How I Learned to Stop Worrying and Love the Redfield Equation: Completely-Positive Resummation of Non-Markovian Dynamics
Abstract
The Redfield equation with time-dependent coefficients (or TCL2 equation) gives one of the simplest perturbative descriptions of open quantum system dynamics derived from a microscopic theory. Its status as an approximation is nevertheless subtle: while perturbation theory fixes the reduced dynamics order-by-order in the system-environment coupling, solving the Redfield equation yields a particular resummation that need not preserve complete positivity. Here we compare Redfield evolution with an exponentiated (Magnus) resummation of the same second-order perturbative map, which is completely positive and trace preserving (CPTP) for a broad class of physical conditions. We also construct a simple CPTP resolvent resummation, illustrating the non-uniqueness of resummations of the same perturbative map. The Redfield and Magnus prescriptions agree to second order but differ through time-ordering corrections at higher orders. We exhibit a weak-coupling criterion which guarantees that Redfield evolution, even when non-Markovian, remains close to a CPTP map. We illustrate the criterion in two examples, focusing on finite-dimensional (qubit) systems. Our results clarify when, and why, Redfield evolution can remain reliable even when it is highly non-Lindbladian.
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BibTeX
@misc{kaplanek2026learned,
title = {{How I Learned to Stop Worrying and Love the Redfield Equation: Completely-Positive Resummation of Non-Markovian Dynamics}},
author = {Greg Kaplanek and Jason Pollack},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03676},
eprint = {2610.03676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03676},
}
RIS
TY - UNPB
TI - How I Learned to Stop Worrying and Love the Redfield Equation: Completely-Positive Resummation of Non-Markovian Dynamics
AU - Greg Kaplanek
AU - Jason Pollack
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03676
UR - https://arxiv.org/abs/2610.03676
AB - The Redfield equation with time-dependent coefficients (or TCL2 equation) gives one of the simplest perturbative descriptions of open quantum system dynamics derived from a microscopic theory. Its status as an approximation is nevertheless subtle: while perturbation theory fixes the reduced dynamics order-by-order in the system-environment coupling, solving the Redfield equation yields a particular resummation that need not preserve complete positivity. Here we compare Redfield evolution with an exponentiated (Magnus) resummation of the same second-order perturbative map, which is completely positive and trace preserving (CPTP) for a broad class of physical conditions. We also construct a simple CPTP resolvent resummation, illustrating the non-uniqueness of resummations of the same perturbative map. The Redfield and Magnus prescriptions agree to second order but differ through time-ordering corrections at higher orders. We exhibit a weak-coupling criterion which guarantees that Redfield evolution, even when non-Markovian, remains close to a CPTP map. We illustrate the criterion in two examples, focusing on finite-dimensional (qubit) systems. Our results clarify when, and why, Redfield evolution can remain reliable even when it is highly non-Lindbladian.
ER -