Optimal error regions for quantum state estimation

TL;DR

This paper introduces the concept of optimal regions in quantum state estimation, focusing on maximum-likelihood and credible regions, with methods for their determination and criteria for prior assignment.

Contribution

It defines and analyzes optimal regions for quantum state estimation, extending point estimators to region-based estimators with specific optimality criteria.

Findings

Optimal regions have constant likelihood on their boundary.

Maximum-likelihood regions maximize likelihood for a given size.

Smallest credible regions minimize size for a given posterior probability.

Abstract

Rather than point estimators, states of a quantum system that represent one's best guess for the given data, we consider optimal regions of estimators. As the natural counterpart of the popular maximum-likelihood point estimator, we introduce the maximum-likelihood region---the region of largest likelihood among all regions of the same size. Here, the size of a region is its prior probability. Another concept is the smallest credible region---the smallest region with pre-chosen posterior probability. For both optimization problems, the optimal region has constant likelihood on its boundary. We discuss criteria for assigning prior probabilities to regions, and illustrate the concepts and methods with several examples.