A New Upper Bound on the Capacity of a Primitive Relay Channel Based on Channel Simulation

An upper bound on the capacity of a primitive three-node discrete memoryless relay channel is considered, where a source X wants to send information to destination Y with the help of a relay Z. The signals at each node are also denoted as X, Y, and Z, respectively, with a slight abuse of notation. Y and Z are independent given X, and the link from Z to Y is lossless with rate R₀. A new inequality is introduced to upper-bound the capacity when the encoding rate is beyond the capacities of both individual links XY and XZ. It is based on generalizing the blowing-up lemma, linking conditional entropy to decoding error, and generalizing channel simulation to the case with side information. The upper bound is strictly better than the well-known cut-set bound in several cases when the latter is C_XY +R₀, with C_XY being the channel capacity between X and Y. One particular case is when the channel is stochastically degraded, i.e., either Y is a stochastically degraded version of Z with respect to X, or Z is a stochastically degraded version of Y with respect to X. Moreover, in this case, the bound is shown to be explicitly computable. The upper bound on the capacity of the binary erasure channel is analyzed in detail and evaluated numerically.