  We consider a two-user Gaussian multiple-input multiple-output (MIMO)
broadcast channel with a common multiple-antenna relay, and a shared digital
(noiseless) link between the relay and the two destinations. For this channel,
this paper introduces an asymptotically sum-capacity-achieving
quantize-and-forward (QF) relay strategy. Our technique to design an
asymptotically optimal relay quantizer is based on identifying a
cross-deterministic relation between the relay observation, the source signal,
and the destination observation. In a relay channel, an approximate cross
deterministic relation corresponds to an approximately deterministic relation,
where the relay observation is to some extent a deterministic function of the
source and destination signals. We show that cross determinism can serve as a
measure for quantization penalty. By identifying an analogy between a
deterministic broadcast relay channel and a Gaussian MIMO relay channel, we
propose a three-stage dirty paper coding strategy, along with receiver
beamforming and quantization at the relay, to asymptotically achieve an
extended achievable rate region for the MIMO broadcast channel with a common
multiple-antenna relay.
