Distillation of Greenberger-Horne-Zeilinger states by combinatorial methods

We prove a lower bound on the rate of Greenberger-Horne-Zeilinger states distillable from pure multipartite states by local operations and classical communication (LOCC). Our proof is based on a modification of a combinatorial argument used in the fast matrix multiplication algorithm of Coppersmith and Winograd. Previous use of methods from algebraic complexity in quantum information theory concerned transformations with stochastic local operations and classical operation (SLOCC), resulting in an asymptotically vanishing success probability. In contrast, our new protocol works with asymptotically vanishing error.