Minimal input sets determining phase-covariant and universal quantum cloning

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 an arbitrary input qubit to two identical copies, however, can be determined completely by only four input states located at the four vertices of a tetrahedron in a Bloch sphere. The phase-covariant quantum cloning machine, which can create two copies from an arbitrary qubit located on the equator of the Bloch sphere, can be determined by three qubits located symmetrically on the equator of the Bloch sphere with equal relative phase. These results sharpen further the well-known results that Bennett-Brassard 1984 protocol (BB84) states and six states used in quantum cryptography can determine completely the phase-covariant and universal quantum cloning machines. This can simplify the testing procedure of whether the quantum clone machines are successful or not; namely, we only need to check that the minimal input sets can be cloned optimally, which can ensure that the quantum clone machines can work well for all input states.