Optimal phase estimation and square root measurement

We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite-dimensional system using a finite number of identical samples. Assuming the uniform a priori distribution for the displacement parameter, an optimal strategy can be constructed by making the square root measurement based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so-called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two-dimensional system.