  We describe a protocol for distilling maximally entangled bipartite states
between random pairs of parties from those sharing a tripartite W state, and
show that, rather surprisingly, the total distillation rate (the total number
of EPR pairs distilled per W, irrespective of who shares them) may be done at a
higher rate than distillation of bipartite entanglement between specified pairs
of parties. Specifically, the optimal distillation rate for specified
entanglement for the W has been previously shown to be the asymptotic
entanglement of assistance of 0.92 EPR pairs per W, while our protocol can
asymptotically distill 1 EPR pair per W between random pairs of parties, which
we conjecture to be optimal. We thus demonstrate a tradeoff between the overall
asymptotic rate of EPR distillation and the distribution of final EPR pairs
between parties. We further show that by increasing the number of parties in
the protocol that there exist states with fixed lower-bounded distillable
entanglement for random parties but arbitrarily small distillable entanglement
for specified parties.
