  This paper is concerned with quantum data compression of asymptotically many
independent and identically distributed copies of ensembles of mixed quantum
states. The encoder has access to a side information system. The figure of
merit is per-copy or local error criterion. Rate-distortion theory studies the
trade-off between the compression rate and the per-copy error. The optimal
trade-off can be characterized by the rate-distortion function, which is the
best rate given a certain distortion. In this paper, we derive the
rate-distortion function of mixed-state compression. The rate-distortion
functions in the entanglement-assisted and unassisted scenarios are in terms of
a single-letter mutual information quantity and the regularized entanglement of
purification, respectively. For the general setting where the consumption of
both communication and entanglement are considered, we present the full
qubit-entanglement rate region. Our compression scheme covers both blind and
visible compression models (and other models in between) depending on the
structure of the side information system.
