Support inclusion under rank-one dephasing
= Support inclusion under rank-one dephasing
{title2=$\operatorname{supp}\rho\subseteq\operatorname{supp}\Delta(\rho)$}
If $p_i=\langle\psi_i|\rho|\psi_i\rangle=0$, positivity gives $\sqrt\rho|\psi_i\rangle=0$, so the corresponding basis vector is in $\ker\rho$. Therefore $\ker\Delta(\rho)\subseteq\ker\rho$, and $\operatorname{supp}\rho\subseteq\operatorname{supp}\Delta(\rho)$. This makes the <quantum relative entropy> of $\rho$ relative to its <rank-one dephasing> finite.