# Open Quantum System Dynamics: recovering positivity of the Redfield equation via Partial-Secular Approximation

## TL;DR

This paper introduces a coarse graining method to restore complete positivity in the Redfield quantum master equation, analyzing the impact of coarse graining time on the dynamics of open quantum systems.

## Contribution

It provides a general framework and bounds for coarse graining time scales that ensure positivity of the Redfield equation, with specific analysis for two-level and harmonic oscillator systems.

## Key findings

- Coarse graining can recover positivity of the Redfield equation.
- Bounds on coarse graining time scale are derived.
- Impact on Lamb shift and non-commutation of generator components.

## Abstract

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