Miniatures on Open Quantum Systems

TL;DR

This paper offers a comprehensive, mathematically rigorous overview of open quantum systems, emphasizing operator algebras, non-equilibrium states, and entropy production, serving as a detailed reference for researchers and students.

Contribution

It consolidates and extends foundational and modern perspectives on open quantum systems within the operator algebra framework, emphasizing infinite systems and non-equilibrium phenomena.

Findings

Analysis of non-equilibrium steady states and entropy production.

Development of systematic treatment for small systems coupled to reservoirs.

Clarification of structural and conceptual links between equilibrium and non-equilibrium quantum phenomena.

Abstract

We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.