  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 Schr\"odinger 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 the 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 both in optical and quantum
communications.
