  Optical telecommunication is at the heart of today's internet and is
currently enabled by the transmission of intense optical signals between remote
locations. As we look to the future of telecommunication, quantum mechanics
promise new ways to be able to transmit and process that information.
Demonstrations of quantum key distribution and quantum teleportation using
multi-photon states have been performed, but only over ranges limited to one
hundred kilometers. To go beyond this, we need repeaters that are compatible
with these quantum multi-photon continuous variables pulses. Here we present a
design for a continuous variable quantum repeaters that can distribute
entangled and pure two-mode squeezed states over arbitrarily long distances
with a success probability that scales only polynomially with distance. The
proposed quantum repeater is composed from several basic known building blocks
such as non-Gaussian operations for entanglement distillation and an iterative
Gaussification protocol (for retaining the Gaussian character of the final
state), but complemented with a heralded non-Gaussian entanglement swapping
protocol, which allows us to avoid extensive iterations of quantum
Gaussification. We characterize the performance of this scheme in terms of key
rates for quantum key distribution and show a secure key can be generated over
thousands of kilometers.
