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:2604.21366

Time-Uniform Error Bound for Temporal Coarse Graining in Markovian Open Quantum Systems

Teruhiro Ikeuchi, Takashi Mori

cond-mat.stat-mechquant-phSubmitted Apr 23, 2026• Announced Apr 24, 2026• 0 citations• Revised May 28, 2026 (v2)
v1 · Apr 23, 2026v2 · May 28, 2026
arXivHTMLPDF

Abstract

Several approximation procedures, such as the full or partial rotating-wave, time-averaging, and geometric-arithmetic approximations, have been proposed to derive Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generators from the Born-Markov quantum master equation (e.g., the Redfield equation). Establishing rigorous error bounds for these approximations is of fundamental and practical importance. However, existing bounds face two major limitations: they are highly specific to individual methods, and, more critically, they diverge in the long-time limit, ensuring the accuracy of the derived GKSL generator only in short-time regimes. In this Letter, we resolve both issues by deriving a unified, rigorous error bound for a general class of approximation methods -- termed temporal coarse graining -- that encompasses all aforementioned schemes. Crucially, our error bound is time-uniform. This guarantees that GKSL generators obtained via temporal coarse graining remain accurate for arbitrarily long times, provided the dissipation timescale is significantly longer than the bath correlation timescale.

Comments
6+2 pages

Related papers

From Redfield Dynamics to Coarse-Grained Completely Positive Generators: Bridging the Gap Between Non-Markovian and GKSL Descriptions

Alexey Alekseev

This manuscript addresses a long-standing challenge in the theory of open quantum systems: the loss of complete positivity (CP) in the non-secular Redfield master equation when energy-level splittings become comparable to relaxation rates. While the standard Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) framework guarantees complete positivity through full secular truncation, it systematically suppresses physically relevant low-frequency quantum coherences. We present a rigorous coarse-graining framework that bridges the gap between exact non-Markovian Nakajima-Zwanzig dynamics and Markovian quantum master equations. By introducing a finite time-averaging window as a physically motivated coarse-graining scale, we demonstrate that partial secularization acts as a controlled frequency filter, preserving the completely positive and trace-preserving (CPTP) structure of the generator while retaining coherence effects that are lost under full secularization. Analytical bounds for the coarse-grained approximation are derived, and the resulting generator is shown to provide a physically consistent and highly accurate description of open-system dynamics in regimes where the standard GKSL approximation becomes overly restrictive. The formalism establishes a mathematically robust intermediate layer between Redfield and GKSL descriptions and offers a natural framework for the investigation of structured environments, near-degenerate quantum systems, and scale-dependent dissipative quantum flows.

•Jun 19, 2026•0 citations

Rotating-Wave and Secular Approximations for Open Quantum Systems

Daniel Burgarth, Paolo Facchi, Giovanni Gramegna +1 more

We derive a nonperturbative bound on the distance between evolutions of open quantum systems described by time-dependent generators. We show how this result can be employed to provide an explicit upper bound on the error of the rotating-wave approximation in the presence of dissipation and decoherence. We apply the derived bound to the strong-coupling limit in open quantum systems and to the secular approximation used to obtain a master equation of the Gorini-Kossakowski-Lindblad-Sudarshan form from the Redfield master equation.

quant-pharXiv:2603.26606•Mar 30, 2026•0 citations

Coarse-graining master equation for periodically driven systems

Ronja Hotz, Gernot Schaller

We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similarly to undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but individually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at the highest computational cost.

arXiv:2102.03063•Physical Review A•Nov 24, 2021•8 citations

From Nakajima-Zwanzig Memory Kernels to Markovian GKSL Dynamics: A Microscopic Derivation and Validity Analysis

Alexey Alekseev

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.

•Jun 15, 2026•0 citations

Universal Lindblad equation for open quantum systems

Frederik Nathan, Mark Spencer Rudner

We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.

arXiv:2004.01469•Physical review. B./Physical review. B•Sep 3, 2020•212 citations

Open-quantum-system dynamics: Recovering positivity of the Redfield equation via the partial secular approximation

Donato Farina, Vittorio Giovannetti

We show how to recover complete positivity (and hence positivity) of the Redfield equation via a coarse-grained averaging technique. We derive general bounds for the coarse graining timescale above which the positivity of the Redfield equation is guaranteed. It turns out that a coarse grain timescale has strong impact on the characteristics of the Lamb shift term and implies, in general, noncommutation between the dissipating and the Hamiltonian components of the generator of the dynamical semigroup. Finally, we specify the analysis to a two-level system or a quantum harmonic oscillator coupled to a fermionic or bosonic thermal environment via dipolelike interaction.

arXiv:1903.07324•Physical Review A•Jul 9, 2019•87 citations
Several approximation procedures, such as the full or partial rotating-wave, time-averaging, and geometric-arithmetic approximations, have been proposed to derive Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generators from the Born-Markov quantum master equation (e.g., the Redfield equation). Establishing rigorous error bounds for these approximations is of fundamental and practical importance. However, existing bounds face two major limitations: they are highly specific to individual methods, and, more critically, they diverge in the long-time limit, ensuring the accuracy of the derived GKSL generator only in short-time regimes. In this Letter, we resolve both issues by deriving a unified, rigorous error bound for a general class of approximation methods -- termed temporal coarse graining -- that encompasses all aforementioned schemes. Crucially, our error bound is time-uniform. This guarantees that GKSL generators obtained via temporal coarse graining remain accurate for arbitrarily long times, provided the dissipation timescale is significantly longer than the bath correlation timescale.
This manuscript addresses a long-standing challenge in the theory of open quantum systems: the loss of complete positivity (CP) in the non-secular Redfield master equation when energy-level splittings become comparable to relaxation rates. While the standard Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) framework guarantees complete positivity through full secular truncation, it systematically suppresses physically relevant low-frequency quantum coherences. We present a rigorous coarse-graining framework that bridges the gap between exact non-Markovian Nakajima-Zwanzig dynamics and Markovian quantum master equations. By introducing a finite time-averaging window as a physically motivated coarse-graining scale, we demonstrate that partial secularization acts as a controlled frequency filter, preserving the completely positive and trace-preserving (CPTP) structure of the generator while retaining coherence effects that are lost under full secularization. Analytical bounds for the coarse-grained approximation are derived, and the resulting generator is shown to provide a physically consistent and highly accurate description of open-system dynamics in regimes where the standard GKSL approximation becomes overly restrictive. The formalism establishes a mathematically robust intermediate layer between Redfield and GKSL descriptions and offers a natural framework for the investigation of structured environments, near-degenerate quantum systems, and scale-dependent dissipative quantum flows.
We derive a nonperturbative bound on the distance between evolutions of open quantum systems described by time-dependent generators. We show how this result can be employed to provide an explicit upper bound on the error of the rotating-wave approximation in the presence of dissipation and decoherence. We apply the derived bound to the strong-coupling limit in open quantum systems and to the secular approximation used to obtain a master equation of the Gorini-Kossakowski-Lindblad-Sudarshan form from the Redfield master equation.
We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similarly to undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but individually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at the highest computational cost.
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 develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.
We show how to recover complete positivity (and hence positivity) of the Redfield equation via a coarse-grained averaging technique. We derive general bounds for the coarse graining timescale above which the positivity of the Redfield equation is guaranteed. It turns out that a coarse grain timescale has strong impact on the characteristics of the Lamb shift term and implies, in general, noncommutation between the dissipating and the Hamiltonian components of the generator of the dynamical semigroup. Finally, we specify the analysis to a two-level system or a quantum harmonic oscillator coupled to a fermionic or bosonic thermal environment via dipolelike interaction.