  Coded source compression, also known as source compression with helpers, has
been a major variant of distributed source compression, but has hitherto
received little attention in the quantum regime. This work treats and solves
the corresponding quantum coded source compression through an observation that
connects coded source compression with channel simulation. First, we consider
classical source coding with quantum side information where the quantum side
information is observed by a helper and sent to the decoder via a classical
channel. We derive a single-letter characterization of the achievable rate
region for this problem. The direct coding theorem of our result is proved via
the measurement compression theory of Winter, a quantum-to-classical channel
simulation. Our result reveals that a helper's scheme which separately conducts
a measurement and a compression is suboptimal, and measurement compression
seems necessary to achieve the optimal rate region. We then study coded source
compression in the fully quantum regime, where two different scenarios are
considered depending on the types of communication channels between the
legitimate source and the receiver. We further allow entanglement assistance
from the quantum helper in both scenarios. We characterize the involved quantum
resources, and derive single-letter expressions of the achievable rate region.
The direct coding proofs are based on well-known quantum protocols, the quantum
state merging protocol and the fully quantum Slepian-Wolf protocol, together
with the quantum reverse Shannon theorem.
