Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2008.09946 · Physical Review A

Quantum limited superresolution of extended sources in one and two dimensions

Sudhakar Prasad

Published Dec 24, 2020• 10 citations
arXivHTMLPDF

Abstract

We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the length of a uniformly bright incoherent line source with a fixed midpoint and the radius of a uniformly bright incoherent disk shaped source with a fixed center. Prolate spheroidal wave functions (PSWFs) on a centered line segment and their generalized versions on a centered disk furnish the respective bases for computing the eigenstates and eigenvalues of the one-photon density operator, from which we calculate QFI with respect to the spatial parameters of the two sources. Zernike polynomials provide a good set into which to project the full source wavefront, and such classical wavefront projection data can realize quantum limited estimation error bound in each case. We subsequently generalize our approach to analyze sources of arbitrary brightness distributions and shapes using a certain class of Bessel Fourier functions that are closely related to the PSWFs. We illustrate the general approach by computing QFI for estimating the lengths of the principal axes of a uniformly bright, centered elliptical disk.

DOI
10.1103/physreva.102.063719

Related papers

Quantum limits on localizing point objects against a uniformly bright disk

Sudhakar Prasad

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.

arXiv:2211.16395•Physical Review A•Mar 31, 2023•3 citations

Quantum Limited Superresolution of an Incoherent Source Pair in Three Dimensions

Zhixian Yu, Sudhakar Prasad

The error in estimating the separation of a pair of incoherent sources from radiation emitted by them and subsequently captured by an imager is fundamentally bounded below by the inverse of the corresponding quantum Fisher information (QFI) matrix. We calculate the QFI for estimating the full three-dimensional pair separation vector, extending previous work on pair separation in one and two dimensions. We also show that the pair-separation QFI is, in fact, identical to source localization QFI, which underscores the fundamental importance of photon-state localization in determining the ultimate estimation-theoretic bound for both problems. We also propose general coherent-projection bases that can attain the QFI in two special cases. We present simulations of an approximate experimental realization of such quantum limited pair superresolution using the Zernike basis, confirming the achievability of the QFI bounds.

arXiv:1805.09227•Physical Review Letters•Oct 31, 2018•72 citations

Quantum limited super-resolution of an unequal-brightness source pair in three dimensions

Sudhakar Prasad

Abstract This paper extends our previous quantum Fisher information (QFI) based analysis of the problem of separating a pair of equal-brightness incoherent point sources in three dimensions to the case of a pair of sources that are unequally bright. When the pair's geometric center is perfectly known in advance, QFI with respect to the estimation of the three separation coordinates remains independent of the degree of brightness asymmetry. For the experimentally more relevant case of perfect prior knowledge of the pair's brightness centroid, however, such QFI becomes dependent on the pair separation vector in a way that is controlled by the degree of its brightness asymmetry. This study yields potentially useful insights into the analysis of a more general superresolution imaging problem involving extended incoherent sources with nontrivial brightness distributions.

arXiv:1908.10949•Physica Scripta•Nov 13, 2019•25 citations

Quantum limited source localization and pair superresolution in two dimensions under finite-emission bandwidth

Sudhakar Prasad

Optically localizing a single quasimonochromatic source to subdiffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width $w$ of the optical system's point-spread function (PSF) and the sought localization precision, $d$, i.e., as $\ensuremath{\alpha}{(w/d)}^{2}$. For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient $\ensuremath{\alpha}$ must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly $\ensuremath{\alpha}$ must increase with increasing width of a flat-top spectral profile of emission of a point source being localized in two dimensions by an imager with a clear circular aperture by calculating the quantum Fisher information (QFI), whose inverse yields the lowest possible unbiased-estimation variance of source-localization error. The novel use of prolate spheroidal wave functions as a basis for obtaining a solution of the eigenvalue problem of the single-photon density operator needed for the QFI calculation helps us develop the notion of an effective dimensionality of the continuous-state problem in terms of the associated space-bandwidth parameter. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.

arXiv:2006.00982•Physical Review A•Sep 28, 2020•7 citations

Enhanced superresolution by entanglement of the unified quantum wave function

Abdelali Sajia, Xiao‐Feng Qian

When two point sources are separated by less than the Abbe-Rayleigh limit, conventional resolution methods fail to distinguish whether there is one source or two. Consequently, it becomes nearly impossible to resolve important relative properties of the two sources, such as mutual coherence, relative phase, and brightness contrast. In this regime, the incoming composite quantum signal must be treated as a single entity described by an overall quantum wave function. In this study, we explore the superresolution of such two-point sources by examining a special wave property, i.e., entanglement between spatial degree of freedom (DoF) and remaining DoFs (both internal and external). Remarkably, we find that increasing this inter-DoF entanglement enhances the precision of superresolution, as quantified by the quantum Fisher information (QFI) in single-parameter estimation. Moreover, we show that this entanglement reliably ensures a nonvanishing QFI in two-parameter estimation, highlighting its role as a valuable resource for quantum superresolution.

