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 → ∞ 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. Thus, quantum memories provide no advantage in this situation. The relationship between our global Bayesian approach and the quantum Cram´er-Rao bound is also discussed. The ultimate goal of quantum state estimation is to determine the value of the parameters that fully char-acterize a given unknown quantum state. However, in practical applications, a partial characterization is often all one needs. Thus, e.g., knowing the purity of a qubit state or the degree of entanglement of a bipartite state be whether it perform