Entropy continuity from Jordan decomposition ($|S(\rho )-S(\sigma )|\le \delta {\log }_{2}d+(1+\delta )h_2(\delta /(1+\delta ))$)

For density operators in dimension $d$, let $\delta =\frac{1}{2}\Vert \rho -\sigma {\Vert }_1$. The positive and negative parts of their difference have the same trace $\delta$. Normalize them as ${\tau }_{+},{\tau }_{-}$ and write $(\rho +\delta {\tau }_{-})/(1+\delta )=(\sigma +\delta {\tau }_{+})/(1+\delta )$. Applying the two entropy bounds for a quantum mixture to this common state proves the displayed bound. The function $g(\delta )=(1+\delta )h_2(\delta /(1+\delta ))$ is increasing and concave, with $g(0)=0$, which also controls average entropy errors in pure-source compression.