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.
@article{arxiv.1904.00557,
title = {Multi-outcome homodyne detection in a coherent-state light interferometer},
author = {J. Z. Wang and Z. Q. Yang and A. X. Chen and W. Yang and G. R. Jin},
journal= {arXiv preprint arXiv:1904.00557},
year = {2019}
}