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Quantum state tomography with finite sample size

The estimation of the density matrix of a k-level quantum system is studied when the parametrization is given by the real and imaginary part of the entries and they are estimated by a finite number of independent measurements. For pure or nearly pure states, a simple and provable correct constraining algorithm is proposed that yields an asymptotically unbiased estimate. The quality of the estimate is characterized by the mean quadratic error (MQE) matrix. The determinant of the averaged MQE matrix is minimal for mutually unbiased measurements. For 2-label systems, an adaptive optimal two-step-measurement scheme is proposed that uses a set of initialization measurements to determine the number of measurements in each of the measurement directions that is applied in the second step. An improved version of the algorithm uses the rotation of the measurement axes in the second step, too.