# Quantum Fisher information matrix and multiparameter estimation

## TL;DR

This paper reviews the quantum Fisher information matrix's properties, calculation methods, and its broader significance in quantum mechanics, along with recent advances in multiparameter quantum estimation and the associated bounds and measurements.

## Contribution

It provides a comprehensive overview of QFIM's properties, calculation techniques, and its role beyond quantum metrology, including recent developments in multiparameter estimation.

## Key findings

- QFIM links to quantum thermodynamics, phase transitions, and entanglement.
- The paper discusses conditions for attaining the quantum Cramér-Rao bound.
- Recent scenarios show quantum advantages in multiparameter estimation.

## Abstract

Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cram\'{e}r-Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cram\'{e}r-Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.