Quantum information is incompressible without errors

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.