The cut-set bound developed by (Cover and El Gamal, 1979) has since remained the best known upper bound on the capacity of the Gaussian relay channel. We develop a new upper bound on the capacity of the Gaussian primitive relay channel which is tighter than the cut-set bound. Combined with a simple tensorization argument proposed by (Courtade and Ozgur, 2015), our result also implies that the current capacity approximations for Gaussian relay networks, which have linear gap to the cut-set bound in the number of nodes, are order-optimal and leads to a lower bound on the pre-constant.
Our approach for developing the new bound significantly deviates from the standard information-theoretic approach for proving upper bounds on the capacity of multi-user channels. We build on measure concentration to analyze the geometric probabilistic relations between the typical sets of the n-letter random variables associated with a reliable code for communicating over the channel. These relations translate to new entropy inequalities between the n-letter random variables involved. Our approach can be extended to the discrete memoryless relay channel, and in particular, can be used to obtain surprising insights on a long-standing open problem posed by (Cover, 1987), namely, what is the minimum required relay-to-destination communication rate in order for the capacity of the relay channel to be equal to that of the broadcast cut.