Non-positive-partial-transpose quantum states of rank four are distillable

Abstract: We show that any bipartite quantum state of rank four is distillable, when the partial transpose of the state has at least one negative eigenvalue, i.e., the state is NPT. For this purpose we prove that if the partial transpose of a two-qutrit NPT state has two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT entangled 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.