# A comparative study of estimation methods in quantum tomography

## TL;DR

This paper compares various quantum state estimation methods, analyzing their theoretical properties and practical performance, revealing insights into their asymptotic behavior, risk reduction, and computational efficiency.

## Contribution

It provides a comprehensive comparison of multiple quantum tomography estimators, including new asymptotic analyses and simulation results highlighting their advantages and limitations.

## Key findings

- Projected least squares estimator shows rank-dependent risk reduction.
- Non-monotonic behavior of Bures risk observed in asymptotic analysis.
- Projected estimators demonstrate high computational efficiency and strong optimality.

## Abstract

As quantum tomography is becoming a key component of the quantum engineering toolbox, there is a need for a deeper understanding of the multitude of estimation methods available. Here we investigate and compare several such methods: maximum likelihood, least squares, generalised least squares, positive least squares, thresholded least squares and projected least squares. The common thread of the analysis is that each estimator projects the measurement data onto a parameter space with respect to a specific metric, thus allowing us to study the relationships between different estimators. The asymptotic behaviour of the least squares and the projected least squares estimators is studied in detail for the case of the covariant measurement and a family of states of varying ranks. This gives insight into the rank-dependent risk reduction for the projected estimator, and uncovers an interesting non-monotonic behaviour of the Bures risk. These asymptotic results complement recent non-asymptotic concentration bounds of \cite{GutaKahnKungTropp} which point to strong optimality properties, and high computational efficiency of the projected linear estimators. To illustrate the theoretical methods we present results of an extensive simulation study. An app running the different estimators has been made available online.