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On Entropy for general quantum systems

W. A. Majewski, L. E. Labuschagne

TL;DR

The paper addresses the challenge of defining entropy for general quantum systems, including those described by type III von Neumann algebras, where standard von Neumann entropy is insufficient. It develops a density-based entropy framework built on noncommutative Radon–Nikodym derivatives, Connes cocycles, and modular dynamics, unified via the Orlicz space pair $\langle L^{\cosh-1}, L\log(L+1)\rangle$ and crossed-product techniques. Key contributions include a general entropy definition via modular derivatives that recovers Araki's relative entropy in finite cases, a density-based formulation for commuting densities with a practical entropy functional $\tilde{S}$, and the result that equilibrium states yield vanishing entropy, supporting an intensive-density interpretation in the thermodynamic limit. The work provides a mathematically robust route to quantum thermodynamics for large systems and offers a bridge between classical H-function-like behavior and quantum entropy concepts in noncommutative settings.

Abstract

In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.

On Entropy for general quantum systems

TL;DR

The paper addresses the challenge of defining entropy for general quantum systems, including those described by type III von Neumann algebras, where standard von Neumann entropy is insufficient. It develops a density-based entropy framework built on noncommutative Radon–Nikodym derivatives, Connes cocycles, and modular dynamics, unified via the Orlicz space pair and crossed-product techniques. Key contributions include a general entropy definition via modular derivatives that recovers Araki's relative entropy in finite cases, a density-based formulation for commuting densities with a practical entropy functional , and the result that equilibrium states yield vanishing entropy, supporting an intensive-density interpretation in the thermodynamic limit. The work provides a mathematically robust route to quantum thermodynamics for large systems and offers a bridge between classical H-function-like behavior and quantum entropy concepts in noncommutative settings.

Abstract

In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.

Paper Structure

This paper contains 7 sections, 10 theorems, 79 equations.

Table of Contents

  1. Boltzmann's H-functional and (classical) entropy
  2. Algebraic preliminaries
  3. The von Neumann entropy and Dirac's formalism.
  4. General quantum case
  5. An alternative approach to the general quantum case
  6. Discussion
  7. Conclusions

Key Result

Theorem 2.1

Let ${\mathfrak{M}}$ be a von Neumann algebra and $\phi$, $\psi$ faithful semifinite normal weights on ${\mathfrak{M}}$. Then there exists a $\sigma$-strongly continuous one parameter family $\{u_t\}$ of unitaries in ${\mathfrak{M}}$ with the following properties: where $\sigma_t^{\varphi}$ ($\sigma_t^{\psi}$) stands for the modular evolution determined by $\varphi$ ($\psi$ respectively).

Theorems & Definitions (29)

  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 19 more