# Optimal measurements for simultaneous quantum estimation of multiple phases

## TL;DR

This paper develops a theoretical framework for identifying optimal measurements in multiphase quantum estimation, crucial for quantum sensing, by deriving conditions for projective measurements to reach fundamental precision bounds.

## Contribution

It introduces necessary and sufficient conditions for projective measurements on pure states to saturate quantum Fisher information bounds in multiphase estimation.

## Key findings

- Derived conditions for optimal projective measurements.
- Applied theory to interferometric phase estimation with photon number measurements.
- Provided insights for future quantum metrology experiments.

## Abstract

A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this letter we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the maximal theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.