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 rate
$R_{0}^{*}$
needed for the
$Z$
–
$Y$
link 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. Existing upper bounds on the capacity of the relay channel, and the cut-set bound in particular, would rather imply
$R_{0}^{*}\to 0$
, while achievability schemes require
$R_{0}^{*}\to 1$
. We conjecture that
$R_{0}^{*}\to 1$
as
$p\to 1/2$
.