  This paper presents a series of results on the interplay between quantum
estimation, cloning and finite de Finetti theorems. First, we consider the
measure-and-prepare channel that uses optimal estimation to convert M copies
into k approximate copies of an unknown pure state and we show that this
channel is equal to a random loss of all but s particles followed by cloning
from s to k copies. When the number k of output copies is large with respect to
the number M of input copies the measure-and-prepare channel converges in
diamond norm to the optimal universal cloning. In the opposite case, when M is
large compared to k, the estimation becomes almost perfect and the
measure-and-prepare channel converges in diamond norm to the partial trace over
all but k systems. This result is then used to derive de Finetti-type results
for quantum states and for symmetric broadcast channels, that is, channels that
distribute quantum information to many receivers in a permutationally invariant
fashion. Applications of the finite de Finetti theorem for symmetric broadcast
channels include the derivation of diamond-norm bounds on the asymptotic
convergence of quantum cloning to state estimation and the derivation of bounds
on the amount of quantum information that can be jointly decoded by a group of
k receivers at the output of a symmetric broadcast channel.
