Entropy comparison inequalities are obtained for the differential entropy $h(X+Y)$ of the sum of two independent random vectors $X,Y$, when one is replaced by a Gaussian. For identically distributed random vectors $X,Y$, these are closely related to bounds on the entropic doubling constant, which quantifies the entropy increase when adding an independent copy of a random vector to itself. Consequences of both large and small doubling are explored. For the former, lower bounds are deduced on the entropy increase when adding an independent Gaussian, while for the latter, a qualitative stability result for the entropy power inequality is obtained. In the more general case of non-identically distributed random vectors $X,Y$, a Gaussian comparison inequality with interesting implications for channel coding is established: For additive-noise channels with a power constraint, Gaussian codebooks come within a $\frac{{\sf snr}}{3{\sf snr}+2}$ factor of capacity. In the low-SNR regime this improves the half-a-bit additive bound of Zamir and Erez (2004). Analogous results are obtained for additive-noise multiple access channels, and for linear, additive-noise MIMO channels.

翻译：对于两个独立随机向量 $X,Y$ 的和 $X+Y$ 的微分熵 $h(X+Y)$，当其中一个向量被高斯分布替代时，本文获得了熵比较不等式。当 $X,Y$ 独立同分布时，这些不等式与熵加倍常数的上界密切相关，该常数量化了将随机向量与其独立副本相加时熵的增量。本文探讨了熵加倍效应较大和较小时的情形：对于前者，推导了加入独立高斯向量时熵增量的下界；对于后者，获得了熵幂不等式的定性稳定性结果。在更一般的非独立同分布随机向量 $X,Y$ 情形下，建立了具有信道编码重要含义的高斯比较不等式：对于带功率约束的加性噪声信道，高斯码本的容量可达性因子为 $\frac{{\sf snr}}{3{\sf snr}+2}$。在低信噪比区域，该结果改进了 Zamir 与 Erez (2004) 提出的半比特加性界。类似结论也适用于加性噪声多址信道及线性加性噪声 MIMO 信道。