  A direct derivation is given for the optimal mean fidelity of quantum state
estimation of a d-dimensional unknown pure state with its N copies given as
input, which was first obtained by M. Hayashi in terms of an infinite set of
covariant positive operator valued measures (POVM's) and by Bruss and
Macchiavello establishing a connection to optimal quantum cloning. An explicit
condition for POVM measurement operators for optimal estimators is obtained, by
which we construct optimal estimators with finite POVM using exact quadratures
on a hypersphere. These finite optimal estimators are not generally universal,
where universality means the fidelity is independent of input states. However,
any optimal estimator with finite POVM for M(>N) copies is universal if it is
used for N copies as input.
