Investigating Bias in Maximum Likelihood Quantum State Tomography

TL;DR

This paper examines the bias inherent in maximum likelihood quantum state tomography, revealing that estimates of higher purity states tend to underestimate the true purity, especially with finite data.

Contribution

It provides a detailed analysis of how bias depends on state purity, measurement number, and measurement bases in quantum state tomography.

Findings

Higher purity states show significant bias towards lower purity estimates.

Bias decreases as the number of measurements increases.

The number of measurement bases influences the bias magnitude.

Abstract

Maximum likelihood quantum state tomography yields estimators that are consistent, provided that the likelihood model is correct, but the maximum likelihood estimators may have bias for any finite data set. The bias of an estimator is the difference between the expected value of the estimate and the true value of the parameter being estimated. This paper investigates bias in the widely used maximum likelihood quantum state tomography. Our goal is to understand how the amount of bias depends on factors such as the purity of the true state, the number of measurements performed, and the number of different bases in which the system is measured. For that, we perform numerical experiments that simulate optical homodyne tomography under various conditions, perform tomography, and estimate bias in the purity of the estimated state. We find that estimates of higher purity states exhibit considerable bias, such that the estimates have lower purity than the true states.