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.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.

quant-pharXiv:2604.06908Apr 9, 20260 citations

A Note on Quantum Entropy

Frank Hansen

arXiv:1511.09189Mathematical Physics Analysis and GeometryMay 12, 20161 citation

Entropies and Relative Entropies

Gilad Gour

Chapter 6 builds upon the foundation of divergences from Chapter 5, advancing into entropies and relative entropies with an axiomatic approach, and the inclusion of the additivity axiom. The chapter delves into the classical and quantum relative entropies, establishing their core properties and revealing the significance of the KL-divergence introduced in Chapter 5, notably characterized by asymptotic continuity. Quantum relative entropies are addressed as generalizations of classical ones, with a focus on the conditions necessary for these measures in the quantum framework. Several variants of relative entropies are discussed, including Renyi relative entropies and their extensions to quantum domain such as the Petz quantum Renyi divergence, minimal quantum Renyi divergence, and the maximal quantum Renyi divergence. This discourse underlines the relevance of continuity and its relation to faithfulness in relative entropies. The concept of entropy is portrayed as a measure with a broad spectrum of interpretations and applications across fields, from thermodynamics and information theory to cosmology and economics.Chapter 6 builds upon the foundation of divergences from Chapter 5, advancing into entropies and relative entropies with an axiomatic approach, and the inclusion of the additivity axiom. The chapter delves into the classical and quantum relative entropies, establishing their core properties and revealing the significance of the KL-divergence introduced in Chapter 5, notably characterized by asymptotic continuity. Quantum relative entropies are addressed as generalizations of classical ones, with a focus on the conditions necessary for these measures in the quantum framework. Several variants of relative entropies are discussed, including Renyi relative entropies and their extensions to quantum domain such as the Petz quantum Renyi divergence, minimal quantum Renyi divergence, and the maximal quantum Renyi divergence. This discourse underlines the relevance of continuity and its relation to faithfulness in relative entropies. The concept of entropy is portrayed as a measure with a broad spectrum of interpretations and applications across fields, from thermodynamics and information theory to cosmology and economics.

Cambridge University Press eBooksApr 10, 20250 citations

Quantum Entropy

Dominik Janzing

Compendium of Quantum PhysicsJan 1, 20090 citations

Quantum Entropy

Stanley Gudder

Springer eBooksNov 2, 20061 citation

Regularized barycentric Rényi divergences

Milán Mosonyi

Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies $D^{q_x}$ . When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all $D^{q_x}$ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the $D^{q_x}$ are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies $D^{q_x}$. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all $D^{q_x}$ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the $D^{q_x}$ are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.

quant-pharXiv:2609.17517Sep 16, 20260 citations

D^{q_x}

. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all

D^{q_x}

are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the

D^{q_x}

are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.