Quantum tomography of pure states with projective measurements generated by MUBs and SIC-POVM

The article undertakes the problem of pure state estimation based on projective measurements. Two generic frames for qubit tomography are considered -- one defined by the vectors from the mutually unbiased bases (MUBs) and the other composed of the elements of the SIC-POVM. Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits. The accuracy of each frame is quantified by the average fidelity and purity. The efficiency of the frames is compared and discussed. The method can be generalized to higher-dimensional states and transferred to other fields where the problem of complex vectors reconstruction appears.