Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective
This work addresses a foundational gap in quantum information theory by providing a new divergence that captures geometric effects, though it is incremental as it builds on known entropy concepts.
The authors tackled the problem that most quantum divergences rely on classical constructions, which obscure quantum geometric effects, by introducing a quantum relative-alpha-entropy that extends Umegaki's relative entropy and falls outside the f-divergence class, revealing it as a geometric notion of quantum distinguishability not captured by existing frameworks.
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.