  We resume the investigation of the problem of independent local compression
of correlated quantum sources, the classical case of which is covered by the
celebrated Slepian-Wolf theorem. We focus specifically on classical-quantum
(cq) sources, for which one edge of the rate region, corresponding to the
compression of the classical part, using the quantum part as side information
at the decoder, was previously determined by Devetak and Winter [Phys. Rev. A
68, 042301 (2003)]. Whereas the Devetak-Winter protocol attains a rate-sum
equal to the von Neumann entropy of the joint source, here we show that the
full rate region is much more complex, due to the partially quantum nature of
the source. In particular, in the opposite case of compressing the quantum part
of the source, using the classical part as side information at the decoder,
typically the rate sum is strictly larger than the von Neumann entropy of the
total source.
  We determine the full rate region in the generic case, showing that, apart
from the Devetak-Winter point, all other points in the achievable region have a
rate sum strictly larger than the joint entropy. We can interpret the
difference as the price paid for the quantum encoder being ignorant of the
classical side information. In the general case, we give an achievable rate
region, via protocols that are built on the decoupling principle, and the
principles of quantum state merging and quantum state redistribution. Our
achievable region is matched almost by a single-letter converse, which however
still involves asymptotic errors and an unbounded auxiliary system.
