Questions tagged [smooth-entropies]

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 positive semidefinite. There is also a smoothed…

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 quantity was introduced in (Renner and Wolf 2004),…

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 B_\epsilon(\rho)} \mathrm D_{\rm…

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 $\alpha\in[\frac12,1)$ such that $$\log_2 m\ge…