Any multipartite entangled state violating the Mermin-Klyshko inequality can be distilled for almost all bipartite splits

Abstract

We study the explicit relation between violation of Bell inequalities and bipartite distillability of multiqubit states. It has been shown that even though for N >= 8 there exist N-qubit bound entangled states which violates a Bell inequality [W. Dur, Rev. Lett. 87, 230402 (2001)], for all the states violating the inequality there exists at least one splitting of the parties into two groups such that pure-state entanglement can be distilled [A. Acin, Rev. Lett. 88, 027901 (2002)]. We here prove that for all N-qubit states violating the inequality the number of distillable bipartite splits increases exponentially with N, and hence the probability that a randomly chosen bipartite split is distillable approaches 1 exponentially with N, as N tends to infinity. We also show that there exists at least one N-qubit bound entangled state violating the inequality if and only if N >= 6.