Strong data processing inequalities (SDPI) are an important object of study in Information Theory and have been well studied for $f$-divergences. Universal upper and lower bounds have been provided along with several applications, connecting them to impossibility (converse) results, concentration of measure, hypercontractivity, and so on. In this paper, we study R\'enyi divergence and the corresponding SDPI constant whose behavior seems to deviate from that of ordinary $\Phi$-divergences. In particular, one can find examples showing that the universal upper bound relating its SDPI constant to the one of Total Variation does not hold in general. In this work, we prove, however, that the universal lower bound involving the SDPI constant of the Chi-square divergence does indeed hold. Furthermore, we also provide a characterization of the distribution that achieves the supremum when $\alpha$ is equal to $2$ and consequently compute the SDPI constant for R\'enyi divergence of the general binary channel.

翻译：强数据处理不等式（SDPI）是信息论中的重要研究对象，且已在$f$-散度框架下得到充分探讨。现有研究提供了通用上下界及其多种应用，涵盖不可能性（逆命题）结论、测度集中性、超压缩性等领域。本文研究Rényi散度及其对应的SDPI常数，该常数的行为与普通$\Phi$-散度存在显著差异。特别地，我们可构造反例证明：将Rényi散度的SDPI常数与全变差散度的SDPI常数相关联的通用上界并非普遍成立。然而，本研究证实涉及卡方散度SDPI常数的通用下界确实成立。进一步地，我们给出了当参数$\alpha=2$时达到上确界的分布特征描述，并据此推导出一般二元信道下Rényi散度的SDPI常数的解析表达式。