Economical Phase-Covariant Cloning of Qudits
  We derive the optimal N to M phase-covariant quantum cloning for equatorial states in dimension d with M=kd+N, k integer. The cloning maps are optimal for both global and single-qudit fidelity. The map is achieved by an ``economical'' cloning machine, which works without ancilla. 
Francesco Buscemi, Giacomo Mauro D'Ariano, and Chiara Macchiavello