Optimal full estimation of qubit mixed states

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number $N$ of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary $N$, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large $N$. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.