Quantum Limits on Localizing Point Objects against a Uniformly Bright Disk

TL;DR

This paper derives quantum limits on localizing point sources against bright backgrounds using quantum Fisher information, compares wavefront projection methods, and assesses their efficiency in high-precision astronomical imaging.

Contribution

It provides the first quantum Fisher information calculations for localizing point sources against bright disks and evaluates wavefront projection methods against quantum limits.

Findings

Quantum Fisher information bounds for source localization are derived.

Wavefront projection methods approach quantum-limited estimation efficiency.

Numerical comparisons show the effectiveness of different basis measurements.

Abstract

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 eigen-basis 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 Cramer-Rao lower bound (CRB) for wavefront projections in three separate bases, those comprised of 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.