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In this article we propose a method to estimate with high accuracy purequantum states of a single qudit. Our method is based on the minimization ofthe squared error between the complex probability amplitudes of the unknownstate and its estimate. We show by means of numerical experiments that theestimation accuracy of the present method, which is given by the expectation ofthe squared error on the sample space of estimates, is state independent.Furthermore, the estimation accuracy delivered by our method is close to twicethe Gill-Massar lower bound, which represents the best achievable accuracy, forall inspected dimensions. The minimization problem is solved via theconcatenation of the Complex simultaneous perturbation approximation, aniterative stochastic optimization method that works within the field of thecomplex numbers, and Maximum likelihood estimation, a well-known statisticalinference method. This can be carried out with the help of a multi-arminterferometric array. In the case of a single qubit, a Mach-Zehnderinterferometer suffices. We also show that our estimation procedure can beeasily extended to estimate unknown unitary transformations acting on a singlequdit. Thereby, the estimation of unitary transformations achieves a higheraccuracy than that achieved by processes based on tomographic methods for mixedstates.

Mean-squared-error-based adaptive estimation of pure quantum states and unitary transformations