2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/92217We 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.10 pages revtex4, 7 figures, replaced with modified versionQuantum Physicstext