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 $\sum_i p_i \Psi_i$, 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 the 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.