Support inclusion under rank-one dephasing ($\mathrm{supp}\rho \subseteq \mathrm{supp}\Delta (\rho )$)

If $p_i=⟨{\psi }_i|\rho |{\psi }_i⟩=0$, positivity gives $\sqrt{\rho }|{\psi }_i⟩=0$, so the corresponding basis vector is in $\ker \rho$. Therefore $\ker \Delta (\rho )\subseteq \ker \rho$, and $\mathrm{supp}\rho \subseteq \mathrm{supp}\Delta (\rho )$. This makes the quantum relative entropy of $\rho$ relative to its rank-one dephasing finite.