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Canadian Journal of Physics

Lectures on open quantum systems

Marco Merkli, Angel E Neira

Published Jan 1, 2026• 0 citations
Publisher

Abstract

These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area that has applications across the quantum sciences dealing with systems subjected to external noise. The idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes–Cummings model the reader will discover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini–Kossakowski–Sudarshan–Lindblad theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.

DOI
10.1139/cjp-2026-0029

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These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area that has applications across the quantum sciences dealing with systems subjected to external noise. The idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes–Cummings model the reader will discover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini–Kossakowski–Sudarshan–Lindblad theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.
The year 2026 marks the fiftieth anniversary of the two seminal papers in which Gorini, Kossakowski, and Sudarshan, and independently Lindblad, characterized the generators of completely positive quantum dynamical semigroups. We revisit the conceptual and historical route leading to the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation and examine several of its early developments and consequences. Our aim is to show how several initially separate mathematical and physical ideas converged in the GKLS structure theorem, imme- diately opening the way to manifold consequences for the description of dissipative quantum phenomena and how that theorem, fifty years later, remains both a foundational result and an active framework for open quantum dynamics.
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
We investigate the mathematical structure of entropy-weighted quantum evolution within the framework of open quantum systems. Starting from a system–reservoir interaction model with weak entropy-coupled dynamics, we derive the reduced evolution of the system density matrix under the Born–Markov approximation. The resulting generatoris shown to be of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form, ensuring complete positivity and trace preservation. We further demonstrate that the associated dynamical semigroup admits both Stinespring dilation and Kraus operator representations, establishing that entropy-weighted reduced dynamics can be embedded into a fully unitary system environment framework. The entropy operator appearing in the effective description is identified with the positive-semidefinite dissipative structure induced by reservoir corre lation functions.The construction does not modify standard quantum mechanics but provides a parameterization of dissipative generators in terms of entropy-weighted couplings. This establishes a mathematically consis tent bridge between entropy-based dynamical formulations and conventional open quantum system theory.
The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.
Open quantum systems describe quantum systems that interact unavoidably with their environments. These notes introduce the physical origins of decoherence and dissipation in realistic settings, developing the theoretical framework of reduced dynamics and quantum master equations within the connection to thermodynamics in the quantum regime. Additionally, a set of relevant exercises are proposed to reinforce the acquisition of knowledge. The text equips readers with the tools to model and analyze open quantum dynamics.