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.

spiridon_the_sun_rotator · Answer

Consider 3 cases : ($alpha > 1$, $alpha < 1$, $alpha = 1$)

$alpha > 1$ $$ sum p_k^{alpha} leq sum p_k = 1 Rightarrow log (text{Tr}(rho^alpha)) leq 0Rightarrow S_alpha geq 0 $$
$alpha < 1$ $$ sum p_k^{alpha} geq sum p_k = 1 Rightarrow log (text{Tr}(rho^alpha)) geq 0Rightarrow S_alpha geq 0 $$
$alpha = 1$ $$ S_alpha =frac{1}{1 - alpha} log (text{Tr}(rho e^{(alpha - 1)log rho})) = - text{Tr}(rho log rho) = - sum p_k log p_k geq 0 $$ And in the last sum we used, that $x log x leq 0$ for $x in [0, 1]$

Answered by spiridon_the_sun_rotator on June 23, 2021

Proof Rényi entropy is non negative

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