Conditional independence given a real valued function of variables

Question

Let $X$, $Y$, $A$, $B$ be random variables and $f:\mathbb{R}^2 \to \mathbb{R}$ measurable. I would like to know under which conditions there exists a function $f$ such that $$X \perp\!\!\!\!\perp Y | f(A,B).$$ For example: Let $X = N_1$, $A= X + N_2$, $B = X + N_3$ and $Y= A + B + N_4$ (where $N_i$ are $i.i.d$), then $$X \perp\!\!\!\!\perp Y | A+B.$$ This is because $X = (A+B)/2 - (N_2 + N_3)/2$ which is independent of $Y= (A + B) + N_4$ given $A+B$.