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Quantitative Convergence and Spectral-Projector Stability for Finite-Dimensional GKSL Semigroups, with Illustrations on Truncated Bosonic Models

Geraldo José Ferraresi de Araújo

Published Jan 1, 2026• 0 citations
Publisher

Abstract

We develop a quantitative framework for analyzing convergence and spectral-projector stability in finite-dimensional open quantum systems governed by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dynamics. Our focus is on computable bounds for the approach to the asymptotic sector and on perturbative stability estimates for the associated spectral projection. Rather than claiming universal constants for arbitrary Lindbladians, we formulate our results under explicit spectral-separation and resolvent assumptions, thereby making the role of non-normality transparent. We distinguish two geometrically different cases: (i) primitive semigroups with a unique steady state, and (ii) non-primitive semigroups with multidimensional steady-state structure relevant to encoded quantum information. We then illustrate the framework on truncated bosonic models motivated by dissipatively stabilized cat-qubit memories. This repository contains: (1) the preprint manuscript (v1.1) with Appendices A–C; (2) a Python validation suite implementing computation of resolvent norms, spectral gaps, projector stability, and primitivity detection; (3) 41 validated tests covering Propositions 1–2, Frigerio criterion, noise bias, and truncation error; (4) four numerical experiments on truncated bosonic models with publication-quality figures; (5) reproducible numerical results with checksums.

DOI
10.17605/osf.io/gp3z9

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We develop a quantitative framework for analyzing convergence and spectral-projector stability in finite-dimensional open quantum systems governed by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dynamics. Our focus is on computable bounds for the approach to the asymptotic sector and on perturbative stability estimates for the associated spectral projection. Rather than claiming universal constants for arbitrary Lindbladians, we formulate our results under explicit spectral-separation and resolvent assumptions, thereby making the role of non-normality transparent. We distinguish two geometrically different cases: (i) primitive semigroups with a unique steady state, and (ii) non-primitive semigroups with multidimensional steady-state structure relevant to encoded quantum information. We then illustrate the framework on truncated bosonic models motivated by dissipatively stabilized cat-qubit memories. This repository contains: (1) the preprint manuscript (v1.1) with Appendices A–C; (2) a Python validation suite implementing computation of resolvent norms, spectral gaps, projector stability, and primitivity detection; (3) 41 validated tests covering Propositions 1–2, Frigerio criterion, noise bias, and truncation error; (4) four numerical experiments on truncated bosonic models with publication-quality figures; (5) reproducible numerical results with checksums.
We develop a quantitative framework for analyzing convergence and spectral-projector stability in finite-dimensional open quantum systems governed by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dynamics. Our focus is on computable bounds for the approach to the asymptotic sector and on perturbative stability estimates for the associated spectral projection. We distinguish two geometrically different cases: (i) primitive semigroups with a unique steady state, and (ii) non-primitive semigroups with multidimensional steady-state structure relevant to encoded quantum information. We then illustrate the framework on truncated bosonic models motivated by dissipatively stabilized cat-qubit memories. Validation code and numerical results available at https://doi.org/10.17605/OSF.IO/GP3Z9
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 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.
We investigate quantum Markov semigroups on bosonic Fock space and identify a broad class of infinite-dimensional dissipative evolutions that exhibit instantaneous Sobolev regularization. Motivated by stability problems in quantum computation, we show that for certain Lindblad operators that are polynomials of creation and annihilation operators, the resulting dynamics immediately transform any initial state into one with finite expectation in all powers of the number operator. A key application is in the bosonic cat code, where we obtain explicit estimates in the trace norm for the speed of convergence. These estimates sharpen existing perturbative bounds at both short and long times, offering new analytic tools for assessing stability and error suppression in bosonic quantum information processing. For example, we improve the strong exponential convergence of the (shifted) 222-photon dissipation to its asymptotic channel to the uniform topology. For multi-mode systems, a generation theorem in concentrated single-sandwich norms supplies the domain properties required for the regularization argument.
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 The aims of this paper are twofold. The first aim is to give a brief review of the most relevant theoretical results concerning the uniqueness of the steady-state solution of the Lindblad–Gorini–Kossakowski–Sudarshan master equation and the criteria which guarantee relaxingness and irreducibility of dynamical semigroups. In particular, we test and discuss their physical meaning by considering their applicability to the characterisation of the simplest open quantum system i.e. a two-level system coupled to a bath of harmonic oscillators at zero temperature. The second aim is to provide a set of sufficient conditions which guarantees the uniqueness of the steady-state solution and its attractivity. Starting from simple assumptions, we derive simple criteria that can be efficiently exploited to characterise the behavior of dissipative systems of spins and bosons (with truncated Fock space), and a wide variety of other open quantum systems recently studied.