5

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)^{\...


3

Assuming everything is finite dimensional. For $S_0$ we have $$S_0(\rho) = \log \mathrm{rank}(\rho).$$ It's pretty straightforward to see this is not continuous. Take $\rho_{\epsilon} = \epsilon |0\rangle \langle 0 | + (1-\epsilon) |1\rangle \langle 1 |$. Then for all $0 < \epsilon < 1$ we have $S_0(\rho) = \log 2$ but for $\epsilon \in \{0,1\}$ we ...


2

Both limits are dealt with in a fair amount of detail in the work that originally defined the sandwiched entropies: On quantum Renyi entropies: a new generalization and some properties. In particular, you'll find the relevant results in section IV.C.


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