Approximate Reduced Lindblad Dynamics via Algebraic and Adiabatic Methods

TL;DR

This paper introduces an algebraic and adiabatic framework for reducing Markovian open quantum system models, ensuring physical validity and providing explicit error bounds for approximate dynamics.

Contribution

It develops a novel algebraic approach for model reduction of Lindblad dynamics that guarantees complete positivity and trace preservation, with perturbative and adiabatic methods linked to existing techniques.

Findings

Projection on the center manifold yields an asymptotically exact unitary reduced dynamics.

Perturbative reduction maintains Lindbladian structure under arbitrary perturbations.

Explicit finite-time error bounds quantify leakage from the unperturbed sector.

Abstract

We present an algebraic framework for approximate model reduction of Markovian open quantum dynamics that guarantees complete positivity and trace preservation by construction. First, we show that projecting a Lindblad generator on its center manifold -- the space spanned by eigenoperators with purely imaginary eigenvalue -- yields an asymptotically exact reduced quantum dynamical semigroup whose dynamics is unitary, with exponentially decaying transient error controlled by the generator's spectral gap. Second, for analytic perturbations of a Lindblad generator with a tractable center manifold, we propose a perturbative reduction that keeps the reduced space fixed at the unperturbed center manifold. The resulting generator is shown to remain a valid Lindbladian for arbitrary perturbation strengths, and explicit finite-time error bounds, that quantify leakage from the unperturbed center sector, are provided. We further clarify the connection to adiabatic elimination methods, by both showing how the algebraic reduction can be directly related to a first-order adiabatic-elimination and by providing sufficient conditions under which the latter method can be applied while preserving complete positivity. We showcase the usefulness of our techniques in dissipative many-body quantum systems exhibiting non-stationary long-time dynamics.