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# Questions tagged [smooth-entropies]

For questions about smoothed entropic quantities, such as smooth min(max) entropies, smoothed conditional min(max) entropies, smoothed min(max) relative entropies, etc.

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### Connection between smooth max-relative entropy and smooth max-information

The max-relative entropy between two states is defined as $$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$ where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
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### What are explicit examples of smoothed conditional min(max) entropies?

Some general discussion of smoothed entropic quantities is found for example in Watrous notes, and an overview and discussion on its operational interpretations in (Koenig et al. 2008). It seems the ...
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### Why are "smooth entropic quantities" useful/necessary?

Consider the $\epsilon$-smoothed relative max-entropy of $\rho$ with respect to $Q$, defined as (following Watrous' notation from these notes): \mathrm D_{\rm max}^{\epsilon}(\rho\|Q) = \min_{\xi\in ...
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In (Tomamichel 2015), in (1.2) the author mentions the result that a source $X$ with probability distribution $\rho\equiv\rho_X$ admits an $(\varepsilon,m)$-code as long as there is some \$\alpha\in[\...