Quantum Fisher Information as the Convex Roof of Variance

TL;DR

This paper establishes that quantum Fisher information equals the convex roof of variance, providing an operational interpretation and explicit construction of pure-state ensembles that achieve this minimal averaged variance.

Contribution

It proves that quantum Fisher information is the convex roof of variance and offers an explicit method to construct pure-state ensembles realizing this minimal variance.

Findings

Quantum Fisher information equals the convex roof of variance.

Explicit construction of pure-state ensembles achieving minimal variance.

Provides an operational meaning for quantum Fisher information.

Abstract

Quantum Fisher information places the fundamental limit to the accuracy of estimating an unknown parameter. Here we shall provide the quantum Fisher information an operational meaning: a mixed state can be so prepared that a given observable has the minimal averaged variance, which equals exactly to the quantum Fisher information for estimating an unknown parameter generated by the unitary dynamics with the given observable as Hamiltonian. In particular we shall prove that the quantum Fisher information is the convex roof of the variance, as conjectured by Toth and Petz based on numerical and analytical evidences, by constructing explicitly a pure-state ensemble of the given mixed state in which the averaged variance of a given observable equals to the quantum Fisher information.