Measuring the purity of a qubit state: entanglement estimation with fully separable measurements

Given a finite number $N$ of copies of a qubit state we compute the maximum fidelity that can be attained using joint-measurement protocols for estimating its purity. We prove that in the asymptotic $N\to\infty$ limit, separable-measurement protocols can be as efficient as the optimal joint-measurement one if classical communication is used. This in turn shows that the optimal estimation of the entanglement of a two-qubit state can also be achieved asymptotically with fully separable measurements. The relationship between our global Bayesian approach and the quantum Cramer-Rao bound is also discussed.