Majorization criterion for distillability of a bipartite quantum state

Bipartite quantum states are classified into three categories: separable states, bound entangled states, and free entangled states. It is of great importance to characterize these families of states for the development of quantum information science. In this paper, I show that the separable states and the bound entangled states have a common spectral property. More precisely, I prove that for undistillable -- separable and bound entangled -- states, the eigenvalue vector of the global system is majorized by that of the local system. This result constitutes a new sufficient condition for distillability of bipartite quantum states. This is achieved by proving that if a bipartite quantum state satisfies the reduction criterion for distillability, then it satisfies the majorization criterion for separability.