= Entropy continuity from Jordan decomposition
{title2=$|S(\rho)-S(\sigma)|\leq\delta\log_2d+(1+\delta)h_2(\delta/(1+\delta))$}
For <density operators> in dimension $d$, let $\delta=\tfrac12\|\rho-\sigma\|_1$. 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.
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