Convex approximations of quantum channels

We address the problem of optimally approximating the action of a desired and unavailable quantum channel Phi having at our disposal a single use of a given set of other channels {Psi(i)}. The problem is recast to look for the least distinguishable channel from Phi among the convex set Sigma(i) p(i) Psi(i), and the corresponding optimal weights {p(i)} provide the optimal convex mixing of the available channels {Psi(i)}. 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.