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 maximum-likelihood principle. In the numerical implementation the constraint of completely positivity can be imposed by exploiting the isomorphism between linear transformations from Hilbert spaces $\mathcal{H}$ to $\mathcal{K}$ and linear operators in $\mathcal{K}\otimes \mathcal{H}.$ 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.