Hedged maximum likelihood estimation

TL;DR

This paper introduces hedged maximum likelihood estimation (HMLE), a quantum state estimation method that improves predictive performance over traditional maximum likelihood estimation by ensuring strictly positive density matrices.

Contribution

It presents HMLE as a simple modification of MLE, inspired by classical

Findings

HMLE outperforms MLE in most cases for quantum state estimation.

HMLE produces strictly positive density matrices, improving predictive behavior.

Neither HMLE nor MLE is optimal for nearly-pure states.

Abstract

This paper proposes and analyzes a new method for quantum state estimation, called hedged maximum likelihood (HMLE). HMLE is a quantum version of Lidstone's Law, also known as the "add beta" rule. A straightforward modification of maximum likelihood estimation (MLE), it can be used as a plugin replacement for MLE. The HMLE estimate is a strictly positive density matrix, slightly less likely than the ML estimate, but with much better behavior for predictive tasks. Single-qubit numerics indicate that HMLE beats MLE, according to several metrics, for nearly all "true" states. For nearly-pure states, MLE does slightly better, but neither method is optimal.