Attainability of the quantum information bound in pure state models

TL;DR

This paper investigates conditions under which the quantum Cramér-Rao bound can be globally saturated in pure state models without prior knowledge of the parameter, focusing on unitary encoding processes.

Contribution

It provides a complete characterization of initial states and measurements that achieve the quantum information bound without prior parameter knowledge.

Findings

Identifies all pure states enabling QIB saturation without prior info.

Determines projective measurements that achieve global QIB saturation.

Focuses on unitary parameter encoding processes.

Abstract

The attainability of the quantum Cram\'er-Rao bound [QCR], the ultimate limit in the precision of the estimation of a physical parameter, requires the saturation of the quantum information bound [QIB]. This occurs when the Fisher information associated to a given measurement on the quantum state of a system which encodes the information about the parameter coincides with the quantum Fisher information associated to that quantum state. Braunstein and Caves [PRL {\bf 72}, 3439 (1994)] have shown that the QIB can always be achieved via a projective measurement in the eigenvectors basis of an observable called symmetric logarithmic derivative. However, such projective measurement depends, in general, on the value of the parameter to be estimated. Requiring, therefore, the previous knowledge of the quantity one is trying to estimate. For this reason, it is important to investigate under which situation it is possible to saturate the QCR without previous information about the parameter to be estimated. Here, we show the complete solution to the problem of which are all the initial pure states and the projective measurements that allow the global saturation of the QIB, without the knowledge of the true value of the parameter, when the information about the parameter is encoded in the system by a unitary process.