Accuracy barriers for positive Lorentzian bath representations

Independently damped modes with positive Lorentzian spectra are a standard way to replace a continuum bath by a tractable model, and the number of modes needed for a prescribed accuracy sets their cost. The known lower bounds on this number concern the frequency domain and particular spectra, so it has been unclear whether the slow convergence of positive Lorentzian fits in time is intrinsic. We bound the spectral mass that a uniformly accurate description can place outside the band of the target spectrum, which forces most of its weight onto narrow lines, and count the narrow lines with a positive windowed energy inequality. For every fixed nonzero compactly supported spectral density in 𝐿^1 ∩ 𝐿^2 we prove that, below an explicit accuracy threshold, all-time uniform error 𝛿 requires at least 𝑐_𝜌/𝛿 lines, for arbitrary centers, weights and linewidths. The all-time error of 𝐺 positive lines therefore decreases no faster than 𝑐_𝜌/𝐺, whereas for a flat spectrum complex exponential decompositions need only polylogarithmically many terms. For a flat spectrum an explicit construction matches the lower bound up to one logarithm. For two-channel baths an exact three-lag filtered-force identity gives the same inverse-error cost without a spectral cutoff, and two baths with identical trace correlations at all times and identical equal-time covariance require two lines and a diverging number of lines, respectively. For undamped descriptions at fixed accuracy, the mode count grows logarithmically with the horizon for weights of either sign, and the optimal linewidth decreases exponentially with the number of modes