Resolvability in Eγ with applications to lossy compression and wiretap channels

We study the amount of randomness needed for an input process to approximate a given output distribution of a channel in the Eγ distance. A general one-shot achievability bound for the precision of such an approximation is developed. In the i.i.d. setting where γ = exp(nE), a (nonnegative) randomness rate above infQU:D(QX||πX)≤E{D(QX||πX) + I(QU, QX|U)−E} is necessary and sufficient to asymptotically approximate the output distribution π⊗n X using the channel Q ⊗n X|U, where QU → QX|U → QX. The new resolvability result is then used to derive a oneshot 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.