The un-polarized bit-channels in the wiretap polar coding scheme

Polar coding theorems state that as the number of channel use, n, tends to infinity, the fraction of un-polarized bit-channels (the bit-channels whose Z parameters are in the interval (δ(n), 1- δ (n)), tends to zero. Consider two BEC channels W(z
₁
) and W(z
₂
). Motivated by polar coding scheme proposed for the wiretap channel, we investigate the number of bit-channels which are simultaneously un-polarized for both of W(z
₁
) and W(z
₂
). We show that for finite values of n, there is a considerable regime of (z
₁
, Z
₂
) where the set of (joint)un-polarized bit-channels is empty. We also show that for γ ≤ 1/2 and δ (n) = 2
^-nγ
, the number of un-polarized bit-channels is lower bounded by 2γ log (n).