  We prove new quantitative limitations on any approximate simultaneous cloning
or broadcasting of mixed states. The results are based on information-theoretic
(entropic) considerations and generalize the well known no-cloning and
no-broadcasting theorems. We also observe and exploit the fact that the
universal cloning machine on the symmetric subspace of $n$ qudits and
symmetrized partial trace channels are dual to each other. This duality
manifests itself both in the algebraic sense of adjointness of quantum channels
and in the operational sense that a universal cloning machine can be used as an
approximate recovery channel for a symmetrized partial trace channel and vice
versa. The duality extends to give control on the performance of generalized
UQCMs on subspaces more general than the symmetric subspace. This gives a way
to quantify the usefulness of a-priori information in the context of cloning.
For example, we can control the performance of an antisymmetric analogue of the
UQCM in recovering from the loss of $n-k$ fermionic particles.
