We show that the nonlocal Fisher information - defined as the entropy dissipation of the Boltzmann entropy for nonlocal heat equations - admits a natural lifting in the sense of Guillen and Silvestre (2025). Important examples include the discrete Fisher information arising in Markov chains and the fractional Fisher information $i_s$ associated with the fractional Laplacian $(-Δ)^{s}$ on $\mathbb{R}^d$, $s\in (0,1)$. We further establish a Blachman-Stam inequality (BSI) for the fractional Fisher information $i_s$, and prove that, for a large class of functions, $i_s$ converges to the classical Fisher information as $s\to 1$. Through this nonlocal-to-local limit, we recover the classical BSI and the lifting property of the classical Fisher information.

翻译：我们证明，非局部Fisher信息——定义为非局部热方程的Boltzmann熵的熵耗散——在Guillen和Silvestre（2025）的意义下具有自然的提升结构。重要实例包括Markov链中出现的离散Fisher信息，以及与$\mathbb{R}^d$上分数阶Laplacian $(-Δ)^{s}$（$s\in (0,1)$）相关联的分数阶Fisher信息 $i_s$。我们进一步建立了分数阶Fisher信息 $i_s$ 的Blachman-Stam不等式（BSI），并证明对于一大类函数，当$s\to 1$时，$i_s$ 收敛于经典Fisher信息。通过这一非局部到局部极限，我们恢复了经典BSI以及经典Fisher信息的提升性质。