  Entanglement distillation is a fundamental information processing task whose
implementation is key to quantum communication and modular quantum computing.
Noise experienced by such communication and computing platforms occurs not only
in the form of Pauli noise such as dephasing (sometimes called $T_2$) but also
non-Pauli noise such as amplitude damping (sometimes called $T_1$). We initiate
a study of practical and asymptotic distillation over what we call the joint
damping-dephasing noise channel. In the practical setting, we propose a
distillation scheme that completely isolates away the damping noise. In the
asymptotic setting we derive lower bounds on the entanglement sharing
capacities including the coherent and reverse coherent information. Like the
protocol achieving the reverse coherent information, our scheme uses only
backward classical communication. However, for realistic damping noise ($T_1
\neq 2T_2$) our strategy can exceed the reverse coherent strategy, which is the
best known for pure damping. In the forward communication setting we
numerically exceed the single-letter coherent information strategy by observing
the channel displays non-additivity at the two-letter level. The work shows
non-additivity can also be found in realistic noise models with magnitudes of
non-additivity similar to those found in more idealized noise channels.
