Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective

Quantum Physics [Submitted on 8 Apr 2026] Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective Sayantan Roy, Atin Gayen, Aditi Kar Gangopadhyay, Sugata Gangopadhyay 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. Comments: 32 Pages, Submitted to a Journal Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph) Cite as: arXiv:2604.06908 [quant-ph]   (or arXiv:2604.06908v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.06908 Focus to learn more Submission history From: Atin Gayen [view email] [v1] Wed, 8 Apr 2026 10:06:28 UTC (232 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.IT hep-th math math-ph math.IT math.MP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)