Boolean and Free Symmetrization of Bernoulli Distributions

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.