We explore the relation between the rank of a bipartite 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 × n Hilbert space. A direct consequence of this result is that there are no bipartite bound entangled states of rank two. We also show that a separability condition in terms of a quantum entropy inequality is associated with the above results. 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.@en

We exhibit a two-parameter family of bipartite mixed states [Formula Presented] in a [Formula Presented] Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in [Formula Presented] can be distilled by local quantum operations and classical communication (LQ+CC). Evidence for this undistillability is provided by the result that, for certain states in this family, we cannot extract entanglement from any arbitrarily large number of copies of [Formula Presented] using a projection on [Formula Presented] These states are canonical NPT states in the sense that any bipartite mixed state in any dimension with NPT can be reduced by LQ+CC operations to a NPT state of the [Formula Presented] form. We show that the main question about the distillability of mixed states can be formulated as an open mathematical question about the properties of composed positive linear maps.@en