### Abstract

Given a number N of identically prepared qubit mixed states, I analyse the optimal estimation protocol, based on collective measurements. I discuss two scenarios: completely general states (3D), and states known to lie on the equatorial plane of the Bloch sphere (2D). I will derive the optimal POVMs for finite number of copies and obtain the analytical expressions of the fidelities in the large N regime. In 3D, I will show that the optimal POVM's are independent of the prior distributions, provided they are isotropic. In 2D this is only true asymptotically. However, for the latter, the fidelity is F=1-1/(2N), independently of the prior.