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：本文针对两个独立随机向量$X,Y$的和的微分熵$h(X+Y)$，当其中一个被高斯随机向量替代时，得到了熵比较不等式。对于同分布的随机向量$X,Y$，这些不等式与熵倍增常数的界密切相关，该常数用于量化随机向量与其独立副本相加时的熵增。本文探讨了大倍增与小倍增两种情形的推论：对于前者，推导了添加独立高斯随机变量时熵增的下界；对于后者，得到了熵幂不等式的一种定性稳定性结果。在更一般的非同分布随机向量$X,Y$情形下，建立了一个对信道编码具有重要意义的高斯比较不等式：对于具有功率约束的加性噪声信道，高斯码本可达到容量的$\frac{{\sf snr}}{3{\sf snr}+2}$倍。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。类似结果也被推广到加性噪声多址信道，以及线性加性噪声MIMO信道中。