Continuity of finite-dimensional quantum entropy

Define $h(t)=-t\ln t$ for $0<t\le 1$ and $h(0)=0$. It is continuous for positive $t$ by [F3]. Since $\ln$ is increasing and onto, $\ln t\to -\mathrm{\infty }$ as $t\downarrow 0$. Put $u=-\ln t$; then $t=\exp (-u)=1\mathrm{/}\exp u$ by the inverse and reciprocal identities in [F3]. Hence $h(t)=u\mathrm{/}\exp u\to 0$ by [F4] with $m=a=1$, so $h$ is continuous on $[0,1]$.

Let ${\lambda }_1(\rho )\ge \cdots \ge {\lambda }_d(\rho )$ and similarly for ${\rho }_k$. By [F5], density eigenvalues lie in $[0,1]$; [F2] gives $\mathrm{\mid }{\lambda }_j({\rho }_k)-{\lambda }_j(\rho )\mathrm{\mid }\le \mathrm{\parallel }{\rho }_k-\rho \mathrm{\parallel }$ for every $j$. For any $\epsilon >0$, continuity of $h$ at the finitely many ${\lambda }_j(\rho )$ gives one positive radius $\delta$ such that $\mathrm{\parallel }{\rho }_k-\rho \mathrm{\parallel }<\delta$ makes $\mathrm{\mid }h({\lambda }_j({\rho }_k))-h({\lambda }_j(\rho ))\mathrm{\mid }<\epsilon \mathrm{/}d$ for all $j$.

Sum the $d$ inequalities in step 2.1 and use [F1]. This gives $\mathrm{\mid }S({\rho }_k)-S(\rho )\mathrm{\mid }<\epsilon$ whenever $\mathrm{\parallel }{\rho }_k-\rho \mathrm{\parallel }<\delta$, proving continuity. The zero-eigenvalue boundary is covered by step 1.1, so no full-rank assumption is used.