Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.

翻译：大多数量子散度的结构源于经典f散度或Rényi型构造，这种依赖关系掩盖了若干量子几何效应。我们提出一种量子相对α熵，它拓展了Umegaki相对熵，同时不归属于f散度类。该散度展现出非线性凸性性质，由此为大于1的α值导出了Petz-Rényi散度的一种广义凸性结果，补足了已知α小于1时的凸性结论。该散度在张量积下具有可加性、在酉变换下保持不变，且仅依赖于量子态的相对几何性质而非其绝对幅值。借助Nussbaum-Szkola型分布，我们还建立了该散度与经典相对α熵的精确对应关系。这揭示了相对α熵是一种未被现有散度框架捕捉的、本质上具有几何意义的量子可区分性概念。