Optimal 1→M phase-covariant cloning in three dimensions

In this paper, we derive the explicit transformations of the optimal 1→3, 4, 5 phase-covariant cloning in three dimensions, and then generalize them to the cases of 1→M = 3n, 3n+1, 3n+2 (n≥1 integer) cloning. The clone fidelities are coincident with the theoretical bounds found.