Improving on the Cut-Set Bound via Geometric Analysis of Typical Sets

We consider the discrete memoryless symmetric primitive relay channel, where, a source $X$ wants to send information to a destination $Y$ with the help of a relay $Z$ and the relay can communicate to the destination via an error-free digital link of rate $R_0$, while $Y$ and $Z$ are conditionally independent and identically distributed given $X$. We develop two new upper bounds on the capacity of this channel that are tighter than existing bounds, including the celebrated cut-set bound. Our approach significantly deviates from the standard information-theoretic approach for proving upper bounds on the capacity of multi-user channels. We build on the blowing-up lemma to analyze the probabilistic geometric relations between the typical sets of the $n$-letter random variables associated with a reliable code for communicating over this channel. These relations translate to new entropy inequalities between the $n$-letter random variables involved. As an application of our bounds, we study an open question posed by (Cover, 1987), namely, what is the minimum needed $Z$-$Y$ link rate $R_0^*$ in order for the capacity of the relay channel to be equal to that of the broadcast cut. We consider the special case when the $X$-$Y$ and $X$-$Z$ links are both binary symmetric channels. Our tighter bounds on the capacity of the relay channel immediately translate to tighter lower bounds for $R_0^*$. More interestingly, we show that when $p\to 1/2$, $R_0^*\geq 0.1803$; even though the broadcast channel becomes completely noisy as $p\to 1/2$ and its capacity, and therefore the capacity of the relay channel, goes to zero, a strictly positive rate $R_0$ is required for the relay channel capacity to be equal to the broadcast bound.