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Improving Accuracy of Estimating Two-Qubit States with Hedged Maximum Likelihood

As a widely used reconstruction algorithm in quantum state tomography, maximum likelihood estimation tends to assign a rank-deficient matrix, which decreases estimation accuracy for certain quantum states. Fortunately, hedged maximum likelihood estimation (HMLE) [Phys. Rev. Lett. 105 (2010) 200504] was proposed to avoid this problem. Here we study more details about this proposal in the two-qubit case and further improve its performance. We ameliorate the HMLE method by updating the hedging function based on the purity of the estimated state. Both performances of HMLE and ameliorated HMLE are demonstrated by numerical simulation and experimental implementation on the Werner states of polarization-entangled photons.