Quantum tele-amplification with a continuous-variable superposition state

Optical coherent states are classical light fields with high purity, and are essential carriers of information in optical networks. If these states could be controlled in the quantum regime, allowing for their quantum superposition (referred to as a Schrödinger-cat state), then novel quantum-enhanced functions such as coherent-state quantum computing (CSQC), quantum metrology and a quantum repeater could be realized in the networks. Optical cat states are now routinely generated in laboratories. An important next challenge is to use them for implementing the aforementioned functions. Here, we demonstrate a basic CSQC protocol, where a cat state is used as an entanglement resource for teleporting a coherent state with an amplitude gain. We also show how this can be extended to a loss-tolerant quantum relay of multi-ary phase-shift keyed coherent states. These protocols could be useful in both optical and quantum communications.