  We derive quantum counterparts of two key theorems of classical information
theory, namely, the rate distortion theorem and the source-channel separation
theorem. The rate-distortion theorem gives the ultimate limits on lossy data
compression, and the source-channel separation theorem implies that a two-stage
protocol consisting of compression and channel coding is optimal for
transmitting a memoryless source over a memoryless channel. In spite of their
importance in the classical domain, there has been surprisingly little work in
these areas for quantum information theory. In the present paper, we prove that
the quantum rate distortion function is given in terms of the regularized
entanglement of purification. We also determine a single-letter expression for
the entanglement-assisted quantum rate distortion function, and we prove that
it serves as a lower bound on the unassisted quantum rate distortion function.
This implies that the unassisted quantum rate distortion function is
non-negative and generally not equal to the coherent information between the
source and distorted output (in spite of Barnum's conjecture that the coherent
information would be relevant here). Moreover, we prove several quantum
source-channel separation theorems. The strongest of these are in the
entanglement-assisted setting, in which we establish a necessary and sufficient
codition for transmitting a memoryless source over a memoryless quantum channel
up to a given distortion.
