Optimal estimation of a physical observable's expectation value for pure states

We study the optimal way to estimate the quantum expectation value of a physical observable when a finite number of copies of a quantum pure state are presented. The optimal estimation is determined by minimizing the squared error averaged over all pure states distributed in a unitary invariant way. We find that the optimal estimation is "biased", though the optimal measurement is given by successive projective measurements of the observable. The optimal estimate is not the sample average of observed data, but the arithmetic average of observed and "default nonobserved" data, with the latter consisting of all eigenvalues of the observable.