$E_{ {\gamma }}$ -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
_γ
-resolvability, in which total variation is replaced by the more general E
_γ
distance. A general one-shot achievability bound for the precision of such an approximation is developed. Let Q
_X|U
be a random transformation, n be an integer, and E ∈ (0, +∞). We show that in the asymptotic setting where γ = exp(nE), a (nonnegative) randomness rate above inf Q
_U
:D
_{(QXIIπX)≤E}
{D(Q
_X
IIπ
_X
) + I(Q
_U
, Q
_X|U
) - E} is sufficient to approximate the output distribution π
_X
^⊗n
using the channel Q
_X|U
^⊗n
, where Q
_U
→ Q
_X|U
→ Q
_X
, and is also necessary in the case of finite U and X . In particular, a randomness rate of inf Q
_U
I(Q
_U
, Q
_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
_γ
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.