# Optimal probes and error-correction schemes in multi-parameter quantum metrology

## TL;DR

This paper establishes conditions for achieving Heisenberg scaling in multi-parameter quantum metrology under noise, and introduces optimal error correction schemes to enhance estimation precision, demonstrating advantages over separate protocols.

## Contribution

It provides a necessary and sufficient condition for Heisenberg scaling and a semidefinite program to find optimal quantum error correction protocols in noisy multi-parameter estimation.

## Key findings

- Heisenberg scaling is achievable under certain conditions with noise.
- Joint-QEC protocols outperform separate-QEC protocols in multi-parameter sensing.
- A semidefinite program identifies optimal error correction strategies.

## Abstract

We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cram\'er-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.