Definition (Simultaneous conditional distance). Let $X,Y,U$ be $G$-valued random variables. The simultaneous conditional distance of $X$ to $Y$ given $U$ is

$$d[X;Y\parallel U]=\sum _u\mathbb{P}[U=u]d[X|U=u;Y|U=u].$$

We say that $X^{\prime }$, $Y^{\prime }$ are conditionally independent trials of $X$, $Y$ given $U$ if:

• $X^{\prime }$ is distributed like $X$.

• $Y^{\prime }$ is distributed like $Y$.

• For each $u\in U$, $X^{\prime }|U=u$ is distributed like $X|U=u$,

• For each $u\in U$, $Y^{\prime }|U=u$ is distributed like $Y|U=u$.

• $X^{\prime }|U=u$ and $Y^{\prime }|U=u$ are independent.

Then

$$d[X;Y\parallel U]=H[X^{\prime }-Y^{\prime }|U]-\frac{1}{2}H[X^{\prime }|U]-\frac{1}{2}H[Y^{\prime }|U]$$

(as can be seen directly from the formula).