  We discuss a problem of parameter estimation for quantum two-level system,
qubit system, in presence of unknown phase parameter. We analyze trade-off
relations for mean-square errors when estimating relevant parameters with
separable measurements based on known precision bounds; the symmetric
logarithmic derivative Cramer-Rao bound and Hayashi-Gill-Massar (HGM) bound. We
investigate the optimal measurement which attains the HGM bound and discuss its
properties. We show that the HGM bound for relevant parameters can be attained
asymptotically by using some fraction of given $n$ quantum states to estimate
the phase parameter. We also discuss the Holevo bound which can be attained
asymptotically by a collective measurement.
