Extremal equation for optimal completely positive maps

We derive an extremal equation for an optimal completely positive map, which most closely approximates a given transformation between pure quantum states. Moreover, we also obtain an upper bound on the maximal mean fidelity that can be attained by the optimal approximate transformation. The developed formalism is applied to a universal-NOT gate, quantum cloning machines, quantum entanglers, and a qubit {ϑ} shifter.