Suppose we have a composite system that is made up of two random variables $A,B$. And suppose also that the mutual information of the system is non-zero, $I(A;B) > 0$. We then apply a stochastic process $\Pi$ to the entire system, whose joint probability distribution is $p(a,b)$.
For lack of better notation, let's use a dash to indicate the state of the system after $\Pi$, so $p(a',b') = \Pi(p(a,b))$. Let $I(A';B')$ be the mutual information of the system after $\Pi$ has been applied.
Under what conditions on $\Pi$ will we have that $I(A';B') \geq I(A;B)$?
I had wondered perhaps if $\Pi$ does not act independently on the two systems, i.e. in matrix form one cannot write the transition elements of $\Pi$ as $p(a',b'|a,b) = p(a'|a)p(b'|b)$, this might be equivalent to creating/not reducing statistical dependence in the output $p(a',b')$? But I also worry that this is so qualitative there may be easy counter examples that I'm overlooking!
If anyone has any insight or can point me toward resources for this topic I'd be very interested. Thank you for your help!