Adaptive identification of coherent states

Abstract

We present methods for efficient characterization of an optical coherent state $|\alpha \rangle$. We choose measurement settings adaptively and stochastically, based on data while it is collected. Our algorithm divides the estimation into two distinct steps: (i) before the first detection of a vacuum state, the probability of choosing a measurement setting is proportional to detecting vacuum with the setting, which makes using too similar measurement settings twice unlikely; and (ii) after the first detection of vacuum, we focus measurements in the region where vacuum is most likely to be detected. In step (i) [(ii)] the detection of vacuum (a photon) has a significantly larger effect on the shape of the posterior probability distribution of $\alpha$. Compared to nonadaptive schemes, our method makes the number of measurement shots required to achieve a certain level of accuracy smaller approximately by a factor proportional to the area describing the initial uncertainty of $\alpha$ in phase space. While this algorithm is not directly robust against readout errors, we make it such by introducing repeated measurements in step (i).