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arXiv:2604.17058

Kramers-Kronig Relations and Causality in Non-Markovian Open Quantum Dynamics: Kernel, State, and Effective Kernel

Kejun Liu

quant-phcond-mat.stat-mechmath-phSubmitted Apr 18, 2026• Announced Apr 21, 2026• 0 citations• Revised Jul 10, 2026 (v5)
v1 · Apr 18, 2026v2 · Apr 21, 2026v3 · Apr 27, 2026v4 · Jun 20, 2026v5 · Jul 10, 2026v6 · Aug 7, 2026
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Abstract

Causality -- that a response cannot precede its cause -- is among the most universal constraints in physics. Yet when a unitary microscopic theory is reduced to an open-system memory-kernel equation, causality is not inherited for free: an upper-half-plane pole of the kernel forces exponential growth of the reduced propagator and is incompatible with any completely positive trace-preserving (CPTP) reduced dynamics. We examine three objects with different causal structure under projection -- the Nakajima-Zwanzig (NZ) memory kernel $\tilde{\mathcal{K}}(z)$, the reduced-state Laplace transform $\tildeσ(z)$, and the force-fit effective kernel $\tilde{\mathcal{K}}_\text{eff}(z)$ -- using the Kramers-Kronig (KK) relations as diagnostic. Under a real-axis spectral hypothesis on the projected generator, $\tilde{\mathcal{K}}(z)$ lies in a vector-valued Hardy space and obeys (subtracted) KK relations, giving a CPTP-consistency criterion, a passivity-analyticity statement, and a Carleman diagnostic. We prove $\tildeσ(z)$ analytic in the upper half-plane for any initial state -- unitarity bounds $\|σ(t)\|_{\mathrm{op}} \leq 1$, so acausality cannot be blamed on the state alone. Yet the force-fit kernel can develop upper-half-plane poles at simple zeros of $\tildeσ(z)$: passive baths sit in a robust regime where these zeros stay real, while near-resonant systems enter a fragile regime in which coherence-channel zeros migrate into the upper half-plane, an intrinsic symmetry property already present for factorized states. We verify the full operator-valued KK relation on the extracted $4\times4$ NZ memory kernel of the Jaynes-Cummings model, the relative $L_2$ residual decreasing under refinement ($3.8\% \to 0.95\%$), consistent with exact matrix-valued KK in the continuum limit.

