We consider cloning transformations of d-dimensional states of the form e^{ i\phi_0}|0> + e^{ i\phi_1}|1> +...+ e^{ i\phi_{ d-1}}|d-1> that are covariant with respect to rotations of the phases \phi_i's. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent 1 -> 2 cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.
Categories: quant-ph