  Promising access to high-speed quantum networks relies on the creation of
high-dimensional entangled memories that provide quantum communication with
higher capacity of noisy quantum channels, thereby reducing the transmission
time of information. Yet, the distribution of multidimensional entanglement
between remote memory nodes is still faintly investigated. We propose an
experimentally feasible protocol of deterministic generation of
high-dimensional entanglement between distant multi-level atoms confined in
high-finesse optical cavities and driven by laser pulses. Three-dimensional
entanglement is generated deterministically between remote atoms by triggering
a two-photon wavepacket, which mediates a superposition of states chosen by the
laser pulse parameters. The efficient transfer of atomic states between remote
nodes allows the construction of a three-dimensional quantum repeater, where
the successful creation of entanglement can be verified by a simple method for
reliable measurement of atomic ground Zeeman states.
