  In present-day quantum communications, one of the main problems is the lack of a quantum repeater design that can simultaneously secure high rates and long distances. Recent literature has established the end-to-end capacities that are achievable by the most general protocols for quantum and private communication within a quantum network, encompassing the case of a quantum repeater chain. However, whether or not a physical design exists to approach such capacities remains a challenging objective. Driven by this motivation, in this work, we put forward a design for continuous-variable quantum repeaters and show that it can actually achieve the feat. We also show that even in a noisy regime our rates surpass the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound. Our repeater setup is developed upon using noiseless linear amplifiers, quantum memories, and continuous-variable Bell measurements. We, furthermore, propose a non-ideal model for continuous-variable quantum memories that we make use of in our design. We then show that potential quantum communications rates would deviate from the theoretical capacities, as one would expect, if the quantum link is too noisy and/or low-quality quantum memories and amplifiers are employed. 
Capacity-approaching quantum repeaters for quantum communications