Introduction to quantum Fisher information

TL;DR

This paper explores the mathematical transition from classical Fisher information to quantum Fisher information, highlighting different quantum versions and their applications in quantum theory.

Contribution

It provides a comprehensive overview of quantum Fisher information, including its parametrizations, relation to covariances, and special cases like skew information and chi-square divergence.

Findings

Multiple quantum Fisher information versions parametrized by functions

One-to-one correspondence between Fisher informations and covariances

Analysis of skew information and chi-square-divergence as special cases

Abstract

The subject of this paper is a mathematical transition from the Fisher information of classical statistics to the matrix formalism of quantum theory. If the monotonicity is the main requirement, then there are several quantum versions parametrized by a function. In physical applications the minimal is the most popular. There is a one-to-one correspondence between Fisher informations (called also monotone metrics) and abstract covariances. The skew information and the chi-square-divergence are treated here as particular cases.