We study the minimal input sets which can determine completely the universal
and the phase-covariant quantum cloning machines. We find that the universal
quantum cloning machine, which can copy arbitrary input qubit equally well,
however can be determined completely by only four input states located at the
four vertices of a tetrahedron. The phase-covariant quantum cloning machine,
which can copy all qubits located on the equator of the Bloch sphere, can be
determined by three equatorial qubits with equal angular distance. These
results sharpen further the well-known results that BB84 states and six-states
used in quantum cryptography can determine completely the phase-covariant and
universal quantum cloning machines. This concludes the study of the power of
universal and phase-covariant quantum cloning, i.e., from minimal input sets
necessarily to full input sets by definition. This can simplify dramatically
the testing of whether the quantum clone machines are successful or not, we
only need to check that the minimal input sets can be cloned optimally.Comment: 7 pages, 4 figure
