Network Simplification: The Gaussian diamond network with multiple antennas

We consider the N-relay Gaussian diamond network when the source and the destination have n_s ≥ 2 and n_d ≥ 2 antennas respectively. We show that when n_s = n_d = 2 and when the individual MISO channels from the source to each relay and the SIMO channels from each relay to the destination have the same capacity, there exists a two relay sub-network that achieves approximately all the capacity of the network. To prove this result, we establish a simple relation between the joint entropies of three Gaussian random variables, which is not implied by standard Shannon-type entropy inequalities.