# Density Matrix Reconstructions in Ultrafast Transmission Electron Microscopy: Uniqueness, Stability, and Convergence Rates

## TL;DR

This paper provides a mathematical analysis of the inverse problem in ultrafast transmission electron microscopy, focusing on the uniqueness, stability, and convergence rates of density matrix reconstructions, complementing prior experimental work.

## Contribution

It offers a theoretical framework analyzing the inverse problem's properties, including conditions for uniqueness and stability, and convergence rates, enhancing understanding of the reconstruction process.

## Key findings

- Analysis of conditions for unique density matrix reconstruction
- Stability estimates under various a-priori information
- Convergence rates of reconstruction algorithms

## Abstract

In the recent paper [17] the first experimental determination of the density matrix of a free electron beam has been reported. The employed method leads to a linear inverse problem with a positive semidefinite operator as unknown. The purpose of this paper is to complement the experimental and algorithmic results in the work mentioned above by a mathematical analysis of the inverse problem concerning uniqueness, stability, and rates of convergence under different types of a-priori information.