An information theoretic perspective over an extremal entropy inequality

This paper focuses on developing an alternative proof for an extremal entropy inequality, originally presented in [1]. The proposed alternative proof is simply based on the classical entropy power inequality and the data processing inequality. Compared with the proofs in [1], the proposed alternative proof is simpler, more direct, and information theoretic, and presents the advantage of providing the structure of the optimal solution covariance matrix. Also, the proposed proof might also be used as a novel method to address applications such as calculation of the vector Gaussian broadcast channel capacity, establishing a lower bound for the achievable rate of distributed source coding with a single quadratic distortion constraint, and the secrecy capacity of the Gaussian wire-tap channel.