  This paper defines the amortized entanglement of a quantum channel as the
largest difference in entanglement between the output and the input of the
channel, where entanglement is quantified by an arbitrary entanglement measure.
We prove that the amortized entanglement of a channel obeys several desirable
properties, and we also consider special cases such as the amortized relative
entropy of entanglement and the amortized Rains relative entropy. These latter
quantities are shown to be single-letter upper bounds on the
secret-key-agreement and PPT-assisted quantum capacities of a quantum channel,
respectively. Of especial interest is a uniform continuity bound for these
latter two special cases of amortized entanglement, in which the deviation
between the amortized entanglement of two channels is bounded from above by a
simple function of the diamond norm of their difference and the output
dimension of the channels. We then define approximately teleportation- and
positive-partial-transpose-simulable (PPT-simulable) channels as those that are
close in diamond norm to a channel which is either exactly teleportation- or
PPT-simulable, respectively. These results then lead to single-letter upper
bounds on the secret-key-agreement and PPT-assisted quantum capacities of
channels that are approximately teleportation- or PPT-simulable, respectively.
Finally, we generalize many of the concepts in the paper to the setting of
general resource theories, defining the amortized resourcefulness of a channel
and the notion of $\nu$-freely-simulable channels, connecting these concepts in
an operational way as well.
