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arXiv · 2608.04539

Coupling Does Not Reduce the Auxiliary-Mode Count for $1/|\omega|$ Spectra in Passive Lindblad Networks

Qing-Ao Xiang·Yan Liu·Xin-Yuan Yang·Ya-Ju Song
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Abstract

Representing continuous environments by finitely many Markovian auxiliary modes is fundamental in non-Markovian open quantum systems, yet a critical question remains: at a fixed mode budget, can coherent intermode coupling reduce the spectral approximation error? We prove that intermode coupling offers no advantage when passive, number-conserving Gaussian Lindblad auxiliary networks approximate a $1/|\omega|$ spectrum over a finite two-sided frequency band. For any mode budget $N$, the general coupled class and its uncoupled diagonal subclass share the same optimal error, which is exactly the degree-$2N$ Zolotarev error for sign approximation. This optimum is attainable by $N$ independent damped auxiliary modes at zero detuning. The result holds when the auxiliary network is in a stationary vacuum state, the system couples to it via a single Hermitian bath operator, and no white-noise feedthrough term is present. Consequently, although a general coupled network has $O(N^{2})$ real parameters, coherent intermode coupling, collective dissipation, and nonnormal structure cannot reduce the number of auxiliary modes required to reach a prescribed tolerance. This exact relation yields both the minimum mode count for a prescribed positive-frequency dynamic range and tolerance, and the maximum dynamic range attainable for a prescribed mode budget and tolerance.

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Qing-Ao Xiang, Yan Liu, Xin-Yuan Yang, Ya-Ju Song. 2026-08-05. Coupling Does Not Reduce the Auxiliary-Mode Count for $1/|\omega|$ Spectra in Passive Lindblad Networks. https://arxiv.org/abs/2608.04539

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