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

TL;DR

This paper explores the monotonicity of the Araki-Uhlmann relative entropy and extends the two-sided Bogoliubov inequality to general von Neumann algebras using advanced operator algebra techniques.

Contribution

It provides a detailed proof of Uhlmann's monotonicity theorem and extends the Bogoliubov inequality to the setting of von Neumann algebras.

Findings

Proof of Uhlmann's monotonicity theorem for relative entropy.

Monotonicity inequalities for relative entropy of normal functionals.

Extension of the two-sided Bogoliubov inequality to von Neumann algebras.

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 Derezi\'{n}ski-Jak\v{s}i\'{c}-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.