  We pose the question whether the asymptotic equivalence between quantum
cloning and quantum state estimation, valid at single-copy level, still holds
when all the copies are examined jointly. For an N-to-M cloner, we consider the
overall fidelity between the state of the M output systems and the state of M
ideal copies, and we ask whether the optimal fidelity is attained by a
measure-and- prepare protocol in the limit of large M. In order to gain
intuition into the general problem, we analyze two concrete examples: i)
cloning qubit states on the equator of the Bloch sphere and ii) cloning
two-qubit maximally entangled states. In the first case, we show that the
optimal measure-and- prepare fidelity converges to the fidelity of the optimal
cloner in the limit of large M. In the second case, we restrict our attention
to economical covariant cloners, and again, we exhibit a measure- and-prepare
protocol that achieves asymptotically the optimal fidelity. Quite
counterintuitively, in both cases the optimal states that have to be prepared
in order to maximize the overall fidelity are not product states corresponding
to M identical copies, but instead suitable M-partite entangled states: the
simple protocol where one estimates the input state and re-prepares M identical
copies of the estimated state is strictly suboptimal, even in the asymptotic
limit.
