(* structural entropy for N=3 *)
S[p1_, p2_, p3_] := (
p1 + p2 + p3 == 1;
S1 = -(p1*Log[p1] + p2*Log[p2] + p3*Log[p3]);
S2 = -Log[(p1^2 + p2^2 + p3^2)];
Sstr = S1 - S2 )
NMaximize[{S[p1, p2, p3], 0 <= p1 <= 1, 0 <= p2 <= 1, 0 <= p3 <= 1, p1 + p2 + p3 == 1},
{p1, p2, p3}]
I get
{0.137934, {p1 -> 0.0320363, p2 -> 0.0223545, p3 -> 0.945609}}
while
S[0.806, 0.097, 0.097]
you can get
0.223654
So I believe 0.137934 is not the maximum value