  We consider the most general (finite-dimensional) quantum mechanical
information source, which is given by a quantum system $A$ that is correlated
with a reference system $R$. The task is to compress $A$ in such a way as to
reproduce the joint source state $\rho^{AR}$ at the decoder with asymptotically
high fidelity. This includes Schumacher's original quantum source coding
problem of a pure state ensemble and that of a single pure entangled state, as
well as general mixed state ensembles. Here, we determine the optimal
compression rate (in qubits per source system) in terms of the Koashi-Imoto
decomposition of the source into a classical, a quantum, and a redundant part.
The same decomposition yields the optimal rate in the presence of unlimited
entanglement between compressor and decoder, and indeed the full region of
feasible qubit-ebit rate pairs.
