Key capacity for product sources with application to stationary Gaussian processes

We show that for product sources, rate splitting is optimal for secret key agreement using limited one-way communication between 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 a wide class of discrete memoryless vector Gaussian sources which subsumes the case without an eavesdropper. Moreover, we derive an explicit formula for the maximum secret key per bit of communication for all discrete memoryless vector Gaussian sources using a tensorization property and a variation on the enhanced channel technique of Weingarten et al. Finally, a one-shot information spectrum achievability bound for key generation is proved from which we characterize the communication-rate-key-rate tradeoff for stationary Gaussian processes.