Building on the recently introduced notion of Boolean entropy, we define the corresponding Boolean Fisher information via a de Bruijn identity. We study the monotonicity of this Fisher information in the Boolean Central Limit Theorem and establish several functional inequalities involving these quantities, including a logarithmic Sobolev inequality. We also develop Non-microstate counterparts and prove the associated functional inequalities. In addition, we introduce a notion of Stein discrepancy in the Boolean setting, which leads to new Berry--Esseen type bounds in the Boolean central limit theorem.

翻译：基于最近提出的布尔熵概念，我们通过de Bruijn恒等式定义了相应的布尔费舍尔信息。我们研究了布尔中心极限定理中该费舍尔信息的单调性，并建立了涉及这些量的若干函数不等式，包括一个对数索伯列夫不等式。我们还发展了非微观态对应物，并证明了相关的函数不等式。此外，我们在布尔框架下引入了斯坦因差异的概念，从而在布尔中心极限定理中得到了新的Berry-Esseen型界限。