As already known by Rana's result, all eigenvalues of any partial-transposed bipartite state fall within the closed interval [-12,1]. In this note, we study a family of bipartite quantum states where the minimal eigenvalues of partial-transposed states are -12. For a two-qubit system, we find that the minimal eigenvalue of its partial-transposed state is -12 if and only if such a two-qubit state is maximally entangled. However this result does not hold in general for a two-qudit system when the dimensions of the underlying space are larger than two.
Keywords: maximally entangled state; moment; positive partial transpose.