measurements acting on multiple state copies. Additionally, we show that spatial mode-demultiplexing (SPADE) often saturates the NH bound solidifying its status as a near-optimal separable measurement strategy that significantly outperforms direct imaging. Finally, we articulate two joint detection receivers implemented with bona fide quantum resources that asymptotically achieve the Helstrom bound.