  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 $[-\frac12,1]$. In this note, we study a family of
bipartite quantum states whose minimal eigenvalues of partial-transposed states
being $-\frac12$. For a two-qubit system, we find that the minimal eigenvalue
of its partial-transposed state is $-\frac12$ 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.
