# Multi-outcome homodyne detection in a coherent-state light interferometer

## TL;DR

This paper develops a family of inversion estimators for multi-outcome homodyne detection in a coherent-light interferometer that nearly saturate the Cramér-Rao bound across all phase ranges, advancing quantum parameter estimation.

## Contribution

It introduces a new class of inversion estimators for multi-outcome measurements that approach the quantum limit for phase estimation.

## Key findings

- Estimators nearly saturate the Cramér-Rao bound across all phase intervals.
- Provides insights into constructing optimal estimators for multi-outcome quantum measurements.
- Enhances understanding of measurement strategies in quantum interferometry.

## Abstract

The Cram\'{e}r-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cram\'{e}r-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any multi-outcome measurement.