If $W$ is an entanglement witness ($W \neq 0$), prove that

(a) $ tr(W) >0$

(b) $ tr(W)^2 > tr(W^2)$

For (a), by definition, since $ |ab\rangle$ is separable, thus $tr(\rho W)=\langle ab | W| ab \rangle \ge 0$. Then $$ tr(W) = \sum_{i,j} \langle a_i b_j | W | a_i b_j \rangle \ge 0 $$ But I don't know how to prove it cannot be 0.

For (b), I have a hint: any state $\rho$ s.t. $tr(\rho^2)\le 1/(d-1)$, where $d$ is the dimension of $\rho$, is separable. But I have no idea how to use this property, and I also can't prove it.