# Bayesian multi-parameter quantum metrology with limited data

## TL;DR

This paper develops a Bayesian framework for multi-parameter quantum metrology that optimally extracts information with limited data, accounting for prior knowledge, and demonstrates its effectiveness in practical quantum sensing scenarios.

## Contribution

It introduces a new Bayesian quantum bound, constructs optimal measurements for single-shot scenarios, and extends to repeated measurements, addressing practical limitations in quantum metrology.

## Key findings

- New Bayesian multi-parameter quantum bound derived
- Optimal measurement strategies constructed for single-shot cases
- Method effectively incorporates measurement count and prior info

## Abstract

A longstanding problem in quantum metrology is how to extract as much information as possible in realistic scenarios with not only multiple unknown parameters, but also limited measurement data and some degree of prior information. Here we present a practical solution to this: we derive a new Bayesian multi-parameter quantum bound, construct the optimal measurement when our bound can be saturated for a single shot and consider experiments involving a repeated sequence of these measurements. Our method properly accounts for the number of measurements and the degree of prior information, and we illustrate our ideas with a qubit sensing network and a model for phase imaging, clarifying the non-asymptotic role of local and global schemes. Crucially, our technique is a powerful way of implementing quantum protocols in a wide range of practical scenarios that tools such as the Helstrom and Holevo Cram\'{e}r-Rao bounds cannot normally access.