  In blind compression of quantum states, a sender Alice is given a specimen of
a quantum state $\rho$ drawn from a known ensemble (but without knowing what
$\rho$ is), and she transmits sufficient quantum data to a receiver Bob so that
he can decode a near perfect specimen of $\rho$. For many such states drawn iid
from the ensemble, the asymptotically achievable rate is the number of qubits
required to be transmitted per state. The Holevo information is a lower bound
for the achievable rate, and is attained for pure state ensembles, or in the
related scenario of entanglement-assisted visible compression of mixed states
wherein Alice knows what state is drawn.
  In this paper, we prove a general and robust lower bound on the achievable
rate for ensembles of classical states, which holds even in the least demanding
setting when Alice and Bob share free entanglement and a constant per-copy
error is allowed. We apply the bound to a specific ensemble of only two states
and prove a near-maximal separation (saturating the dimension bound in leading
order) between the best achievable rate and the Holevo information for constant
error. This also implies that the ensemble is incompressible -- compression
does not reduce the communication cost by much. Since the states are classical,
the observed incompressibility is not fundamentally quantum mechanical. We
lower bound the difference between the achievable rate and the Holevo
information in terms of quantitative limitations to clone the specimen or to
distinguish the two classical states.
