  Ask how the quantum compression of ensembles of pure states is affected by
the availability of entanglement, and in settings where the encoder has access
to side information. We find the optimal asymptotic quantum rate and the
optimal tradeoff (rate region) of quantum and entanglement rates. It turns out
that the amount by which the quantum rate beats the Schumacher limit, the
entropy of the source, is precisely half the entropy of classical information
that can be extracted from the source and side information states without
disturbing them at all ("reversible extraction of classical information").
  In the special case that the encoder has no side information, or that she has
access to the identity of the states, this problem reduces to the known
settings of blind and visible Schumacher compression, respectively, albeit here
additionally with entanglement assistance. We comment on connections to
previously studied and further rate tradeoffs when also classical information
is considered.
