Distillability of non-positive-partial-transpose bipartite quantum states of rank four

We show that a bipartite quantum state of rank four is distillable if its partial transpose has at least one negative eigenvalue; that is, the state is a non-positive-partial transpose (NPT). For this purpose we prove that if the partial transpose of a two-qutrit NPT state has at least two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT state of rank five which is not 1-distillable and show that it is not n-distillable for any given n when the parameter is sufficiently small. This state has the smallest rank among all 1-undistillable NPT states. We conjecture that the state is not distillable.