Functional inequalities for Boolean entropy

1. Problem Statement and Context

The paper addresses the development of a comprehensive information-theoretic framework for Boolean independence, the third pillar of non-commutative probability alongside classical and free independence (established by Speicher). While Boolean entropy was recently introduced by Pan as the rate function for large deviations in specific random matrix models, a full theory of associated Fisher information and functional inequalities (analogous to those in classical and free probability) was missing.

The authors aim to:

1. Define Boolean Fisher information via a de Bruijn identity.
2. Establish monotonicity properties in the Boolean Central Limit Theorem (CLT).
3. Prove functional inequalities (Log-Sobolev, Talagrand, HWI, HSI) relating entropy, Fisher information, and transport distances.
4. Develop a "Non-microstates" approach (intrinsic definitions independent of random matrix approximations) and compare it with the "Microstates" approach.
5. Apply Stein's method to derive quantitative Berry-Esseen bounds for the Boolean CLT.

2. Methodology

The authors employ a dual approach, mirroring Voiculescu's distinction between Microstates and Non-microstates in free probability:

• Microstates Approach:

  • Defines Boolean entropy $\Gamma(\mu)$ as the logarithmic integral $\int \log|x| d\mu(x)$.
  • Defines Boolean Fisher information $\Psi(\mu)$ via a de Bruijn identity along the Boolean heat semigroup (convolution with a scaled Rademacher variable).
  • Relies on the analytical structure of the Boolean convolution and the properties of the Cauchy transform.
  • Focuses primarily on symmetric probability measures to ensure analytical tractability and positivity.

• Non-microstates Approach:

  • Derives definitions from the Boolean Schwinger-Dyson equation.
  • Identifies $x \mapsto 1/x$ as the "score function" for the Rademacher distribution.
  • Defines Non-microstates Fisher information $\Psi^*(\mu) = \int x^{-2} d\mu(x)$ and entropy $\Gamma^*(\mu) = -\frac{1}{2}\log(\int x^{-2} d\mu(x))$.
  • This approach yields exact algebraic identities rather than inequalities in many cases due to the linear behavior of these quantities under Boolean convolution.

• Stein's Method:

  • Adapts Stein's method to the Rademacher target distribution.
  • Introduces a Boolean Stein discrepancy $D^*(\mu|b)$ based on the deviation of the second moment from 1.
  • Uses this discrepancy to bound Wasserstein distances and derive convergence rates.

3. Key Contributions and Results

A. Definitions and Fundamental Identities

• Boolean Fisher Information (Microstates):
  $$\Psi(\mu) = \iint \frac{\log x^2 - \log y^2}{x^2 - y^2} d\mu(x)d\mu(y)$$
  Satisfies the de Bruijn identity: $\frac{d}{dt}\Gamma(X+\sqrt{t}B) = \frac{1}{2}\Psi(X+\sqrt{t}B)$.
• Boolean Fisher Information (Non-microstates):
  $$\Psi^*(\mu) = \int \frac{1}{x^2} d\mu(x)$$
  Satisfies a simpler de Bruijn identity and behaves linearly under Boolean convolution.
• Relative Quantities: The authors define relative entropy and Fisher information with respect to the Rademacher distribution $b = \frac{1}{2}\delta_{-1} + \frac{1}{2}\delta_{1}$, which serves as the equilibrium state (analogous to the Gaussian in classical/free settings).

B. Monotonicity in the Central Limit Theorem

• Fisher Information Monotonicity: The paper proves that the Boolean Fisher information $\Psi(S_n)$ is non-increasing along the Boolean CLT sequence $S_n = \frac{1}{\sqrt{n}}\sum X_i$. Equality holds if and only if the variables are scaled Rademacher distributions.
• Entropy Monotonicity: As a consequence, the Boolean entropy $\Gamma(S_n)$ is non-decreasing, providing a new proof of the monotonicity established in previous work.

C. Functional Inequalities

The authors establish a hierarchy of inequalities connecting Transport ($W_2$), Entropy ($\Gamma$), and Fisher Information ($\Psi$).

1. Talagrand Inequality: $W_2(\mu, b)^2 \leq 2\Gamma(\mu|b)$.
2. Log-Sobolev Inequality (LSI): $\Gamma(\mu|b) \leq \frac{1}{2}\Psi(\mu|b)$.
   • Proven for both Microstates and Non-microstates settings.
   • Implies exponential decay of relative entropy along the Boolean Ornstein-Uhlenbeck process.
3. HWI Inequality (Otto-Villani):
   $$\Gamma(\mu|b) \leq W_2(\mu, b)\sqrt{\Psi^*(\mu|b)} - \frac{1}{2}W_2(\mu, b)^2$$
   This interpolates between transport and Fisher information.
4. HSI Inequality (Ledoux-Nourdin-Peccati):
   $$2\Gamma(\mu|b) \leq D^*(\mu|b)^2 \cdot \log\left(1 + \frac{\Psi^*(\mu|b)}{D^*(\mu|b)^2}\right)$$
   This relates entropy directly to the Stein discrepancy.

Key Distinction: The paper highlights a structural difference between Microstates and Non-microstates quantities. In the Non-microstates setting, many inequalities (like Stam and Entropy Power) become exact equalities due to the linearity of $\Psi^*$ under Boolean convolution. Furthermore, while Microstates entropy converges to the limit in the CLT, Non-microstates entropy remains constant.

D. Stein's Method and Berry-Esseen Bounds

• Stein Discrepancy: Defined as $D^*(\mu|b) = \sqrt{\int (x^2-1)^2 d\mu(x)}$.
• Transport Control: The authors prove $W_1(\mu, b) \leq D^*(\mu|b)$ and $W_2(\mu, b) \leq D^*(\mu|b)$ (for symmetric measures).
• Quantitative CLT: For a symmetric measure $\mu$ with finite fourth moment, the convergence rate in Wasserstein distance is $O(n^{-1/2})$:
  $$W_2(\mu_n, b) \leq \frac{1}{\sqrt{n}}\sqrt{m_4(\mu) - 1}$$
  This matches the optimal rate in classical and free CLTs.
• Entropic Convergence: Using the HSI inequality, they derive the rate of convergence for relative entropy:
  $$\Gamma(\mu_n|b) = O\left(\frac{\log n}{n}\right)$$

4. Significance and Impact

• Completing the Triptych: This work solidifies Boolean independence as a fully developed branch of non-commutative probability, establishing that it possesses a rich structure of functional inequalities parallel to classical and free probability.
• New Analytical Tools: The introduction of Boolean Stein discrepancy and the adaptation of Stein's method provide powerful new tools for quantitative limit theorems in the Boolean setting.
• Microstates vs. Non-microstates: The paper provides a clear comparative analysis, showing that while the Non-microstates approach offers algebraic simplicity (exact equalities), the Microstates approach captures the more complex analytic behavior (inequalities) similar to the classical case.
• Optimal Rates: The derivation of $O(n^{-1/2})$ convergence rates for Wasserstein distance and $O(\log n / n)$ for entropy fills a gap in the literature, surpassing previous results that were limited to weaker metrics or required stronger moment assumptions.

5. Limitations and Future Work

The current theory is restricted to symmetric probability measures. The authors note that the non-symmetric case leads to significantly more complex expressions for Fisher information and leaves the translation invariance properties undefined. Future work will address non-symmetric measures and multivariate extensions. Additionally, the authors conjecture a direct HWI inequality relating the Microstates entropy $\Gamma$ and Microstates Fisher information $\Psi$ (rather than the mixed $\Gamma$ and $\Psi^*$ currently proven).