# Quantum limits of localisation microscopy

## TL;DR

This paper establishes quantum limits on the precision of localizing multiple incoherent point sources in microscopy, revealing fundamental bounds and comparing measurement techniques against these limits.

## Contribution

It introduces a quantum framework for localization microscopy, deriving bounds on estimation precision and comparing measurement strategies.

## Key findings

- Quantum Fisher information bounds the localization precision.
- Maximum independent parameters estimable is two, regardless of sources.
- Spatial-mode demultiplexing can approach quantum limits.

## Abstract

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. 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.