Geometry of optimal estimation scheme for SU(D) channels

The problem of estimating an unknown SU(D) channel Gamma(U) : rho /-> U rho U* is studied based on the quantum Cramer-Rao inequality. It is shown that the minimum estimation error is of O(1/n(2)), where n is the degree of extension of the channel. The mechanism behind this asymptotic behaviour is investigated from a differential geometrical point of view.