  Quantum communication relies on the existence of high quality quantum
channels to exchange information. In practice, however, all communication links
are affected by noise from the environment. Here we investigate the ability of
quantum channels to perform quantum communication tasks by restricting the
participants to use only local operations and one-way classical communication
(one-way LOCC) along with the available quantum channel. In particular, a
channel can be used to distill a highly entangled state between two parties,
which further enables quantum or private communication. In this work, we invoke
the framework of superchannels to study the distillation of a resourceful
quantum state, such as a maximally entangled state or a private state, using
multiple instances of a point-to-point quantum channel. We use the idea of
$k$-extendibility to obtain a semidefinite relaxation of the set of one-way
LOCC superchannels and define a class of entanglement measures for quantum
channels that decrease monotonically under such superchannels; therefore these
measures, dubbed collectively the ``unextendible entanglement of a channel'',
yield upper bounds on several communication-theoretic quantities of interest in
the regimes of resource distillation and zero error. We then generalize the
formalism of $k$-extendibility to bipartite superchannels, thus obtaining
functions that are monotone under two-extendible superchannels. This allows us
to analyze probabilistic distillation of ebits or secret key bits from a
bipartite state when using a resourceful quantum channel. Moreover, we propose
semidefinite programs to evaluate several of these quantities, providing a
computationally feasible method of comparison between quantum channels for
resource distillation.
