Convex approximations of quantum channels

We address the problem of optimally approximating the action of a desired and unavailable quantum channel Φ having at our disposal a single use of a given set of other channels {Ψi}. The problem is recast to look for the least distinguishable channel from Φ among the convex set ?ipiΨi, and the corresponding optimal weights {pi} provide the optimal convex mixing of the available channels {Ψi}. For single-qubit channels we study specifically cases where the available convex set corresponds to covariant channels or to Pauli channels, and the desired target map is an arbitrary unitary transformation or a generalized damping channel.