  The distortion-rate function of output-constrained lossy source coding with
limited common randomness is analyzed for the special case of squared error
distortion measure. An explicit expression is obtained when both source and
reconstruction distributions are Gaussian. This further leads to a partial
characterization of the information-theoretic limit of quadratic Gaussian
rate-distortion-perception coding with the perception measure given by
Kullback-Leibler divergence or squared quadratic Wasserstein distance.
