Quantum State Tomography of a Single Qubit: Comparison of Methods

TL;DR

This paper compares various quantum state tomography methods for single qubits, finding Bayesian mean estimation most accurate, and discusses how to incorporate experimental errors and priors for improved results.

Contribution

It provides a comprehensive comparison of multiple tomography methods and evaluates their accuracy, highlighting the effectiveness of Bayesian mean estimation for single qubit state reconstruction.

Findings

Bayesian mean estimation yields the most accurate density matrix estimates.

MLE is slightly less accurate than BME in the studied scenarios.

Using appropriate priors improves the accuracy of state estimation.

Abstract

The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance, and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography errors. As a measure of the accuracy of these methods we average the trace distance between a given density matrix and the tomographic density matrices it can give rise to through experimental measurements. We find that the BME provides the most accurate estimate of the density matrix, and suggest using either the pure-state prior, if the system is known to be in a rather pure state, or the Bures prior if any state is possible. The MLE is found to be slightly less accurate. We comment on the extrapolation of these results to larger systems.