Least-bias state estimation with incomplete unbiased measurements

TL;DR

This paper introduces a method for quantum state estimation using incomplete measurements that maximizes Shannon entropy, ensuring physically meaningful results and addressing limitations of traditional linear estimators.

Contribution

It proposes a novel entropy-maximizing estimator for quantum tomography with incomplete unbiased measurements, improving physical validity and uncertainty quantification.

Findings

Maximizes Shannon entropy for unmeasured outcomes

Produces positive definite density matrices

Enhances physical interpretability of estimators

Abstract

Measuring incomplete sets of mutually unbiased bases constitutes a sensible approach to the tomography of high-dimensional quantum systems. The unbiased nature of these bases optimizes the uncertainty hypervolume. However, imposing unbiasedness on the probabilities for the unmeasured bases does not generally yield the estimator with the largest von Neumann entropy, a popular figure of merit in this context. Furthermore, this imposition typically leads to mock density matrices that are not even positive definite. This provides a strong argument against perfunctory applications of linear estimation strategies. We propose to use instead the physical state estimators that maximize the Shannon entropy of the unmeasured outcomes, which quantifies our lack of knowledge fittingly and gives physically meaningful statistical predictions.