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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Superresolved Localisation of Blinking Sources of Light and Convexity of Fisher Information Matrix

D. B. Horoshko, Alexander B. Mikhalychev, Fedor Jelezko, P. Kuzhir

Published Jun 23, 2025• 0 citations
Publisher

Abstract

In the last two decades, a number of techniques have been developed for localisation of stochastically activated (blinking) fluorophores such as photoactivated localisation microscopy (PALM), stochastic optical reconstruction microscopy (STORM), points accumulation for imaging in nanoscale topography (PAINT), and their variants. These techniques are know by the general name of single-molecule localisation microscopy (SMLM) and have found numerous applications in cell biology [1]. These techniques can be considered in the framework of the quantum theory of multiple parameter estimation [2]. In this theory, the ultimate limit of precision is set by the quantum Cramér-Rao bound, Cov$(\hat{\theta})\geq \mathbf{F}^{-1}\geq \mathbf{Q}^{-1}$, where$\hat{\theta}$is an unbiased estimator for the vector of parameters$\theta=(\theta_{1}, \ldots, \theta_{M})$, Cov stands for covariance matrix, and the$M\times M$matrices$\mathbf{F}$and$\mathbf{Q}$are the Fisher information matrix and quantum Fisher information matrix, respectively.

DOI
10.1109/cleo/europe-eqec65582.2025.11111470

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A group of techniques known by the general name of single-molecule localization microscopy reaches a nanometer-scale spatial resolution of point light emitters, well below the diffraction limit of traditional microscopy. The key feature of these techniques is blinking, alternation of bright and dark states, of each emitter so that no more than one emitter is bright within the width of the point-spread function of the microscope during a time sufficient for its localization. We give a formulation of the optical part of these techniques in terms of quantum metrology, where the limit of precision is determined by the Fisher information on the emitters' positions contained in the measurement data. We show that the advantage in resolution provided by making the emitters blink is a consequence of the fundamental property of Fisher information, its convexity. In particular, we prove the weak matrix convexity and the trace convexity of the Fisher information matrix—two fundamental results in the multiparameter estimation theory. We show also that the advantage in information is inherent to the quantum state of light itself before the measurement and is related to the convexity of the quantum Fisher information matrix.
A group of techniques known by the general name of single-molecule localization microscopy reaches a nanometer-scale spatial resolution of point light emitters, well below the diffraction limit of traditional microscopy. The key feature of these techniques is blinking, alternation of bright and dark states, of each emitter so that no more than one emitter is bright within the width of the point-spread function of the microscope during a time sufficient for its localization. We give a formulation of the optical part of these techniques in terms of quantum metrology, where the limit of precision is determined by the Fisher information on the emitters' positions contained in the measurement data. We show that the advantage in resolution provided by making the emitters blink is a consequence of the fundamental property of Fisher information, its convexity. In particular, we prove the weak matrix convexity and the trace convexity of the Fisher information matrix—two fundamental results in the multiparameter estimation theory. We show also that the advantage in information is inherent to the quantum state of light itself before the measurement and is related to the convexity of the quantum Fisher information matrix.
Abstract Localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. This enables us to bound the multi-parameter covariance matrix for an unbiased estimator for the locations in terms of the quantum Fisher information matrix, which we obtained analytically. In the regime of arbitrarily small separations we find it to be no more than rank two—implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits.
Point source localization is a problem of persistent interest in optical imaging. In particular, a number of widely used biological microscopy techniques rely on precise three-dimensional localization of single fluorophores. As emitter depth localization is more challenging than lateral localization, considerable effort has been spent on engineering the response of the microscope in a way that reveals increased depth information. Here, we prove the (sub)optimality of these approaches by deriving and comparing to the measurement-independent quantum Cramér-Rao bound (QCRB). We show that existing methods for depth localization with single-objective collection exceed the QCRB, and we gain insight into the bound by proposing an interferometer arrangement that approaches it. We also show that for light collection with two opposed objectives, an established interferometric technique globally reaches the QCRB in all three dimensions simultaneously, and so this represents an interesting case study from the point of view of quantum multiparameter estimation.
In recent years quantum metrology and quantum sensing have enabled the advancement of quantum technologies and research in fundamental physics. Through statistical analysis, in the context of parameter estimation, the amount of information about a parameter of interest encoded in a quantum state can be quantified. In this thesis we focus on the application of quantum estimation theory to imaging and spectroscopy with quantum light. We study the fundamental bound of the mean square error for an unbiased estimator in terms of the quantum Fisher information (QFI) for two problems, one in microscopy and one in spectroscopy. Firstly, we study the problem of localising multiple point sources below the diffraction limit. We show that localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. We obtain the QFI matrix elements analytically. In the regime of arbitrarily small separations we find it to be no more than rank two – implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits. Secondly, we study the estimation of the electric dipole moment (EDM) of a two-level atom through its interaction with quantum pulses of light in free space. We derive analytical expressions for the states of one-photon wavepackets and entangled photon pairs (EPP). We numerically calculate the QFI for different states of light, including coherent and squeezed pulses, and compare their performance in estimating the EDM. We find that the one-photon wavepacket and the EPP have similar performances, while the performance of the entangled photon pair is not improved by increasing the entanglement of the EPP. Our results indicate that using Fock states to estimate the EDM of a two-level atom is preferable to using entangled light.
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.