Resolvability in Eγ with Applications to Lossy Compression and Wiretap Channels

Abstract: We study the amount of randomness needed for an input process to approximate a given output distribution of a channel in the $E_{\gamma}$ distance. A general one-shot achievability bound for the precision of such an approximation is developed. In the i.i.d.~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 necessary and sufficient to asymptotically approximate the output distribution $\pi_{\sf X}^{\otimes n}$ using the channel $Q_{\sf X|U}^{\otimes n}$, where $Q_{\sf U}\to Q_{\sf X|U}\to Q_{\sf X}$. The new resolvability result is then used to derive a one-shot upper bound on the error probability in the rate distortion problem, and a lower bound on the size of the eavesdropper list to include the actual message in the wiretap channel problem. Both bounds are asymptotically tight in i.i.d.~settings.