Teleportation stretching for lossy Gaussian channels

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.