Phase-covariant quantum cloning of qudits
  We study the phase-covariant quantum cloning machine for qudits, i.e. the input states in d-level quantum system have complex coefficients with arbitrary phase but constant module. A cloning unitary transformation is proposed. After optimizing the fidelity between input state and single qudit reduced density opertor of output state, we obtain the optimal fidelity for 1 to 2 phase-covariant quantum cloning of qudits and the corresponding cloning transformation. 
Heng Fan, Hiroshi Imai, Keiji Matsumoto, Xiang-Bin Wang