  We revisit the task of visible compression of an ensemble of quantum states
with entanglement assistance in the one-shot setting. The protocols achieving
the best compression use many more qubits of shared entanglement than the
number of qubits in the states in the ensemble. Other compression protocols,
with potentially larger communication cost, have entanglement cost bounded by
the number of qubits in the given states. This motivates the question as to
whether entanglement is truly necessary for compression, and if so, how much of
it is needed.
  Motivated by questions in communication complexity, we lift certain
restrictions that are placed on compression protocols in tasks such as
state-splitting and channel simulation. We show that an ensemble of the form
designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds
on the sum of communication and entanglement costs, even with the relaxed
compression protocols we study.
  The ensemble and the associated one-way communication protocol have several
remarkable properties. The ensemble is incompressible by more than a constant
number of qubits without shared entanglement, even when constant error is
allowed. Moreover, in the presence of shared entanglement, the communication
cost of compression can be arbitrarily smaller than the entanglement cost. The
quantum information cost of the protocol can thus be arbitrarily smaller than
the cost of compression without shared entanglement. The ensemble can also be
used to show the impossibility of reducing, via compression, the shared
entanglement used in two-party protocols for computing Boolean functions.
