AI 中文总结
本文解决Yu和Tan的两个猜想，确定ρ相关高斯源的精确共同信息表达式与精确信道合成的容许区域，证明其严格大于Wyner共同信息及总变差版本所需速率，采用最优传输等方法完成证明。
AI 中文摘要
本文解决了Yu和Tan在2020年提出的两个猜想（分别发表于《IEEE Transactions on Information Theory》的两篇独立论文中）。具体而言，我们证明了：1）一对ρ相关高斯源的精确共同信息由猜想表达式$\frac{1}{2}\frac{1+\rho}{1-\rho}+\frac{\rho}{1+\rho}$给出；2）精确信道合成中共享随机率与通信率的容许区域恰好是猜想的区域。这些结果带来两个重要推论：第一，对于任意ρ>0，相关高斯对的精确共同信息严格大于Wyner的共同信息；第二，对于ρ>0，该对的精确信道合成需要比总变差版本严格更高的速率。证明结合了最坏情况高斯交叉熵的精确最优传输表示、Fathi高斯传输不等式，以及来自条件均值协方差结构的行列式不等式。
英文摘要
In this paper, we resolve two conjectures posed by Yu and Tan in 2020 (in two separate papers published in the IEEE Trans. Inf. Theory). Specifically, we establish that: 1) the exact common information for a pair of $ρ$-correlated Gaussian sources is given by the conjectured expression $\frac{1}{2}\log\frac{1+ρ}{1-ρ}+\fracρ{1+ρ}$; and 2) the admissible region for the shared randomness rate and the communication rate in exact channel synthesis is exactly the conjectured one. These results yield two important consequences. First, for any $ρ>0$, the exact common information of a correlated Gaussian pair strictly exceeds Wyner's common information. Second, for $ρ>0$, the exact channel synthesis of such a pair requires strictly higher rates than the total-variation version. The proof combines an exact optimal-transport representation of the worst-case Gaussian cross-entropy, Fathi's Gaussian transport inequality, and a determinant inequality arising from the covariance structure of the conditional means.
Comments16 pages, 2 figures
