# About the Mutual (Conditional) Information

## Renato Renner and Ueli Maurer

```
```In general, the mutual information between two random
variables $X$ and $Y$, $I(X;Y)$, might be larger or smaller than
their mutual information conditioned on some additional information
$Z$, $I(X;Y|Z)$. Such additional information $Z$ can be seen as
output of a channel $C$ taking as input $X$ and $Y$. It is thus a
natural question, with applications in fields such as information
theoretic cryptography, whether conditioning on the output $Z$ of a
fixed channel $C$ can potentially increase the mutual information
between the inputs $X$ and $Y$.

In this paper, we give a necessary, sufficient, and easily
verifiable criterion for the channel $C$, i.e., the conditional
probability distribution $P_{Z|XY}$, such that $I(X;Y) \geq
I(X;Y|Z)$ for every distribution of the random variables $X$ and
$Y$. Furthermore, the result is generalized to channels with $n$
inputs (for $n \in \mathbb{N}$), that is, to conditional probability
distributions of the form $P_{Z | X_1 \cdots X_n}.