Entanglement distillation on symmetric two-qutrit entangled states of rank five

Entanglement distillation is a key step in quantum information, both theoretically and practically. It has been proven that nonpositive-partial transpose (NPT) entangled states of rank at most four is 1-distillable under local operation and classical communications. In this paper, we investigate the distillation of a more complex family of NPT entangled states, namely, a family of symmetric two-qutrit states ρ of rank five with given eigenvectors. We explicitly construct five families of such states by requiring four of the five eigenvalues to be the same. We, respectively, show that some of them are 1-distillable. It turns out that such states may be not 1-distillable for some interval of eigenvalues. We provide some conditions for eigenvalues that allow ρ to be 1-distillable or to be 1-undistillable.