Abstract
As already known by Rana's result \href{https://doi.org/10.1103/PhysRevA.87.054301}{[\pra {\bf87} (2013) 054301]}, all eigenvalues of any partial-transposed bipartite state fall within the closed interval [-1/2,1]. In this note, we study a family of bipartite quantum states whose minimal eigenvalues of partial-transposed states being -1/2. For a two-qubit system, we find that the minimal eigenvalue of its partial-transposed state is -1/2 if and only if such two-qubit state must be 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.