  The entanglement cost of a quantum channel is the minimal rate at which
entanglement (between sender and receiver) is needed in order to simulate many
copies of a quantum channel in the presence of free classical communication. In
this paper we show how to express this quantity as a regularised optimisation
of the entanglement formation over states that can be generated between sender
and receiver. Our formula is the channel analog of a well-known formula for the
entanglement cost of quantum states in terms of the entanglement of formation;
and shares a similar relation to the recently shattered hope for additivity.
The entanglement cost of a quantum channel can be seen as the analog of the
quantum reverse Shannon theorem in the case where free classical communication
is allowed. The techniques used in the proof of our result are then also
inspired by a recent proof of the quantum reverse Shannon theorem and feature
the one-shot formalism for quantum information theory, the post-selection
technique for quantum channels as well as Sion's minimax theorem. We discuss
two applications of our result. First, we are able to link the security in the
noisy-storage model to a problem of sending quantum rather than classical
information through the adversary's storage device. This not only improves the
range of parameters where security can be shown, but also allows us to prove
security for storage devices for which no results were known before. Second,
our result has consequences for the study of the strong converse quantum
capacity. Here, we show that any coding scheme that sends quantum information
through a quantum channel at a rate larger than the entanglement cost of the
channel has an exponentially small fidelity.
