Collective versus local measurements in a qubit mixed-state estimation

We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as $1/N.$ For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as ${1/N}^{1/4},$ while our approach leads to an error proportional to ${1/N}^{3/4}.$