We investigate our ability to determine the mean position, or centroid, of a linear array of equally bright incoherent point sources of light whose continuum limit is the problem of estimating the center of a uniformly radiating object. We consider two receivers, an image-plane ideal direct-detection imager and a receiver that employs Hermite-Gaussian (HG) spatial-mode demultiplexing in the image plane, prior to shot-noise-limited photon detection. We compare the Fisher information (FI) for estimating the centroid achieved by these two receivers, which quantifies the information-accrual rate per photon, and compare those with the quantum Fisher information (QFI): the maximum attainable FI by any choice of measurement on the collected light allowed by physics. We find that focal-plane direct imaging is strictly suboptimal, although not by a large margin. We also find that not only is the HG mode sorter, which is the optimal measurement for estimating the separation between point sources (or the length of a line object), suboptimal, but it performs worse than direct imaging. We study the scaling behavior of the QFI and direct imaging's FI for a continuous uniformly bright object in terms of its length and find that both are inversely proportional to the object's length when it is sufficiently larger than the Rayleigh length. Finally, we propose a two-stage adaptive modal receiver design that attains the QFI for centroid estimation.
We investigate our ability to determine the mean position, or centroid, of a linear array of equally bright incoherent point sources of light whose continuum limit is the problem of estimating the center of a uniformly radiating object. We consider two receivers, an image-plane ideal direct-detection imager and a receiver that employs Hermite-Gaussian (HG) spatial-mode demultiplexing in the image plane, prior to shot-noise-limited photon detection. We compare the Fisher information (FI) for estimating the centroid achieved by these two receivers, which quantifies the information-accrual rate per photon, and compare those with the quantum Fisher information (QFI): the maximum attainable FI by any choice of measurement on the collected light allowed by physics. We find that focal-plane direct imaging is strictly suboptimal, although not by a large margin. We also find that not only is the HG mode sorter, which is the optimal measurement for estimating the separation between point sources (or the length of a line object), suboptimal, but it performs worse than direct imaging. We study the scaling behavior of the QFI and direct imaging's FI for a continuous uniformly bright object in terms of its length and find that both are inversely proportional to the object's length when it is sufficiently larger than the Rayleigh length. Finally, we propose a two-stage adaptive modal receiver design that attains the QFI for centroid estimation.
Passive imaging receivers that demultiplex an incoherent optical field into a set of orthogonal spatial modes prior to detection can surpass canonical diffraction limits on spatial resolution. However, these mode-sorting receivers exhibit sensitivity to contextual nuisance parameters (e.g., the centroid of a clustered or extended object), raising questions on their viability in realistic scenarios where prior information about the scene is limited. We propose a multistage detection strategy that segments the total recording time between different physical measurements to build up the required prior information for near quantum-optimal imaging performance at sub-Rayleigh length scales. We show, via Monte Carlo simulations, that an adaptive two-stage scheme that dynamically allocates recording time between a conventional direct detection measurement and a binary mode sorter outperforms idealized direct detection alone when no prior knowledge of the object centroid is available, achieving one to two orders of magnitude improvement in mean squared error for simple estimation tasks. Our scheme can be generalized for more sophisticated tasks involving multiple parameters and/or minimal prior information.
We consider estimating the length of an incoherently radiating quasimonochromatic extended object of length much smaller than the traditional diffraction limit, the Rayleigh length. This is the simplest abstraction of the problems of estimating the diameter of a star in astronomical imaging or the dimensions of a cellular feature in biological imaging. We find, as expected by the Rayleigh criterion, that the Fisher information (FI) of the object's length, per integrated photon, vanishes in the limit of small sub-Rayleigh length for an ideal image-plane direct-detection receiver. With an image-plane Hermite-Gaussian (HG) mode sorter followed by direct detection, we show that this normalized FI does not diminish with decreasing object length. The FI per photon of both detection strategies gradually decreases as the object length greatly exceeds the Rayleigh limit, due to the relative inefficiency of information provided by photons emanating from near the center of the object about its length. We evaluate the quantum Fisher information per unit integrated photon and find that the HG mode sorter exactly achieves this limit at all values of the object length. Further, a simple binary mode sorter maintains the advantage of the full mode sorter at highly sub-Rayleigh lengths. In addition to this FI analysis, we quantify improvement in terms of the actual mean-square error of the length estimate using predetection mode sorting. We consider the effect of imperfect mode sorting and show that the performance improvement over direct detection is robust over a range of sub-Rayleigh lengths.
We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the two-dimensional location of a point source against a uniformly bright disk of known center and radius in the ideal photon-counting limit. We present both a perturbative calculation of the QFI in powers of the background-to-source brightness ratio and a numerically exact calculation of the QFI in the eigenbasis of the one-photon density operator. A related problem of the quantum limit on estimating the location of a small-area brightness hole in an otherwise uniformly bright disk, a problem of potential interest to the extrasolar planet detection community, is also treated perturbatively in powers of the ratio of the areas of the hole and the background disk. We then numerically evaluate the Cram\'er-Rao lower bound (CRB) for wavefront projections in three separate bases, those comprising Zernike, Fourier-Bessel, and localized point-source modes, for unbiased estimation of the two position coordinates of the point source and of the brightness hole center, respectively, for the two problems. By comparing these CRBs with the corresponding quantum-limited minimum error variances, given by inverting the QFI matrix, and with the CRBs associated with direct imaging, we assess the maximum efficiency of these wavefront projections in performing such estimations.
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.
We consider the problem, where a camera is tasked with determining one of two hypotheses: first with an incoherently-radiating quasi-monochromatic point source and the second with two identical closely spaced point sources. We are given that the total number of photons collected over an integration time is assumed to be the same under either hypothesis. For the one-source hypothesis, the source is taken to be on-axis along the line of sight and for the two-source hypothesis, we give ourselves the prior knowledge of the angular separation of the sources, and they are assumed to be identical and located symmetrically off-axis. This problem was studied by Helstrom in 1973, who evaluated the probability of error achievable using a sub-optimal optical measurement, with an unspecified structured realization. In this paper, we evaluate the quantum Chernoff bound, a lower bound on the minimum probability of error achievable by any physically-realizable receiver, which is exponentially tight in the regime that the integration time is high. We give an explicit structured receiver that separates three orthogonal spatial modes of the aperture field followed by quantum-noise-limited time-resolved photon measurement and show that this achieves the quantum Chernoff bound. In other words, the classical Chernoff bound of our mode-resolved detector exactly matches the quantum Chernoff bound for this problem. Finally, we evaluate the classical Chernoff bound on the error probability achievable using an ideal focal plane array---a signal shot-noise limited continuum photon-detection receiver with infinitely many infinitesimally-tiny pixels---and quantify its performance gap with the quantum limit.
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.