The binarization criterion for bound secrecy

Let $X,Y,Z$ be jointly distributed, and let $\overline{X}$ and $\overline{Y}$ be binary outputs of arbitrary binary output channels applied to $X$ and $Y$. For $N\geq 1$, let $X^N,Y^N,Z^N$ denote $N$ independent copies and let $\overline{X^N},\overline{Y^N}$ be binary outputs of arbitrary binary output channels applied to $X^N$ and $Y^N$. The binarization criterion for bound secrecy. If

$$I(\overline{X}:\overline{Y} \downarrow Z)=0$$

for every pair of binary output channels, then

$$I(\overline{X^N}:\overline{Y^N} \downarrow Z^N)=0$$

for every $N$ and every pair of binary output channels on $X^N$ and $Y^N$.