Rank Two Bound Entangled States Do Not Exist

We explore the relation between the rank of a density matrix and the existence of bound entanglement. We show a relation between the rank, marginal ranks, and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n øtimes n Hilbert space. A direct consequence of this result is that there are no bound entangled states of rank two. We explore the idea of how many pure states are needed in a mixture to cancel the distillable entanglement of a Schmidt rank n pure state and provide a lower bound of n-1. We also prove that a mixture of a non-zero amount of any pure entangled state with a pure product state is distillable