  We give a characterization of $C^*$-independence for $\mathbb{Z}_2$-graded
$C^*$-subalgebras of $\mathcal{B}(\mathcal{H})$ based on the Schlieder
property. We provide other notions of statistical independence for
$\mathbb{Z}_2$-graded von Neumann algebras and prove some relationships between
them. We provide a characterization for the graded nuclearity property.
