The GKSL master equation for N-level systems provides a necessary and sufficient form for the generator of a quantum dynamical semigroup in the Schrodinger picture where the underlying Hilbert space is $\mathbb{C}^N$. In this paper we provide a detailed, self-contained, and elementary construction of the GKSL master equation for an N-level system. We also provide necessary and sufficient conditions for forms of generators of semigroups which have some, but not all, of the defining properties of quantum dynamical semigroups. We do this in such a way to illuminate how each defining property of a quantum dynamical semigroup contributes to the form of the generators.
The GKSL master equation for N-level systems provides a necessary and sufficient form for the generator of a quantum dynamical semigroup in the Schrodinger picture where the underlying Hilbert space is CN. In this paper we provide a detailed, self-contained, and elementary construction of the GKSL master equation for an N-level system. We also provide necessary and sufficient conditions for forms of generators of semigroups which have some, but not all, of the defining properties of quantum dynamical semigroups. We do this in such a way to illuminate how each defining property of a quantum dynamical semigroup contributes to the form of the generators.
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.
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.
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.
This preprint presents a self-contained and mathematically explicit reconstruction of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation starting from exact microscopic unitary dynamics through the Nakajima-Zwanzig projection operator formalism. Beginning with the exact non-local generalized master equation, we systematically analyze the hierarchy of approximations required for the emergence of a Markovian quantum dynamical semigroup. The Born (weak-coupling), Markov (time-localization), and secular approximations are derived and interpreted as successive information-reduction steps, each associated with clearly identifiable mathematical assumptions and physical consequences. Particular attention is devoted to the role of secularization in transforming the Redfield generator into a completely positive GKSL structure. To provide a concrete worked example, the formalism is applied to a damped two-level system coupled to a bosonic thermal reservoir, where the Redfield and GKSL equations are derived explicitly and compared at the microscopic level. Beyond the derivation itself, the work develops a systematic map of validity limits and failure regimes. The breakdown of the Born, Markov, secular, factorization, and stationarity assumptions is analyzed in terms of environmental correlation times, spectral-density structure, non-Markovian memory effects, information backflow, and loss of semigroup behavior. The resulting framework clarifies the precise operational domain in which GKSL dynamics can be regarded as a physically reliable effective description and provides a rigorous foundation for applications involving entropy-production theorems, quantum thermodynamics, and open-system information theory.
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.
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to a GKSL master equation for the full density matrix.