  We investigate entanglement distribution in pure-state quantum networks. We
consider the case when non-maximally entangled two-qubit pure states are shared
by neighboring nodes of the network. For a given pair of nodes, we investigate
how to generate the maximal entanglement between them by performing local
measurements, assisted by classical communication, on the other nodes. We find
optimal measurement protocols for both small and large 1D networks. Quite
surprisingly, we prove that Bell measurements are not always the optimal ones
to perform in such networks. We generalize then the results to simple small 2D
networks, finding again counter-intuitive optimal measurement strategies.
Finally, we consider large networks with hierarchical lattice geometries and 2D
networks. We prove that perfect entanglement can be established on large
distances with probability one in a finite number of steps, provided the
initial entanglement shared by neighboring nodes is large enough. We discuss
also various protocols of entanglement distribution in 2D networks employing
classical and quantum percolation strategies.
