  The theory of the asymptotic manipulation of pure bipartite quantum systems
can be considered completely understood: The rates at which bipartite entangled
states can be asymptotically transformed into each other are fully determined
by a single number each, the respective entanglement entropy. In the
multi-partite setting, similar questions of the optimally achievable rates of
transforming one pure state into another are notoriously open. This seems
particularly unfortunate in the light of the revived interest in such questions
due to the perspective of experimentally realizing multi-partite quantum
networks. In this work, we report substantial progress by deriving surprisingly
simple upper and lower bounds on the rates that can be achieved in asymptotic
multi-partite entanglement transformations. These bounds are based on ideas of
entanglement combing and state merging. We identify cases where the bounds
coincide and hence provide the exact rates. As an example, we bound rates at
which resource states for the cryptographic scheme of quantum secret sharing
can be distilled from arbitrary pure tripartite quantum states, providing
further scope for quantum internet applications beyond point-to-point.
