Gradient-based stopping rules for maximum-likelihood quantum-state tomography

TL;DR

This paper introduces a gradient-based stopping rule for maximum-likelihood quantum-state tomography, providing a reliable way to determine when iterative algorithms have sufficiently converged, thus improving the efficiency and confidence in quantum state estimation.

Contribution

It presents a novel gradient-based upper bound on likelihood ratios that aids in formulating effective stopping criteria for quantum-state tomography algorithms.

Findings

The bound is independent of the specific optimization algorithm used.

It enables the formulation of stopping rules based on likelihood differences.

The approach improves confidence region determination in quantum-state estimation.

Abstract

When performing maximum-likelihood quantum-state tomography, one must find the quantum state that maximizes the likelihood of the state given observed measurements on identically prepared systems. The optimization is usually performed with iterative algorithms. This paper provides a gradient-based upper bound on the ratio of the true maximum likelihood and the likelihood of the state of the current iteration, regardless of the particular algorithm used. This bound is useful for formulating stopping rules for halting iterations of maximization algorithms. We discuss such stopping rules in the context of determining confidence regions from log-likelihood differences when the differences are approximately chi-squared distributed.