# Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$

The Renyi entropy of order $$\beta$$, for a discrete probability distribution $$p$$ is given by $$\begin{equation} H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right), \end{equation}$$ where $$S$$ is the set of all strings in the support of $$p$$.

As is mentioned here, for two discrete distributions $$p$$ and $$r$$ the Renyi entropy of the product distribution $$p \times r$$ is

$$\begin{equation} H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r). \end{equation}$$

What might be a proof of this fact?

• what applications does Renyi entropy have in quantum Dec 16, 2020 at 4:24
• A quantum variant of the classical Renyi entropy appear in a lot of places in quantum information theory. Properties of the same can be found here: scholarpedia.org/article/Quantum_entropies. Dec 16, 2020 at 9:34

This only holds if the two distributions are independent. In this case \begin{aligned} H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\ &= \frac{1}{1-\beta} \log\left( \left(\sum_{i}p(i)^{\beta}\right) \left(\sum_jq(j)^{\beta}\right) \right) \\ &= \frac{1}{1-\beta} \left(\log \left(\sum_{i}p(i)^{\beta}\right) + \log \left(\sum_{j}q(j)^{\beta}\right)\right) \\ &= H_{\beta}(p) + H_{\beta}(q) . \end{aligned}