  In two-dimensional quantum site-percolation square lattice models, the von
Neumann entropy is extensively studied numerically. At a certain eigenenergy,
the localization-delocalization transition is reflected by the derivative of
von Neumann entropy which is maximal at the quantum percolation threshold
$p_q$. The phase diagram of localization-delocalization transitions is deduced
in the extrapolation to infinite system sizes. The non-monotonic eigenenergies
dependence of $p_q$ and the lowest value $p_q\simeq0.665$ are found. At
localized-delocalized transition points, the finite scaling analysis for the
von Neumann entropy is performed and it is found the critical exponents $\nu$
not to be universal. These studies provide a new evidence that the existence of
a quantum percolation threshold $p_q<1$ in the two-dimensional quantum
percolation problem.
