  We present a "hybrid quantum repeater" protocol for the long-distance
distribution of atomic entangled states beyond qubits. In our scheme, imperfect
noisy entangled pairs of two qudits, i.e., two discrete-variable $d$-level
systems, each of, in principle, arbitrary dimension $d$, are initially shared
between the intermediate stations of the channel. This is achieved via local,
sufficiently strong light-matter interactions, involving optical coherent
states and their transmission after these interactions, and optical
measurements on the transmitted field modes, especially (but not restricted to)
efficient continuous-variable homodyne detections ("hybrid" here refers to the
simultaneous exploitation of discrete and continuous variable degrees of
freedom for the local processing and storage of entangled states as well as
their non-local distribution, respectively). For qutrits we quantify the
light-matter entanglement that can be effectively shared through an elementary
lossy channel, and for a repeater spacing of up to 10 km we show that the
realistic (lossy) qutrit entanglement is even larger than any ideal (loss-free)
qubit entanglement. After including qudit entanglement purification and
swapping procedures, we calculate the long-distance entangled-pair distribution
rates and the final entangled-state fidelities for total communication
distances of up to 1280 km. With three rounds of purification, entangled qudit
pairs of near-unit fidelity can be distributed over 1280 km at rates of the
order of, in principle, 100 Hz.
