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

TL;DR

This paper derives a time-uniform error bound for a broad class of approximation methods in quantum systems, ensuring long-term accuracy of GKSL generators derived via temporal coarse graining.

Contribution

It provides the first unified, rigorous, time-uniform error bound for various approximation schemes in Markovian open quantum systems.

Findings

The error bound is valid for arbitrarily long times.

It guarantees the accuracy of GKSL generators when dissipation timescales exceed bath correlation times.

The bound applies to a general class of temporal coarse graining methods.

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.