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Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras

Benedikt M. Reible

TL;DR

This paper develops a rigorous operator-algebraic framework for the Araki-Uhlmann relative entropy, establishing its standard- and spatial-form definitions and proving a detailed version of Uhlmann’s monotonicity theorem, with applications to vector-functionals on von Neumann algebras. It then extends monotonicity to vector-level settings and advances unbounded perturbation theory for KMS-states, culminating in a general two-sided Bogoliubov inequality for arbitrary von Neumann algebras. By leveraging the standard form, relative modular operators, and spatial derivatives, the work unifies quantum information concepts with modular theory to yield robust entropy inequalities under a broad class of transformations. The results significantly broaden the applicability of relative entropy techniques in algebraic quantum field theory and quantum statistical mechanics, enabling variational and perturbative analyses beyond the finite-dimensional setting.

Abstract

This text studies, on the one hand, certain monotonicity properties of the Araki-Uhlmann relative entropy and, on the other hand, unbounded perturbation theory of KMS-states which facilitates a proof of the two-sided Bogoliubov inequality in general von Neumann algebras. After introducing the necessary background from the theory of operator algebras and Tomita-Takesaki modular theory, the relative entropy functional is defined and its basic properties are studied. In particular, a full and detailed proof of Uhlmann's important monotonicity theorem for the relative entropy is provided. This theorem will then be used to derive a number of monotonicity inequalities for the relative entropy of normal functionals induced by vectors of the form $V \varOmega, V \varPhi \in \mathcal{H}$, where $V \in \mathscr{B}(\mathcal{H})$ is a suitable transformation. After that, an introduction to perturbation theory in von Neumann algebras is given, with an emphasis on unbounded perturbations of KMS-states following the framework of Dereziński-Jakšić-Pillet. This mathematical apparatus will then be used to extend the two-sided Bogoliubov inequality for the relative free energy, which was very recently proved for quantum-mechanical systems, to arbitrary von Neumann algebras.

Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras

TL;DR

This paper develops a rigorous operator-algebraic framework for the Araki-Uhlmann relative entropy, establishing its standard- and spatial-form definitions and proving a detailed version of Uhlmann’s monotonicity theorem, with applications to vector-functionals on von Neumann algebras. It then extends monotonicity to vector-level settings and advances unbounded perturbation theory for KMS-states, culminating in a general two-sided Bogoliubov inequality for arbitrary von Neumann algebras. By leveraging the standard form, relative modular operators, and spatial derivatives, the work unifies quantum information concepts with modular theory to yield robust entropy inequalities under a broad class of transformations. The results significantly broaden the applicability of relative entropy techniques in algebraic quantum field theory and quantum statistical mechanics, enabling variational and perturbative analyses beyond the finite-dimensional setting.

Abstract

This text studies, on the one hand, certain monotonicity properties of the Araki-Uhlmann relative entropy and, on the other hand, unbounded perturbation theory of KMS-states which facilitates a proof of the two-sided Bogoliubov inequality in general von Neumann algebras. After introducing the necessary background from the theory of operator algebras and Tomita-Takesaki modular theory, the relative entropy functional is defined and its basic properties are studied. In particular, a full and detailed proof of Uhlmann's important monotonicity theorem for the relative entropy is provided. This theorem will then be used to derive a number of monotonicity inequalities for the relative entropy of normal functionals induced by vectors of the form , where is a suitable transformation. After that, an introduction to perturbation theory in von Neumann algebras is given, with an emphasis on unbounded perturbations of KMS-states following the framework of Dereziński-Jakšić-Pillet. This mathematical apparatus will then be used to extend the two-sided Bogoliubov inequality for the relative free energy, which was very recently proved for quantum-mechanical systems, to arbitrary von Neumann algebras.
Paper Structure (80 sections, 133 theorems, 356 equations)

This paper contains 80 sections, 133 theorems, 356 equations.

Table of Contents

  1. Notation
  2. Introduction
  3. The Relative Entropy in Mathematical Physics
  4. Outline of the Text
  5. Aspects of the Theory of Operator Algebras
  6. C*-Algebras
  7. Definitions and Examples
  8. Positive Elements
  9. Positive Linear Maps
  10. Basic Properties
  11. Completely Positive Maps
  12. Schwarz Mappings
  13. von Neumann Algebras
  14. Operator Topologies
  15. Definition, Bicommutant Theorem, Examples
  16. ...and 65 more sections

Key Result

Theorem III.1.5

Let $\mathfrak{A}$ be a unital $C^\ast$-algebra. For an element $A \in \mathfrak{A}$, the following assertions are equivalent:

Theorems & Definitions (275)

  • Definition III.1.2: $C^\ast$-algebra
  • Definition III.1.4: Positive element
  • Theorem III.1.5: Characterization of positive elements
  • Proposition III.1.7: BR1
  • Lemma III.1.8
  • Proof
  • Lemma III.1.10
  • Proof
  • Example III.1.11: Positive elements in $\mathscr{B}(\mathcal{H})$
  • Definition III.2.1: Positive mapping
  • ...and 265 more