# Evaluating the Holevo Cram\'er-Rao bound for multi-parameter quantum metrology

## TL;DR

This paper develops a method to evaluate the Holevo Cramér-Rao bound for multi-parameter quantum estimation, revealing conditions under which it can be attained and exploring its implications in optical and magnetometry systems.

## Contribution

It recasts the evaluation of the Holevo Cramér-Rao bound as a semidefinite program for finite-dimensional systems and applies this to practical quantum metrology scenarios.

## Key findings

- HCRB can be attained with optimal measurements in certain regimes.
- Numerical evidence shows projective measurements attain HCRB in noiseless two-qubit systems.
- Interplay between HCRB and measurement incompatibility is demonstrated.

## Abstract

Only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed. This is due to the incompatibility of observables. The fundamental bound for multi-parameter quantum estimation is the Holevo Cram\'er-Rao bound (HCRB) whose evaluation has so far remained elusive. For finite-dimensional systems we recast its evaluation as a semidefinite program, with reduced size for rank-deficient states. We use this result to study phase and loss estimation in optical interferometry and three-dimensional magnetometry with noisy multi-qubit systems. For the former, we show that, in some regimes, it is possible to attain the HCRB with the optimal (single-copy) measurement for phase estimation. For the latter, we show a non-trivial interplay between the HCRB and incompatibility, and provide numerical evidence that projective single-copy measurements attain the HCRB in the noiseless two-qubit case.