  The field of quantum communications promises the faithful distribution of
quantum information, quantum entanglement, and absolutely secret keys, however,
the highest rates of these tasks are fundamentally limited by the transmission
distance between quantum repeaters. The ultimate end-to-end rates of quantum
communication networks are known to be achievable by an optimal entanglement
distillation protocol followed by teleportation. In this work, we give a
practical design for this achievability. Our ultimate design is an iterative
approach, where each purification step operates on shared entangled states and
detects loss errors at the highest rates allowed by physics. As a simpler
design, we show that the first round of iteration can purify completely at high
rates. We propose an experimental implementation using linear optics and
photon-number measurements which is robust to inefficient operations and
measurements, showcasing its near-term potential for real-world practical
applications.
