  We show that for product sources, rate splitting is optimal for secret key
agreement using limited one-way communication at two terminals. This yields an
alternative proof of the tensorization property of a strong data processing
inequality originally studied by Erkip and Cover and amended recently by
Anantharam et al. We derive a `water-filling' solution of the
communication-rate--key-rate tradeoff for two arbitrarily correlated vector
Gaussian sources, for the case with an eavesdropper, and for stationary
Gaussian processes.
