  Distribution and distillation of entanglement over quantum networks is a
basic task for Quantum Internet applications. A fundamental question is then to
determine the ultimate performance of entanglement distribution over a given
network. Although this question has been extensively explored for bipartite
entanglement-distribution scenarios, less is known about multipartite
entanglement distribution.
  Here we establish the fundamental limit of distributing multipartite
entanglement, in the form of GHZ states, over a quantum network. In particular,
we determine the multipartite entanglement distribution capacity of a quantum
network, in which the nodes are connected through lossy bosonic quantum
channels. This setting corresponds to a practical quantum network consisting of
optical links. The result is also applicable to the distribution of
multipartite secret key, known as common key, for both a fully quantum network
and trusted-node based quantum key distribution network. Our results set a
general benchmark for designing a network topology and network quantum
repeaters (or key relay in trusted nodes) to realize efficient GHZ state/common
key distribution in both fully quantum and trusted-node-based networks. We show
an example of how to overcome this limit by introducing a network quantum
repeater.
  Our result follows from an upper bound on distillable GHZ entanglement
introduced here, called the "recursive-cut-and-merge" bound, which constitutes
major progress on a longstanding fundamental problem in multipartite
entanglement theory. This bound allows for determining the distillable GHZ
entanglement for a class of states consisting of products of bipartite pure
states.
