The Cram\'{e}r-Rao approach and global quantum estimation of bosonic states

TL;DR

This paper investigates the applicability of the Cramér-Rao approach to global quantum estimation of bosonic states, revealing conditions where it is valid or invalid in non-IID scenarios with limited copies.

Contribution

It identifies when the Cramér-Rao approach can be applied to global quantum estimation of bosonic states and highlights potential pitfalls in its use.

Findings

Cramér-Rao approach works in certain non-IID bosonic estimation scenarios.

It does not always apply for global estimation with limited copies.

Highlights caution in extrapolating local bounds to global quantum estimation.

Abstract

Quantum state estimation is a fundamental task in quantum information theory, where one estimates real parameters continuously embedded in a family of quantum states. In the theory of quantum state estimation, the widely used Cram\'er Rao approach which considers local estimation gives the ultimate precision bound of quantum state estimation in terms of the quantum Fisher information. However practical scenarios need not offer much prior information about the parameters to be estimated, and the local estimation setting need not apply. In general, it is unclear whether the Cram\'er-Rao approach is applicable for global estimation instead of local estimation. In this paper, we find situations where the Cram\'er-Rao approach does and does not work for quantum state estimation problems involving a family of bosonic states in a non-IID setting, where we only use one copy of the bosonic quantum state in the large number of bosons setting. Our result highlights the importance of caution when using the results of the Cram\'er-Rao approach to extrapolate to the global estimation setting.