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arXiv:2610.20258

Gaussian and Qubit Semigroup Dynamics beyond GKSL

Nadine Diesel, Charlotte Bäcker, Walter T. Strunz

quant-phSubmitted Oct 14, 2026• Announced Oct 15, 2026• 0 citations
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Abstract

A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.

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arXiv:2204.07734•Symmetry•Apr 6, 2022•3 citations
A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.
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.
We give a new nonstandard proof of the well-known theorem that the generator LLL of a quantum dynamical semigroup exp⁡(tL)\exp(tL)exp(tL) on a finite-dimensional quantum system has a specific form called a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator (also known as a Lindbladian) and vice versa. The proof starts from the Kraus representation of the quantum channel exp⁡(δtL)\exp (δt L)exp(δtL) for an infinitesimal hyperreal number δt>0δt>0δt>0 and then estimates the orders of the traceless components of the Kraus operators. The jump operators naturally arise as the standard parts of the traceless components of the Kraus operators divided by δt\sqrt{δt}δt​. We also give a nonstandard proof of a related fact that close completely positive maps have close Kraus operators.
This note is written for the special issue of OSID dedicated to the 50th anniversary of the Gorini--Kossakowski--Lindblad--Sudarshan equation. Its purpose is not to give a comprehensive historical review, but rather to reconstruct, in a self-contained way, one logical route leading to the GKLS generator. The emphasis is placed on Kossakowski's structural insight: the combination of Markovianity, complete positivity, trace preservation, and the infinitesimal structure of quantum dynamical semigroups naturally leads to the celebrated form of the generator. I also include a few personal recollections of Andrzej Kossakowski, under whose guidance I spent one year as a postdoctoral researcher in 2003--2004.
We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn–Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and the jump operators. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.
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.
Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension 2. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to 1, and find a moment of time starting from which the FGKLS equation can be used—the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system.