Proof Rényi entropy is non-negative

Question

The Rényi entropy is defined as: \begin{equation} S_\alpha = \dfrac{1}{1-\alpha}\log(\text{Tr}(\rho^\alpha)) \end{equation} for $\alpha \geq 0$. This can be rewrited in terms of $\rho$ eigenvalues, $\rho_k$, which verify $0 \leq \rho_k \leq 1$, as: \begin{equation} S_\alpha = \dfrac{1}{1-\alpha}\log(\sum_k \rho_k^\alpha) \end{equation} How can one proof rigurously that $S_\alpha \geq 0$? I am having trouble with this proof eventhough it seems pretty easy.

Answer 1

Let $\|\rho\|_{\infty}=\sup_{k\geqslant 0}\rho_k$, then $$ S_{\alpha}=\frac{1}{1-\alpha}\log\left(\sum_{k\geqslant 0}\left(\frac{\rho_k}{\|\rho\|_{\infty}}\right)^{\alpha}\right)+\frac{\alpha}{1-\alpha}\log\|\rho\|_{\infty} $$ Let $p$ be the number of $k\geqslant 0$ such that $\rho_k=\|\rho\|_{\infty}$ (there is a finite number of such $k$ because $\rho_k<\|\rho\|_{\infty}$ for $k$ large enough). We thus have $$ S_{\alpha}=\frac{1}{1-\alpha}\log\left(p+o(1)\right)+\frac{\alpha}{1-\alpha}\log\|\rho\|_{\infty} $$ Taking the limit as $\alpha\rightarrow +\infty$ gives $\lim\limits_{\alpha\rightarrow +\infty}S_{\alpha}=-\log\|\rho\|_{\infty}$ and thus $S_{\alpha}\geqslant -\log\|\rho\|_{\infty}\geqslant 0$ for all $\alpha\geqslant 0$.