  The potential impact of future quantum networks hinges on high-quality
quantum entanglement shared between network nodes. Unavoidable real-world
imperfections necessitate means to improve remote entanglement by local quantum
operations. Here we realize entanglement distillation on a quantum network
primitive of distant electron-nuclear two-qubit nodes. We demonstrate the
heralded generation of two copies of a remote entangled state through
single-photon-mediated entangling of the electrons and robust storage in the
nuclear spins. After applying local two-qubit gates, single-shot measurements
herald the distillation of an entangled state with increased fidelity that is
available for further use. In addition, this distillation protocol
significantly speeds up entanglement generation compared to previous
two-photon-mediated schemes. The key combination of generating, storing and
processing entangled states demonstrated here opens the door to exploring and
utilizing multi-particle entanglement on an extended quantum network.
