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.