Economical phase-covariant cloning of qudits

Abstract

We derive the optimal $N\ensuremath{\rightarrow}M$ 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.