Network Simplification: The Gaussian Diamond Network with Multiple Antennas

We consider the N relay Gaussian diamond network when the source and the destination have ns = 2 and nd = 2 antennas respectively. We show that when ns = nd = 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.