$E_γ$-Resolvability

Abstract: The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper we study $E_{\gamma}$-resolvability, in which total variation is replaced by the more general $E_{\gamma}$ distance. A general one-shot achievability bound for the precision of such an approximation is developed. Let $Q_{\sf X|U}$ be a random transformation, $n$ be an integer, and $E\in(0,+\infty)$. We show that in the asymptotic setting where $\gamma=\exp(nE)$, a (nonnegative) randomness rate above $$\inf_{Q_{\sf U}: D(Q_{\sf X}|{{\pi}}<em>{\sf X})\le E} {D(Q</em>{\sf X}|{{\pi}}<em>{\sf X})+I(Q</em>{\sf U},Q_{\sf X|U})-E}$$ is sufficient to approximate the output distribution ${{\pi}}<em>{\sf X}^{\otimes n}$ using the channel $Q</em>{\sf X|U}^{\otimes n}$, where $Q_{\sf U}\to Q_{\sf X|U}\to Q_{\sf X}$, and is also necessary in the case of finite $\mathcal{U}$ and $\mathcal{X}$. In particular, a randomness rate of $\inf_{Q_{\sf U}}I(Q_{\sf U},Q_{\sf X|U})-E$ is always sufficient. We also study the convergence of the approximation error under the high probability criteria in the case of random codebooks. Moreover, by developing simple bounds relating $E_{\gamma}$ and other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and smooth R\'{e}nyi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive 1) a one-shot upper bound on the probability of excess distortion in lossy compression, which is exponentially tight in the i.i.d.~setting, 2) a one-shot version of the mutual covering lemma, and 3) a lower bound on the size of the eavesdropper list to include the actual message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight.