# Optimal control for multi-parameter quantum estimation with time-dependent Hamiltonians

## TL;DR

This paper develops optimal control strategies for multi-parameter quantum estimation in time-dependent systems, deriving maximal quantum Fisher information and proposing methods to handle parameter incompatibility for improved estimation precision.

## Contribution

It analytically derives the maximal quantum Fisher information matrix for two-parameter estimation in time-dependent three-level systems and proposes a variance method for optimal simultaneous estimation.

## Key findings

- Optimal coherent control schemes enhance estimation precision.
- Incompatibility of optimal Hamiltonians for different parameters is identified.
- A variance method achieves optimal simultaneous estimation in a Landau-Zener model.

## Abstract

We investigate simultaneous estimation of multi-parameter quantum estimation with time-dependent Hamiltonians. We analytically obtain the maximal quantum Fisher information matrix for two-parameter in time-dependent three-level systems. The optimal coherent control scheme is proposed to increase the estimation precisions. In a example of a spin-1 particle in a uniformly rotating magnetic field, the optimal coherent Hamiltonians for different parameters can be chosen to be completely same. However, in general, the optimal coherent Hamiltonians for different parameters are incompatibility. In this situation, we suggest a variance method to obtain the optimal coherent Hamiltonian for estimating multiple parameters simultaneously, and obtain the optimal simultaneous estimation precision of two-parameter in a three-level Landau-Zener Hamiltonian.