Performing homodyne detection at one port of squeezed-state light interferometer and then binarzing measurement data are important to achieve super-resolving and super-sensitive phase measurements. Here we propose a new data-processing technique by d
ividing the measurement quadrature into three bins (equivalent to a multi-outcome measurement), which leads to a higher improvement in the phase resolution and the phase sensitivity under realistic experimental condition. Furthermore, we develop a new phase-estimation protocol based on a combination of the inversion estimators of each outcome and show that the estimator can saturate the Cramer-Rao lower bound, similar to asymptotically unbiased maximum likelihood estimator.
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. H
ere 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.
