General Mixed-State Quantum Data Compression With and Without Entanglement Assistance

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.