Conditional entropic Ruzsa distance ($d_R(X\mid U;Y\mid V)$)

For finitely supported random variables, the conditional entropic Ruzsa distance isThus the two conditioning values are sampled independently. If only one variable is conditioned, write $d_R(X\mid U;Y)$.

The simultaneous conditional distance samples one value of the conditioning variable for both arguments:It differs in general from $d_R(X\mid U;Y\mid U)$, which uses two independently sampled conditioning values.

If $U,V,X$ are independent random variables in ${\mathbb{F}}_2^n$, thenIndeed, conditioning reduces entropy and independence giveIn ${\mathbb{F}}_2^n$, $V=U+(U+V)$, so conditional entropy under a deterministic change of variables shows that the left-hand side is unchanged when $U$ and $V$ are exchanged. Averaging the displayed bound with its exchanged version gives the result.