  A classical random variable can be faithfully compressed into a sequence of
bits with its expected length lies within one bit of Shannon entropy. We
generalize this variable-length and faithful scenario to the general quantum
source producing mixed states $\rho_i$ with probability $p_i$. In contrast to
the classical case, the optimal compression rate in the limit of large block
length differs from the one in the fixed-length and asymptotically faithful
scenario. The amount of this gap is interpreted as the genuinely quantum part
being incompressible in the former scenario.
