Optimal phase estimation and square root measurement

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