= Eigenbasis proof of quantum relative entropy nonnegativity
Diagonalize two <density operators> with eigenvalues $r_i,s_j$, and put $q_{ij}=|\langle i|j\rangle|^2$. This overlap matrix has row and column sums one. The scalar inequality $u\ln(u/v)\ge u-v$ yields $D(\rho\|\sigma)\ge\sum_{ij}q_{ij}(r_i-s_j)=0$ in natural units; division by $\ln2$ gives bits. Equality forces $r_i=s_j$ whenever $q_{ij}>0$, so $\sigma|i\rangle=r_i|i\rangle$ and $\rho=\sigma$. Zero eigenvalues use limits, and failure of support inclusion gives infinite <quantum relative entropy>.
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