Continuity of the quantum Fisher information

TL;DR

This paper proves the continuity of the quantum Fisher information (QFI) with respect to small changes in quantum states and their derivatives, ensuring stability of quantum parameter estimation across various dynamics and state evolutions.

Contribution

It establishes a general continuity property for QFI, including special cases involving quantum channels and noisy dynamics, and introduces a regularized symmetric logarithmic derivative representation.

Findings

QFI is continuous for states with close derivatives

Continuity holds under quantum channel evolutions

Regularized symmetric logarithmic derivative maintains continuity

Abstract

In estimating an unknown parameter of a quantum state the quantum Fisher information (QFI) is a pivotal quantity, which depends on the state and its derivate with respect to the unknown parameter. We prove the continuity property for the QFI in the sense that two close states with close first derivatives have close QFIs. This property is completely general and irrespective of dynamics or how states acquire their parameter dependence and also the form of parameter dependence---indeed this continuity is basically a feature of the classical Fisher information that in the case of the QFI naturally carries over from the manifold of probability distributions onto the manifold of density matrices. We demonstrate that in the special case where the dependence of the states on the unknown parameter comes from one dynamical map (quantum channel), the continuity holds in its reduced form with respect to the initial states. In addition, we show that when one initial state evolves through two different quantum channels, the continuity relation applies in its general form. A situation in which such scenario can occur is an open-system metrology where one of the maps represents the ideal dynamics whereas the other map represents the real (noisy) dynamics. In the making of our main result, we also introduce a regularized representation for the symmetric logarithmic derivative which works for general states even with incomplete rank, and its features continuity similarly to the QFI.