  A class of lower bounds for the entanglement cost of any quantum state was
recently introduced in [arXiv:2111.02438] in the form of entanglement monotones
known as the tempered robustness and tempered negativity. Here we extend their
definitions to point-to-point quantum channels, establishing a lower bound for
the asymptotic entanglement cost of any channel, whether finite or infinite
dimensional. This leads, in particular, to a bound that is computable as a
semidefinite program and that can outperform previously known lower bounds,
including ones based on quantum relative entropy. In the course of our proof we
establish a useful link between the robustness of entanglement of quantum
states and quantum channels, which requires several technical developments such
as showing the lower semicontinuity of the robustness of entanglement of a
channel in the weak*-operator topology on bounded linear maps between spaces of
trace class operators.
