Sharp Data-Processing Region for Quantum Conditional Rényi Entropies

We study the three-parameter family of quantum conditional Rényi entropies $H_{\alpha ,z}^{\lambda }$ introduced in $\mathit{\text{Commun.}}\mathit{\text{Math.}}\mathit{\text{Phys.}}$ 407, 180 (2026). We prove that the previously established data-processing region is sharp. We then characterize the club-sandwiched family through a classical coarse-graining principle: within the full data-processing region, universal monotonicity under deterministic coarse-graining of the classical conditioned register holds if and only if $z=\alpha$. Finally, we establish rigidity of entropic duality by showing that any universal duality relation for finite-dimensional tripartite pure states must satisfy the known parameter relations. Thus, the data-processing region, the club-sandwiched subfamily, and the duality transformation are each uniquely characterized by their respective universal properties.