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.Comment: 3 page