Attainability of the quantum information bound in pure-state models

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 [Phys. Rev. Lett. 72, 3439 (1994)] have shown that the QIB can always be achieved via a projective measurement in the eigenvectors basis of an observable called the symmetric logarithmic derivative. However, such projective measurement depends, in general, on the value of the parameter to be estimated, therefore requiring 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 the initial pure states and which projective measurements 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.