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We show that any bipartite quantum state of rank four is distillable, whenthe 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 atwo-qutrit NPT state has two non-positive eigenvalues, then the state isdistillable. We further construct a parametrized two-qutrit NPT entangled stateof 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. Weconjecture that the state is not distillable.

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