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Functional inequalities for Boolean entropy

Guillaume Cébron, Kewei Pan

Abstract

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.

Functional inequalities for Boolean entropy

Abstract

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.
Paper Structure (13 sections, 37 theorems, 246 equations, 1 figure)

This paper contains 13 sections, 37 theorems, 246 equations, 1 figure.

Table of Contents

  1. The Microstates approach
  2. Boolean entropy and Fisher information
  3. Information-theoretic properties
  4. Functional inequalities
  5. The Non-microstates approach
  6. The Boolean Schwinger-Dyson equation
  7. Boolean entropy and Fisher information
  8. Information-theoretic properties
  9. Functional inequalities
  10. Stein's method for Boolean independence
  11. Stein's method
  12. Stein discrepancy
  13. Entropic CLT

Key Result

Theorem 1.3

Let $X$ be a self-adjoint operator of a non-commutative probability space $(\mathcal{M},\tau)$, with law $\mu\in\mathcal{M}^{sym}(\mathbb{R})$, and $B\in \mathcal{M}$ be Boolean independent from $X$ and distributed according to $\mathrm{b}$. Fix $t\geq 0$ such that $\Gamma(X+\sqrt{s}B)<\infty,\ \Psi In particular, $\Gamma(X+\sqrt{t}B)$ is increasing with respect to $t$.

Figures (1)

  • Figure 1: Hierarchy of Boolean functional inequalities.

Theorems & Definitions (77)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • proof
  • Proposition 1.4
  • Corollary 1.5
  • proof
  • Theorem 1.6: Monotonicity of the Fisher information
  • proof
  • Theorem 1.7: Monotonicity of entropy
  • ...and 67 more