Kramers-Kronig Relations for Vector-Valued Hardy Spaces in Non-Markovian Open Quantum Dynamics

• URL: http://arxiv.org/abs/2604.17058v2
• Date: Tue, 21 Apr 2026 17:25:24 GMT
• Title: Kramers-Kronig Relations for Vector-Valued Hardy Spaces in Non-Markovian Open Quantum Dynamics
• Abstract summary: We organize the analysis around three objects -- the memory kernel, the reduced-state propagator, and the effective kernel.<n>For the kernel, under standard bath hypotheses, we prove membership in the vector-valued Hardy space Hp_+ and derive rigorous KK relations.<n>For the effective kernel, we derive a perturbative modified KK relation with UHP corrections at zeros of the reduce d-state propagator.
• Score: 1.8867357521563244
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Kramers-Kronig (KK) relations have not been rigorously established for the Nakajima-Zwanzig memory kernel, the central object of non-Markovian open quantum dynamics. We organize the analysis around three objects -- the memory kernel, the reduced-state propagator, and the effective kernel -- which have different sensitivity to initial system-bath correlations. For the kernel, under standard bath hypotheses (continuous spectrum, local L^p regularity), we prove membership in the vector-valued Hardy space H^p_+ and derive rigorous KK relations. Three consequences follow: a CPTP-Hardy consistency criterion (UHP poles in approximate kernels imply non-physical dynamics); a passivity-analyticity link for passive bosonic baths (dissipative kernels automatically satisfy KK via the Herglotz-Nevanlinna class); and a Carleman criterion guaranteeing that Pade reconstructions from MKCT moments satisfy KK. For the reduced state, we show analyticity in the upper half-plane for any initial state. For the effective kernel, we derive a perturbative modified KK relation with UHP corrections at zeros of the reduce d-state propagator. We verify the framework on solvable models and show that initial correlations can double the KK residual in a Jaynes-Cummings model, indicating force-fit kernel acausality.

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