Boolean and Free Symmetrization of Bernoulli Distributions

TL;DR

This paper explores variance bounds for Bernoulli distributions and their noncommutative analogues across classical, free, and Boolean probability frameworks, revealing sharp bounds, unique symmetrization properties, and phenomena specific to Boolean convolution.

Contribution

It establishes unified variance bounds under symmetry constraints in multiple probabilistic frameworks and uncovers unique properties of Boolean symmetrization.

Findings

Variance bounds of pq for symmetrizers are sharp and achieved by reflection law.

Boolean convolution can produce symmetric measures from non-symmetric ones.

Symmetrizers may be non-unique in Boolean probability.

Abstract

We investigate variance bounds under symmetry constraints in classical, free, and Boolean probability, focusing on Bernoulli distributions and their noncommutative analogues, projections with trace \(p\). We show that symmetrizers under classical, free, and Boolean convolution satisfy a sharp variance bound of \(pq\), with equality for the reflection law. Additionally, we highlight phenomena specific to Boolean convolution, demonstrating that non-symmetric measures can produce symmetric convolutions and that symmetrizers may be non-unique for certain measures. These results unify variance inequalities across probabilistic frameworks and offer insights for quantum information and noncommutative stochastic modeling.