# Coarse-graining on a second channel decreases mutual information?

Let $X_1,B_1,X_2,B_2$ and $Y_1,A_1,Y_2,A_2$ and $C_1$ and $C_2$ be binary random variables.

Suppose:

$I(X_2:B_2|C_2=0)+I(Y_2:A_2|C_2=1) \leq 1$.

This can be thought of as a bound on the capacity of a quantum channel called 2 (for example, you either perfectly correlate input $B_2$ with output $X_2$ (measurement result), or input $A_2$ with output $Y_2$, depending on the value of a bit $C_2$).

Do we have:

$I(X_1:B_1|C_1=0) \geq I(X_1\oplus X_2:B_1\oplus B_2|C_1=0, C_2=0)+ I(X_1\oplus Y_2:B_1\oplus A_2|C_1=0, C_2=1)$

?

$I$ is the mutual information and $\oplus$ the sum modulo 2.

If this inequality is true, how would you show it?