5
votes
Accepted
Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$
This only holds if the two distributions are independent. In this case
$$
\begin{aligned}
H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\
&= \...
- 4,516
3
votes
Accepted
Continuity of Renyi entropies - limiting cases
Assuming everything is finite dimensional.
For $S_0$ we have
$$S_0(\rho) = \log \mathrm{rank}(\rho).$$
It's pretty straightforward to see this is not continuous. Take $\rho_{\epsilon} = \epsilon |0\...
- 4,516
3
votes
What are explicit examples of smoothed conditional min(max) entropies?
I'll just give a classical example, which is a typical motivating example for these smooth quantities. Consider an $n$-bit distribution of the form
$$
p(x) = \begin{cases} 1-\delta \qquad \text{ if }x=...
- 4,516
2
votes
How to take the limits of the sandwiched Renyi divergences?
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, ...
- 4,516
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