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 infQ_U:D(Q_X||π_X)≤E{D(Q_X||π_X) + I(Q_U, Q_X|U) - E} is necessary and sufficient to asymptotically approximate the output distribution π_X^⊗n using the channel Q_X|U^⊗n, where Q_U → Q_X|U → Q_X. 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.