A Counterexample to the Vector Generalization of Costa's EPI, and Partial Resolution

We give a counterexample to the vector generalization of Costa’s entropy power inequality (EPI) due to Liu, Liu, Poor and Shamai. In particular, the claimed inequality can fail if the matix-valued parameter in the convex combination does not commute with the covariance of the additive Gaussian noise. Conversely, the inequality holds if these two matrices commute. For a random vector X with density on R, we let h(X) denote its differential entropy. Let Z ∼ N(0,ΣZ) be a Gaussian vector in R independent of X , and let A be a (real symmetric) positive semidefinite n × n matrix satisfying A I with respect to the positive semidefinite ordering, where I denotes the identity matrix. In [1, Theorem 1], Liu, Liu, Poor and Shamai claim the following generalization of Costa’s EPI [2]: e 2 nh(X+A Z) ≥ |I−A|e 2 nh(X) + |A|e 2 n. (1) Liu et al. apply (1) in their investigation of the secrecy capacity region of the degraded vector Gaussian broadcast channel with layered confidential messages. The purpose of this note is to demonstrate that (1) can fail for n ≥ 2, and also to offer a partial resolution. Toward the first goal, consider n = 2 and let us define