Gaussian broadcast channel with lossy state reconstruction

We consider a state reconstruction problem where the additive state process of a Gaussian Broadcast Channel (GBC) is to be reconstructed at both receivers, subject to some distortion constraints. The state process we consider is Gaussian, independently and identically distributed, and known fully a-priori at the encoder. Each receiver has to perform a lossy estimation, to within some prescribed distortion tolerance under a squared error distortion criterion. The set-up also involves communicating messages across to the receivers, in addition to the estimation. The optimal trade-off between the distortions incurred at the receivers and the rates at which messages can be communicated, is of interest. We provide a complete characterization of the optimal rate vs. distortion performance.