A new class of states of reversible entanglement manipulation under positive partial transpose operations

We first study the reversibility for a class of states under the operations which completely preserve the positivity of partial transpose (PPT) and show that the entanglement cost is equal to the distillable entanglement for a rank-two mixed state on the 4⊗4 antisymmetric subspace under PPT operations. By using a similar method in finding the irreversibility, we also find that the value of a new efficiently computable additive lower bound E_η(ρ) for the asymptotic PPT-relative entropy of entanglement presented inPhys. Rev. Lett. 119, 180506 (2017) is equal to the regularized Rains' bound and an upper bound E_N(ρ) for distillable entanglement for these states. Furthermore, we find states on the symmetric subspace satisfying the relation mentioned above, generalize the antisymmetric states and symmetric states in higher dimensions, and give a specific value for distillable entanglement and entanglement cost for these states under the PPT operations.