# Yet Another Inequality in Information Theory

Consider the following Markov chain:

$$X^n \rightarrow W \rightarrow Y^n$$

The notation $X^n$ is read as $X^n := (X_1, X_2, \ldots, X_n)$. Let $W$ take values in a set of cardinality $2^{nR}$ and let $X_1, X_2, \ldots, X_n$ be distributed independently according to $X$. I am interested in whether the following statement is true:

$$\max_i I(X_i;Y_i) \leq R$$

Note that proving

$$\min_i I(X_i;Y_i) \leq R$$

is fairly straight-forward. First, by means of the data processing inequality, we can argue that

$$I(X^n;Y^n) \leq I(X^n;W) \leq H(W) \leq nR$$

After that it suffices to produce a lower bound on $I(X^n;Y^n)$ as follows:

\begin{align} I(X^n;Y^n) &= H(X^n) - H(X^n \mid Y^n) \\ &= nH(X) - H(X^n \mid Y^n) \\ &= nH(X) - \sum_{i=1}^n H(X_i \mid X^{i-1}, Y^n) \\ &\geq nH(X) - \sum_{i=1}^n H(X_i \mid Y_i) \\ &= \sum_{i=1}^n I(X_i;Y_i) \\ &\geq n \min_i I(X_i;Y_i) \end{align}

Thus, $nR \geq n \min_i I(X_i;Y_i)$ and the second statement follows. Can we produce a similar result for the first claim?