  Given an entanglement measure $E$, the entanglement of a quantum channel is
defined as the largest amount of entanglement $E$ that can be generated from
the channel, if the sender and receiver are not allowed to share a quantum
state before using the channel. The amortized entanglement of a quantum channel
is defined as the largest net amount of entanglement $E$ that can be generated
from the channel, if the sender and receiver are allowed to share an arbitrary
state before using the channel. Our main technical result is that amortization
does not enhance the entanglement of an arbitrary quantum channel, when
entanglement is quantified by the max-Rains relative entropy. We prove this
statement by employing semi-definite programming (SDP) duality and SDP
formulations for the max-Rains relative entropy and a channel's max-Rains
information, found recently in [Wang et al., arXiv:1709.00200]. The main
application of our result is a single-letter, strong-converse, and efficiently
computable upper bound on the capacity of a quantum channel for transmitting
qubits when assisted by positive-partial-transpose preserving (PPT-P) channels
between every use of the channel. As the class of local operations and
classical communication (LOCC) is contained in PPT-P, our result establishes a
benchmark for the LOCC-assisted quantum capacity of an arbitrary quantum
channel, which is relevant in the context of distributed quantum computation
and quantum key distribution.
