Error Bounds for Open Quantum Systems with Harmonic Bosonic Bath

TL;DR

The paper addresses how perturbations in the bath correlation function $B(\cdot,\cdot)$ affect observables in open quantum systems with a harmonic bosonic bath. It develops a rigorous, diagrammatic and combinatorial framework that yields a general error bound for $|\Delta\langle O(t)\rangle|$ in terms of $\Delta B$, without relying on differential inequalities. The main contributions are a general bound $|\Delta\langle O(t)\rangle| \leq \|O_s\|\left( \exp\big(\|W_s\|^2 \int_0^{2t}\int_0^{s_2} |\Delta B(s_1,s_2)| \mathrm{d}s_1 \mathrm{d}s_2\big) - 1 \right)$ along with two auxiliary lemmas establishing a diagrammatic identity and reduction of integrals, and an explicit application to the Spin-Boson model that recovers Mascherpa et al.’s bound. The results provide a rigorous foundation for error control in simulations and spectral-density learning of non-Markovian dynamics, with potential implications for HEOM, pseudomode methods, and inchworm Monte Carlo approaches.

Abstract

We investigate the dependence of physical observable of open quantum systems with Bosonic bath on the bath correlation function. We provide an error estimate of the difference of physical observable induced by the variation of bath correlation function, based on diagrammatic and combinatorial arguments. This gives a mathematically rigorous justification of the result in [Mascherpa et al, Phys Rev Lett 2017].

Figures (3)

• Figure 1: Keldysh contour
• Figure 2: Unfolded Keldysh contour
• Figure 3: Diagrammatic representation for the integral of $(-1)^{\#\{\boldsymbol{s} < t\}} \mathrm{i}^m \tr_s(\rho_s\mathcal{U}^{(0)}(s_{\mathrm{f}}, \boldsymbol{s}, s_{\mathrm{i}})) \mathcal{L}(\mathfrak{q})$ when $m = 6$ and $\mathfrak{q} = \{(s_1,s_6), (s_2,s_4), (s_3,s_5)\}$.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Proposition 1
• proof
• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['thm:bound']}