We address the general problem of removing correlations from quantum states while preserving local quantum information as much as possible. We provide a complete solution in the case of two qubits, by evaluating the minimum amount of noise that is necessary to decorrelate covariant sets of bipartite states. We show that two harmonic oscillators in arbitrary Gaussian state can be decorrelated by a Gaussian covariant map. Finally, for finite-dimensional Hilbert spaces, we prove that states obtained from most cloning channels (e.g., universal and phase-covariant cloning) can be decorrelated only at the expense of a complete erasure of information about the copied state. More generally, in finite dimension, cloning without correlations is impossible for continuous sets of states. On the contrary, for continuos variables cloning, a slight modification of the customary set-up for cloning coherent states allows one to obtain clones without correlations.

11 pages, 2 figures, RevTex4