Optimal symmetric economical phase-covariant quantum cloning

We derive the explicit transformation of the optimal symmetric economical 1 -> M phase-covariant quantum cloning machine (EPCCM), which works without ancilla, as well as the corresponding fidelities in 2-dimension. The fidelity (1 -> M = 2k + 1 EPCCM) is covered with previous contributions [G.M. D'Ariano, et al., Phys. Rev. A 71 (2005) 042327]. Meanwhile, the explicit transformation of the optimal symmetric 1 -> M phase-covariant quantum cloning machine (PCCM) (working with ancilla) as well as the corresponding fidelities in 2-dimension is also derived. The results of the PCCM agree with previous contributions [H. Fan, et al., Phys. Rev. A 65 (2001) 012304]. (c) 2006 Elsevier B.V. All rights reserved.