Single-shot Coherent-State Optical Phase Estimation with Adaptive Photon Counting Measurements*

Single-shot phase estimation for coherent states has a wide variety of applications for quantum information sciences including quantum metrology, enhanced quantum sensing, and quantum optics. The optimal measurement for phase estimation corresponds to the canonical phase measurement, which minimizes the error of phase estimators. However, physical implementations of the canonical phase measurement for the optical phase are unknown. Here, we investigate a practical realization of non-Gaussian measurements based on adaptive photon-counting and optimized displacements operations. We show that the optimization of coherent displacement operations by a suitable cost function, such as mutual information, allows these non-Gaussian measurements to surpass the standard heterodyne limit and outperform adaptive homodyne measurements. We numerically show that adaptive photon counting measurements maximizing information gain approach the canonical phase measurement in the asymptotic limit with a high convergence rate.