# Maximal Quantum Fisher Information for Mixed States

## TL;DR

This paper derives analytic solutions for optimal quantum metrology using mixed states, generalizing known results and revealing the full potential of mixed states for measurement precision.

## Contribution

It provides a rigorous generalization of optimal state preparation and measurement bounds for mixed states in quantum metrology.

## Key findings

- Analytic solutions for optimal state preparation from mixed states.
- Generalized upper bounds on measurement precision for mixed states.
- Optimal Hamiltonian control can saturate bounds with mixed states.

## Abstract

We study quantum metrology for unitary dynamics. Analytic solutions are given for both the optimal unitary state preparation starting from an arbitrary mixed state and the corresponding optimal measurement precision. This represents a rigorous generalization of known results for optimal initial states and upper bounds on measurement precision which can only be saturated if pure states are available. In particular, we provide a generalization to mixed states of an upper bound on measurement precision for time-dependent Hamiltonians that can be saturated with optimal Hamiltonian control. These results make precise and reveal the full potential of mixed states for quantum metrology.