$E_{\gamma}$-Resolvability

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 $\\inf _{Q_{\\sf U}: D(Q_{\\sf X}\\|{\\pi }_{\\sf X})\\le E} \\{D(Q_{\\sf X}\\|{\\pi }_{\\sf X})+I(Q_{\\sf U},Q_{\\sf X|U})-E\\}$ is sufficient to approximate the output distribution ${\\pi }_{\\sf X}^{\\otimes n}$ using the channel $Q_{\\sf X|U}^{\\otimes n}$ , 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 Rényi 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.