  If a quantum channel is commutable with a certain set of operators, a Choi
state of the channel becomes a sufficient resource for any entanglement
generation achieved by a single use of the channel with the help of local
operation and classical communications (LOCC). This property, called the
teleportation stretchable, could be useful to determine an upper bound of a
fundamental rate-loss trade-off for optical quantum key distribution.
Unfortunately, the known formulation of the teleportation-stretching expansion
for lossy Gaussian channels is based on an additional assumption of the Choi
state with infinite energy. In this paper, we present an entanglement-assisted
LOCC protocol with a bounded energy entangled state, and identify a resource
state being sufficient for a general entanglement generation protocol through a
single use of lossy Gaussian channels. We also extend this protocol for the
cases of general single-mode Gaussian channels. Our results provide a regular
path to determine an ultimate limit of quantum communication.

Quantum Physics