The pre-log of Gaussian broadcast with feedback can be two

A generic intuition says that the pre-log, or multi-plexing gain, cannot be larger than the minimum of the number of transmit and receive dimensions. This suggests that for the scalar broadcast channel, the pre-log cannot exceed one. By contrast, in this note, we show that when the noises are anti-correlated and feedback is present, then a pre-log of two can be attained. In other words, in this special case, in the limit of high SNR, the scalar Gaussian broadcast channel turns into two parallel AWGN channels. Achievability is established via a coding strategy due to Schalkwijk, Kailath, and Ozarow.