Geometry of optimal estimation scheme for SU(D) channels

The problem of estimating an unknown SU(D) channel ΓU:ρ UρU* is studied based on the quantum Cramer–Rao inequality. It is shown that the minimum estimation error is of O(1/n2), 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.