Optimal phase-covariant cloning for qubits and qutrits

We consider cloning transformations of equatorial qubits $|{\psi }_{\phi }〉=1/\sqrt{2}\left(|0〉{+e}^{i\phi }|1〉\right)$ and qutrits $|{\psi }_{\phi ,\theta }〉=1/\sqrt{3}\left(|0〉{+e}^{i\phi }|1〉{+e}^{i\theta }|2〉\right),$ with the transformation covariant for rotation of the phases $\phi$ and $\theta .$ The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.