Maximum-likelihood reconstruction of completely positive maps

We present a method for the determination of the completely positive ~CP! map describing a physical device based on random preparation of the input states, random measurements at the output, and the maximumlikelihood principle. In the numerical implementation the constraint of completely positivity can be imposed by exploiting the isomorphism between linear transformations from Hilbert spaces H to K and linear operators in K^ H. The effectiveness of the method is shown on the basis of some examples of reconstruction of CP maps related to quantum communication channels for qubits.

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