Symmetric Petz-R\'enyi relative entropy uncertainty relation

Holevo introduced a fidelity between quantum states that is symmetric and as effective as the trace norm in evaluating their similarity. This fidelity is bounded by a function of the trace norm, a relationship to which we will refer as Holevo's inequality. More broadly, Holevo's fidelity is part of a one-parameter family of symmetric Petz-R\'enyi relative entropies, which in turn satisfy a Pinsker's-like inequality with respect to the trace norm. Although Holevo's inequality is tight, Pinsker's inequality is loose for this family. We show that the symmetric Petz-R\'enyi relative entropies satisfy a tight inequality with respect to the trace norm, improving Pinsker's and reproducing Holevo's as a specific case. Additionally, we show how this result emerges from a symmetric Petz-R\'enyi uncertainty relation, a result that encompasses several relations in quantum and stochastic thermodynamics.