  The study of quantum cryptography and quantum entanglement has traditionally
been based on two-level quantum systems (qubits) and more recently on
three-level systems (qutrits). We investigate several classes of
state-dependent quantum cloners for four-level systems (quartits). These
results apply to symmetric as well as asymmetric cloners, so that the balance
between the fidelity of the two clones can also be analyzed. We extend Cerf's
formalism for cloning states in order to derive cloning machines that remain
invariant under certain unitary transformations. Our results show that a
different cloner has to be used for two mutually unbiased bases which are
related by a double Hadamard transformation, than for two mutually unbiased
bases that are related by a Fourier transformation. This different cloner is
obtained thanks to a redefinition of Bell states that respects the intrinsic
symmetries of the Hadamard transformation.
