  Bu\v{z}ek and Hillery proposed a universal quantum-copying machine (UQCM)
(i.e., transformation) to analyze the possibility of cloning arbitrary states.
The UQCM copies quantum-mechanical states with the quality of its output does
not depend on the input. We propose a slightly different transformation to
analyze a restricted set of input states. We impose the conditions (I) the
density matrices of the two output states are the same, and that (II) the
distance between input density operator and the output density operators is
input state independent. Using Hilbert-Schmidt norm and Bures fidelity, we show
that our transformation can achieves the bound of the fidelity.