Physical Review Research•Jul 28, 2025•0 citations

Quantum-Limited Distance Estimation in Three-Dimensional Optical Superresolution

Junyan Li, Shengshi Pang

Quantum superresolution reveals that the vanishing of separation sensitivity in conventional imaging below the Rayleigh limit does not necessarily indicate a fundamental loss of information in the optical field. However, the quantum limit for estimating the physical distance between two incoherent point sources in three-dimensional imaging systems and its dependence on the spatial structure of the point-spread function remains largely unknown. In this work, we derive the quantum-limited precision for estimating the full distance between two incoherent point sources with arbitrary intensity imbalance in a three-dimensional spatially invariant imaging system. We show that the distance information remains finite in the sub-Rayleigh regime and is governed by the second-order displacement-response tensor of the point-spread function. The eigensystem of this tensor determines the optimal relative orientation between the two sources, and reflection symmetries of the point-spread function can further provide a simplified means of identifying the optimal orientation. This geometric structure is coordinate invariant and provides a direct strategy for improving resolution by physically rotating an anisotropic imaging system to align its optimal principal response direction with the source displacement. For a general three-dimensional Gaussian point-spread function, the response tensor is proportional to the inverse spatial covariance, establishing a direct connection between quantum-limited distance precision and the geometry of Gaussian distribution.

quant-pharXiv:2608.04876•Aug 6, 2026•0 citations
We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the length of a uniformly bright incoherent line source with a fixed midpoint and the radius of a uniformly bright incoherent disk shaped source with a fixed center. Prolate spheroidal wave functions (PSWFs) on a centered line segment and their generalized versions on a centered disk furnish the respective bases for computing the eigenstates and eigenvalues of the one-photon density operator, from which we calculate QFI with respect to the spatial parameters of the two sources. Zernike polynomials provide a good set into which to project the full source wavefront, and such classical wavefront projection data can realize quantum limited estimation error bound in each case. We subsequently generalize our approach to analyze sources of arbitrary brightness distributions and shapes using a certain class of Bessel Fourier functions that are closely related to the PSWFs. We illustrate the general approach by computing QFI for estimating the lengths of the principal axes of a uniformly bright, centered elliptical disk.
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.
The error in estimating the separation of a pair of incoherent sources from radiation emitted by them and subsequently captured by an imager is fundamentally bounded below by the inverse of the corresponding quantum Fisher information (QFI) matrix. We calculate the QFI for estimating the full three-dimensional pair separation vector, extending previous work on pair separation in one and two dimensions. We also show that the pair-separation QFI is, in fact, identical to source localization QFI, which underscores the fundamental importance of photon-state localization in determining the ultimate estimation-theoretic bound for both problems. We also propose general coherent-projection bases that can attain the QFI in two special cases. We present simulations of an approximate experimental realization of such quantum limited pair superresolution using the Zernike basis, confirming the achievability of the QFI bounds.
Abstract This paper extends our previous quantum Fisher information (QFI) based analysis of the problem of separating a pair of equal-brightness incoherent point sources in three dimensions to the case of a pair of sources that are unequally bright. When the pair's geometric center is perfectly known in advance, QFI with respect to the estimation of the three separation coordinates remains independent of the degree of brightness asymmetry. For the experimentally more relevant case of perfect prior knowledge of the pair's brightness centroid, however, such QFI becomes dependent on the pair separation vector in a way that is controlled by the degree of its brightness asymmetry. This study yields potentially useful insights into the analysis of a more general superresolution imaging problem involving extended incoherent sources with nontrivial brightness distributions.
Optically localizing a single quasimonochromatic source to subdiffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width www of the optical system's point-spread function (PSF) and the sought localization precision, ddd, i.e., as $\ensuremath{\alpha}{(w/d)}^{2}$. For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient $\ensuremath{\alpha}$ must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly $\ensuremath{\alpha}$ must increase with increasing width of a flat-top spectral profile of emission of a point source being localized in two dimensions by an imager with a clear circular aperture by calculating the quantum Fisher information (QFI), whose inverse yields the lowest possible unbiased-estimation variance of source-localization error. The novel use of prolate spheroidal wave functions as a basis for obtaining a solution of the eigenvalue problem of the single-photon density operator needed for the QFI calculation helps us develop the notion of an effective dimensionality of the continuous-state problem in terms of the associated space-bandwidth parameter. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.
When two point sources are separated by less than the Abbe-Rayleigh limit, conventional resolution methods fail to distinguish whether there is one source or two. Consequently, it becomes nearly impossible to resolve important relative properties of the two sources, such as mutual coherence, relative phase, and brightness contrast. In this regime, the incoming composite quantum signal must be treated as a single entity described by an overall quantum wave function. In this study, we explore the superresolution of such two-point sources by examining a special wave property, i.e., entanglement between spatial degree of freedom (DoF) and remaining DoFs (both internal and external). Remarkably, we find that increasing this inter-DoF entanglement enhances the precision of superresolution, as quantified by the quantum Fisher information (QFI) in single-parameter estimation. Moreover, we show that this entanglement reliably ensures a nonvanishing QFI in two-parameter estimation, highlighting its role as a valuable resource for quantum superresolution.
Quantum superresolution reveals that the vanishing of separation sensitivity in conventional imaging below the Rayleigh limit does not necessarily indicate a fundamental loss of information in the optical field. However, the quantum limit for estimating the physical distance between two incoherent point sources in three-dimensional imaging systems and its dependence on the spatial structure of the point-spread function remains largely unknown. In this work, we derive the quantum-limited precision for estimating the full distance between two incoherent point sources with arbitrary intensity imbalance in a three-dimensional spatially invariant imaging system. We show that the distance information remains finite in the sub-Rayleigh regime and is governed by the second-order displacement-response tensor of the point-spread function. The eigensystem of this tensor determines the optimal relative orientation between the two sources, and reflection symmetries of the point-spread function can further provide a simplified means of identifying the optimal orientation. This geometric structure is coordinate invariant and provides a direct strategy for improving resolution by physically rotating an anisotropic imaging system to align its optimal principal response direction with the source displacement. For a general three-dimensional Gaussian point-spread function, the response tensor is proportional to the inverse spatial covariance, establishing a direct connection between quantum-limited distance precision and the geometry of Gaussian distribution.