Anisotropic efficiency of adaptive measurements in two-parameter estimation for states in manifolds with nonzero Berry curvature

The efficiency of quantum measurements in distinguishing neighboring quantum states is a central topic in quantum information sciences. We study the efficiency of vertex measurement in two-parameter estimation from the information geometry perspective, where the vertex measurement is a rare typical measurement recognized as optimal for both single-parameter estimation and multiparameter estimation of states in manifolds with vanishing Berry curvature. This article focuses explicitly on scenarios with nonzero Berry curvature. We quantify the precision of an estimation scheme with the density of distinguishable states and measure the efficiency of a given vertex measurement by comparing its precision to the maximum value. Exemplified by the three-level system with two parameters, we show that the efficiency is highly anisotropic concerning the direction of a small mismatch vector between the applied measurement and the estimated parameter. It may even vanish in some specific directions. Furthermore, the direction is hardly acknowledged prior, hindering the practical application of vertex measurement in multiparameter estimations. To remove this obstacle, we further introduce a scheme that can adaptively approach the optimal vertex measurements.