Existence of unbiased resilient estimators in discrete quantum systems

TL;DR

This paper investigates the existence and properties of unbiased resilient estimators in discrete quantum systems, analyzing divergence issues and conditions for their validity, with implications for quantum parameter estimation.

Contribution

It introduces conditions under which unbiased resilient estimators exist in discrete quantum systems and compares different bounds in quantum estimation.

Findings

Unbiased resilient estimators often do not exist when constraints exceed measurement outcomes.

Divergences in bounds affect their applicability in quantum estimation.

Comparison of bounds reveals trade-offs between robustness and accuracy.

Abstract

The Cram\'er-Rao bound serves as a crucial lower limit for the mean squared error of an estimator in frequentist parameter estimation. Paradoxically, it requires highly accurate prior knowledge of the estimated parameter for constructing the optimal unbiased estimator. In contrast, Bhattacharyya bounds offer a more robust estimation framework with respect to prior accuracy by introducing additional constraints on the estimator. In this work, we examine divergences that arise in the computation of these bounds and establish the conditions under which they remain valid. Notably, we show that when the number of constraints exceeds the number of measurement outcomes, an estimator with finite variance typically does not exist. Furthermore, we systematically investigate the properties of these bounds using paradigmatic examples, comparing them to the Cram\'er-Rao and Bayesian approaches.