Lower bound for entanglement cost of antisymmetric states

This report gives a lower bound of entanglement cost for antisymmetric states of bipartite d-level systems to be log 2 d d−1 ebit (for d = 3, Ec ≥ 0.585 . . . ). The paper [1] claims that the value is equal to one ebit for d = 3 , since all of the eigenvalues of reduced matrix of any pure states affiliating to H − is not greater than 2 −N thus the von Neumann entropy is not less than N , but the proof is not true. Hence whether the value is equal to or less than one ebit is not clear at this moment.