Super-resolution techniques in microscopy are opening new frontiers in biology, and super-resolution for telescope imaging may provide similar benefits for astronomy and space domain awareness. Telescope super-resolution techniques such as multi-plane light conversion and photonic lanterns have been inspired by the ability of mode sorting measurements to extract classical Fisher information closer to the quantum bound than standard telescope imaging. Fisher information analyses for super-resolution imaging systems commonly consider the case of imaging point sources, but often only consider estimating a small subset of the parameters. Furthermore, there has been no demonstration of an estimation algorithm with performance that follows the limit set by the Fisher information when all point source parameters are considered. We provide a framework for defining and analyzing super-resolution using the classical and quantum Fisher information for the multi-parameter estimation problem and demonstrate an algorithm that can achieve the classical Fisher information bound for all the parameters simultaneously.
Super-resolution techniques in microscopy are opening new frontiers in biology, and super-resolution for telescope imaging may provide similar benefits for astronomy and space domain awareness. Telescope super-resolution techniques such as multi-plane light conversion and photonic lanterns have been inspired by the ability of mode sorting measurements to extract classical Fisher information closer to the quantum bound than standard telescope imaging. Fisher information analyses for super-resolution imaging systems commonly consider the case of imaging point sources, but often only consider estimating a small subset of the parameters. Furthermore, there has been no demonstration of an estimation algorithm with performance that follows the limit set by the Fisher information when all point source parameters are considered. We provide a framework for defining and analyzing super-resolution using the classical and quantum Fisher information for the multi-parameter estimation problem and demonstrate an algorithm that can achieve the classical Fisher information bound for all the parameters simultaneously.
Telescope systems with distributed apertures are a well-established approach for boosting resolution in astronomical imaging. However, theoretical limits on quantitative imaging precision, and the fundamentally best possible mode-sorting, beam-combining, and detection schemes to use with such arrays, remain largely unexplored. Using mathematical tools of the quantum and classical Crámer-Rao bounds, we perform analyses showing the fundamental origins of the enhancement provided by distributed imaging systems, over and above a single monolithic telescope, and consider the precision with which one can estimate any desired parameter embedded in a scene's incoherent radiation with a multi-aperture imaging system. We show how quantum-optimal measurements can be realized via beam-combination strategies of two classes: (1) , where light from different apertures is directed to a common focal plane, e.g., of a segmented-aperture telescope; and (2) , where light collected at each aperture, e.g., telescope sites of a long-baseline array, is routed to an optical interferometer. As an example, we show an explicit calculation of the quantum Fisher information (QFI) for estimating the angular separation between two-point emitters using two identical apertures separated by a baseline distance. We show that this QFI splits instructively into additive contributions from the single apertures and from the baseline. We quantify the relative benefits of intratelescope (e.g., spatial-mode) optical processing and intertelescope beam combination. We show how both receiver designs can be used to capture both sources of information and discuss how similar methods could be extended to more general imaging tasks. We discuss translating QFI-attaining measurements to explicit receiver designs, and the use of preshared entanglement to achieve the QFI when it is impractical to colocate and combine light collected by the apertures. Published by the American Physical Society 2024
In Bayesian estimation theory, the estimator θ^=E[θ∣l] attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest θ from the observation of l through a noisy channel Pl∣θ, given a prior Pθ on θ. In quantum sensing tasks, the user gets ρθ, the quantum state that encodes θ. They choose a measurement, a positive-operator valued measure (POVM) Πl, which induces the channel Pl∣θ=Tr(ρθΠl) to the measurement outcome l, on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM Πl that minimizes the MMSE over all possible measurements, and that MMSE. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements, when Pθ is unavailable. For multi-parameter estimation, i.e., when θ is a vector, in Fisher quantum estimation theory, the inverse of the QFI matrix provides an operator lower bound to the covariance of an unbiased estimator. However, there has been little work on quantifying quantum limits and measurement designs, for multi-parameter quantum estimation in the {\em Bayesian} setting. In this paper, we build upon Personick's result to construct a Bayesian adaptive measurement scheme for multi-parameter estimation when N copies of ρθ are available. We illustrate an application to localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs far superior to quantum-noise-limited focal-plane direct imaging.
Classically, the angular resolution of a telescope is limited by its diffraction limit: the ratio of its observing wavelength to its diameter. Developments in quantum information theory over the past decade have uncovered quantum-optimal measurement methods that enable super-resolution imaging beyond the diffraction limit, and many laboratory-scale demonstrations of these methods have been performed over the past decade, culminating in the recent first on-sky demonstration of quantum-enhanced astronomical imaging. In this Review, we detail the current progress and achievements of these theoretical and experimental results, discuss the astronomical science cases motivating the development of higher resolution telescopes, summarise the technological challenges facing the practical application of quantum imaging technologies to astronomy, and present a vision for the realization of \emph{quantum telescopes} with imaging resolutions orders of magnitude better than current observatories.
For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.
Optical precision measurement is evolving from conventional intensity and phase readout toward a paradigm centered on mode selection, statistical estimation, and information extraction. Conventional intensity- and phase-based schemes can achieve high sensitivity, but their performance is often constrained by finite-aperture diffraction, reference stability, optical path control, and environmental perturbations in systems. Quantum estimation theory provides a physical framework for optical resolution and measurement precision by optimizing the measurement basis. Inspired by spatial mode demultiplexing, this paper extends quantum parameter estimation from incoherent point-source resolution to small-displacement measurement. Metasurface enabled mode control converts displacement induced optical-field variations into measurable changes in orthogonal modal coefficients, providing a route for extracting sub-diffraction-scale displacement information. This paper introduces Rayleigh’s curse, Fisher information, and spatial mode projection, discusses their applications in one-dimensional displacement measurement and two-dimensional super-resolution imaging, and outlines future directions in digital optics, Engineering Optics 2.0, mode-basis optimization, two-dimensional information reconstruction, and intelligent optical measurement systems for engineering applications.
Recent theoretical and experimental work has shown that the quantum Fisher information associated with estimating the separation between two optical point sources remains finite at small separations, effectively opening new routes to super-resolution imaging of simultaneously emitting sources. Most studies to date, however, implicitly invoke the scalar approximation, which is not appropriate in the context of high-numerical-aperture microscopy. Utilizing parameter estimation theory, here we consider the estimation of separation between two closely spaced dipole emitters, a commonly employed model for single-molecule optical beacons. We consider two limiting cases: one in which the orientations of the emitters are fixed and equal, and another in which both dipoles freely sample all of orientation space over the course of the measurement. We quantify precision limits using quantum and classical variants of the Fisher information and Cramér-Rao bound. In all cases, the vectorial nature of the emission complicates the analyses, but with appropriate filtering of the collected light in the azimuthal-radial polarization basis, a previously proposed scheme to saturate the quantum Fisher information via image inversion interferometry can be salvaged.