Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1810.05933 · Journal of Physics A Mathematical and Theoretical

GKSL generators and digraphs: computing invariant states*

George Androulakis, Alexander Wiedemann

Published Jun 7, 2019• 6 citations
arXivHTMLPDF

Abstract

Abstract In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. In this article we define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be traceless as demanded by the GKSL theorem). Moreover, we consider the converse construction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corresponding semigroup.

DOI
10.1088/1751-8121/ab27f6

Related papers

On The Generators of Quantum Dynamical Semigroups

Alexander Wiedemann

In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.

•Jan 1, 2019•0 citations

Fifty Years of the GKLS Master Equation: Foundations and Early Developments

Fabio Benatti, Dariusz Chruściński, Saverio Pascazio

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.

quant-pharXiv:2609.19404•Open Systems & Information Dynamics•Sep 18, 2026•0 citations

Another Legacy of Andrzej Kossakowski: A Self-Contained Derivation of the GKLS Equation

Gen Kimura

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.

quant-pharXiv:2606.08579•Open Systems & Information Dynamics•Jun 9, 2026•0 citations

Gaussian and Qubit Semigroup Dynamics beyond GKSL

Nadine Diesel, Charlotte Bäcker, Walter T. Strunz

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.

quant-pharXiv:2610.20258•Oct 15, 2026•0 citations

Nonstandard derivation of the Gorini-Kossakowski-Sudarshan-Lindblad master equation of a quantum dynamical semigroup from the Kraus representation

Yui Kuramochi

We give a new nonstandard proof of the well-known theorem that the generator $L$ of a quantum dynamical semigroup $\exp(tL)$ 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 $\exp (δt L)$ for an infinitesimal hyperreal number $δt>0$ 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 $\sqrt{δt}$. We also give a nonstandard proof of a related fact that close completely positive maps have close Kraus operators.

arXiv:2406.03775•Jun 6, 2024•0 citations

Quantum Corridor II: Continuous-Time Admissibility, Quantum Semigroups, and Lindblad Structure

Kenneth Searcy

Continuous-time quantum admissibility forces the generator lane. This paper is the second quantum installment in the Forced Structure corridor series. It begins from the map-level result of Quantum Corridor I, where admissible noncommutative state transformations were fixed as completely positive trace-preserving maps, and asks what additional structure is required to pass to continuous-time evolution. The paper makes those assumptions explicit: time-indexed admissibility, identity at zero, semigroup composition, time-homogeneity, and norm continuity at the origin. Under those assumptions, the admissible evolution lies in the standard uniformly continuous quantum dynamical semigroup setting. In finite dimension, this places the generator in the setting of the Gorini–Kossakowski–Sudarshan–Lindblad theorem, so the admissible generator class is characterized by GKSL/Lindblad structure. In the commuting limit, the restricted generator reduces to the classical rate-matrix generator, showing that the quantum generator corridor is a continuation of the classical branch rather than a detached formalism. The paper is explicit about scope: it does not rederive the GKSL theorem from scratch, and it does not yet address quantum path space, trajectories, or history-level irreversibility.

•Mar 9, 2026•0 citations
Abstract In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. In this article we define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be traceless as demanded by the GKSL theorem). Moreover, we consider the converse construction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corresponding semigroup.
In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.
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.
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.
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.
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.
Continuous-time quantum admissibility forces the generator lane. This paper is the second quantum installment in the Forced Structure corridor series. It begins from the map-level result of Quantum Corridor I, where admissible noncommutative state transformations were fixed as completely positive trace-preserving maps, and asks what additional structure is required to pass to continuous-time evolution. The paper makes those assumptions explicit: time-indexed admissibility, identity at zero, semigroup composition, time-homogeneity, and norm continuity at the origin. Under those assumptions, the admissible evolution lies in the standard uniformly continuous quantum dynamical semigroup setting. In finite dimension, this places the generator in the setting of the Gorini–Kossakowski–Sudarshan–Lindblad theorem, so the admissible generator class is characterized by GKSL/Lindblad structure. In the commuting limit, the restricted generator reduces to the classical rate-matrix generator, showing that the quantum generator corridor is a continuation of the classical branch rather than a detached formalism. The paper is explicit about scope: it does not rederive the GKSL theorem from scratch, and it does not yet address quantum path space, trajectories, or history-level irreversibility.