# joint entropy

$-\sum_i p_i \log(p_i) = -2 \frac{a}{2} \log\left(\frac{a}{2}\right) - 2 \frac{1-a}{2} \log\left(\frac{1-a}{2}\right) = 2 H\left(\frac{a}{2}\right)$
where $H(x) = -x\log(x)-(1-x)\log(1-x)$ is the so-called binary entropy function, which goes to zero only for $x=0$ or $x=1$. So your entropy expression can only go to zero for $a=0$ or $a=2$.