Optimal estimation of quantum dynamics

We construct the optimal strategy for the estimation of an unknown unitary transformation $U\in \mathrm{SU}\mathrm{}\left(d\right).$ This includes, in addition to a convenient measurement on a probe system, finding which is the best initial state on which U is to act. When $U\in \mathrm{SU}\mathrm{}\left(2\right),$ such an optimal strategy can be applied to simultaneously estimate both the direction and the strength of a magnetic field, and shows how to use a spin 1/2 particle to transmit information about a whole coordinate system instead of only a direction in space.