Definition (Entropic Ruzsa distance). Let $G$ be an abelian group and let $X$, $Y$ be $G$-valued random variables. The entropic Ruzsa distance $d[X;Y]$ is

$$H[X^{\prime }-Y^{\prime }]-\frac{1}{2}H[X]-\frac{1}{2}H[Y]$$

where $X^{\prime }$, $Y^{\prime }$ are independent copies of $X$ and $Y$.