Entanglement Distillation from Greenberger–Horne–Zeilinger Shares

We study the problem of converting a product of Greenberger–Horne–Zeilinger (GHZ) states shared by subsets of several parties in an arbitrary way into GHZ states shared by every party. Such a state can be described by a hypergraph on the parties as vertices and with each hyperedge corresponding to a GHZ state shared among the parties incident with it. Our result is that if SLOCC transformations are allowed, then the best asymptotic rate is the minimum of bipartite log-ranks of the initial state, which in turn equals the minimum cut of the hypergraph. This generalizes a result by Strassen on the asymptotic subrank of the matrix multiplication tensor.