# When heterodyning beats homodyning: an assessment with quadrature moments

## TL;DR

This paper compares homodyne and heterodyne measurement schemes for quantum state reconstruction, demonstrating the advantages of heterodyne measurements in estimating moments, especially for non-classical states, with optimized estimators reaching maximum-likelihood efficiency.

## Contribution

It introduces optimized moment estimators for both schemes and provides a comprehensive analysis showing heterodyne measurement's superiority in reconstructing quantum state moments.

## Key findings

- Heterodyne measurement outperforms homodyne in moment reconstruction.
- Optimized estimators achieve maximum-likelihood efficiency in large data limits.
- Heterodyne is particularly effective for non-classical states.

## Abstract

We examine the moment-reconstruction performance of both the homodyne and heterodyne (double-homodyne) measurement schemes for arbitrary quantum states and introduce moment estimators that optimize the respective schemes for any given data. In the large-data limit, these estimators are as efficient as the maximum-likelihood estimators. We then illustrate the superiority of the heterodyne measurement for the reconstruction of the first and second moments by analyzing Gaussian states and many other significant non-classical states.