Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation

TL;DR

This paper compares the Cramer-Rao and mini-max bounds in quantum channel estimation, introducing a local asymptotic mini-max bound and analyzing their relationships and conditions for optimal error decay.

Contribution

It introduces the local asymptotic mini-max bound and analyzes its relation to the Cramer-Rao bound in quantum channel estimation.

Findings

The local asymptotic mini-max bound is larger than the Cramer-Rao bound in phase estimation.

Both bounds coincide when the mean square error decreases as O(1/n).

A sufficient condition for O(1/n) error decay is derived.

Abstract

In a unified viewpoint in quantum channel estimation, we compare the Cramer-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramer-Rao bound in the phase estimation case while the both bounds coincide when the minimum mean square error decreases with the order O(1/n). We also derive a sufficient condition for that the minimum mean square error decreases with the order O(1/n).