Theorem

The capacity $C(P)$ of the discrete-time memoryless additive noise channel with input power constraint $P$, whose noise $\{Z_{i}\}$ has mean $0$ and variance $\sigma^{2}$, satisfies $$C_{G}(P)+D(Z\|Z_{G})\ge C(P)$$where $Z_{G}\sim\mathcal{N}(0,\sigma^{2})$ (i.e. $Z_{G}$ is Gaussian noise) and $D(Z\|Z_{G})$ is the divergence between channel noise $Z$ and $Z_{G}$ called the “non-Gaussianness of the noise $Z$”.