Rate-Distortion Theory for Mixed States

In this paper we consider the compression of asymptotically many i.i.d. copies of ensembles of mixed quantum states where the encoder has access to a side information system. This source is equivalently defined as a classical-quantum state, namely, a quantum system correlated with a classical system playing the role of an inaccessible reference system. The figure of merit is evaluated based on per-copy or local error criterion. Under this set-up, known as a rate-distortion set-up, one can study the trade-off between the compression rate and the error. The optimal trade-off can be characterized by the rate-distortion function, which is the best rate given a certain distortion. We find the rate-distortion functions in the entanglement-assisted and unassisted scenarios, in terms of a single-letter mutual information quantity and the regularized entanglement of purification, respectively. We also consider the general case when both communication and entanglement are charged, and 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.