Entanglement of assistance is the maximum rate of distilling EPR states between A and B from a tripartite pure state psi_ABC, when all three players collaborate. We prove that asymptotically, this is the smaller of A's and B's entropy, and that in fact this rate can be obtained as the sum of an EPR rate (the entanglement cost of the reduced state of AB) and a GHZ rate (the rest). There are also extensions to more than three players. Then we show that this result has implications for two multi-user quantum information tasks.
First, Gregoratti and Werner's "quantum lost and found", where the environment of a channel helps the sender and receiver to transmit quantum states by giving classical information. It turns out that the quantum capacity is the smaller of the input and output entropies, maximised over all input states.
Second, a fully quantum version of the Slepian-Wolf distributed source compression problem, with free classical side communication. For this we are able to determine the full rate region, in analogy to the classical Slepian-Wolf theorem, which limits individual rates by conditional entropies and the rate sum by the total entropy of the source. [This is joint work with Frank Verstraete, John A Smolin (part 1), Jonathan Oppenheim and Michal Horodecki (part 2).]