Comments
v5: minor reference updates

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Causality -- that a response cannot precede its cause -- is among the most universal constraints in physics. Yet when a unitary microscopic theory is reduced to an open-system memory-kernel equation, causality is not inherited for free: an upper-half-plane pole of the kernel forces exponential growth of the reduced propagator and is incompatible with any completely positive trace-preserving (CPTP) reduced dynamics. We examine three objects with different causal structure under projection -- the Nakajima-Zwanzig (NZ) memory kernel K~(z)\tilde{\mathcal{K}}(z)K~(z), the reduced-state Laplace transform σ~(z)\tildeσ(z)σ~(z), and the force-fit effective kernel K~eff(z)\tilde{\mathcal{K}}_\text{eff}(z)K~eff​(z) -- using the Kramers-Kronig (KK) relations as diagnostic. Under a real-axis spectral hypothesis on the projected generator, K~(z)\tilde{\mathcal{K}}(z)K~(z) lies in a vector-valued Hardy space and obeys (subtracted) KK relations, giving a CPTP-consistency criterion, a passivity-analyticity statement, and a Carleman diagnostic. We prove σ~(z)\tildeσ(z)σ~(z) analytic in the upper half-plane for any initial state -- unitarity bounds ∥σ(t)∥op≤1\|σ(t)\|_{\mathrm{op}} \leq 1∥σ(t)∥op​≤1, so acausality cannot be blamed on the state alone. Yet the force-fit kernel can develop upper-half-plane poles at simple zeros of σ~(z)\tildeσ(z)σ~(z): passive baths sit in a robust regime where these zeros stay real, while near-resonant systems enter a fragile regime in which coherence-channel zeros migrate into the upper half-plane, an intrinsic symmetry property already present for factorized states. We verify the full operator-valued KK relation on the extracted 4×44\times44×4 NZ memory kernel of the Jaynes-Cummings model, the relative L2L_2L2​ residual decreasing under refinement (3.8%→0.95%3.8\% \to 0.95\%3.8%→0.95%), consistent with exact matrix-valued KK in the continuum limit.
Kramers-Kronig (KK) relations rest on a binary premise: a response function is either analytic in the upper half-plane or it is not. We show that a single reduced-state transform organizes both outcomes into a sharp dichotomy controlled by microscopic unitarity. One closed-form function carries, simultaneously, a zero and a pole in the upper half-plane; the spectral abscissa alpha of the reduced propagator decides which is realized. For alpha < 0 (unitary reduction) the upper-half-plane object is a protected zero: KK holds, yet the zero is directly measurable from a finite-time coherence record by a damped Fourier transform (no analytic continuation), obeying a closed law Im(zeta) = 0.3092 g. For alpha > 0 (gain-driven non-unitary reduction) the zero is replaced by a genuine pole, the Blaschke winding number jumps from 0 to 1, and KK acquires a Lorentzian residue correction scaling as a power law with negative exponent nu ~ -1.08, peaking at threshold. The protected zero is not inert: any scalar single-channel kernel extraction is forced to reproduce a phantom resonance -- a refractive feature with no absorptive origin, at a protocol-independent frequency -- without any initial system-bath correlation. We give the closed-form criteria, a measurable terahertz signature (31-1391 GHz), and the solvable dimer and Jaynes-Cummings models that realize both sides of the boundary.
The memory kernel of an open quantum system obeys Kramers--Kronig (KK) relations if and only if its Laplace transform is analytic in the upper half-plane -- a property known as Hardy-space analyticity. Here we show that non-unitary exchange statistics, the defining property of paraparticles, intrinsically breaks Hardy-space analyticity. The metric ηηη that guarantees a real closed-system spectrum for these particles necessarily differs from the physical Born inner product (∥η−I∥F/∥I∥F=0.51\|η- I\|_F / \|I\|_F = 0.51∥η−I∥F​/∥I∥F​=0.51) -- a mathematical consequence of the R-matrix's non-unitarity, not a parameter choice. This metric is a "shadow metric": Schur's lemma forces it to commute with every bilinear observable, making the distortion physically invisible in the closed system. But when the paraparticle is coupled to a bath, any coupling operator that lies outside the symmetry algebra -- that is, any interaction that sees the internal flavour structure -- exposes the distortion. The memory kernel then develops upper-half-plane poles at coupling gc≈0.1g_c \approx 0.1gc​≈0.1, breaking standard dispersion relations before the closed-system spectrum complexifies. Fermions and bosons, whose exchange is unitary (η=Iη= Iη=I as an analytic fact of the canonical anticommutation algebra), are immune at any coupling, because there is no distortion to expose. The violation is intrinsic: it distinguishes non-unitary exchange statistics from ordinary particle statistics at the level of the memory kernel's analytic structure.
We develop a data-driven framework for identifying non-Markovian equations of motion for open quantum systems, demonstrated here for qubit-environment dynamics. Starting from the Nakajima-Zwanzig formalism, we vectorize the reduced density matrix into a four-dimensional state vector and cast the dynamics as a Volterra integro-differential equation with an operator-valued memory kernel. The learning task is then formulated as a constrained optimization problem over the admissible operator space, where correlation functions are approximated by rational functions using Pade approximants. We establish well-posedness of the learning problem, ensuring existence of minimizers. To assess performance, we construct synthetic data sets from representative test problems of increasing complexity: (i) exactly solvable pure dephasing, with correlation functions expressed in terms of special functions, (ii) a damped Jaynes-Cummings model with an analytic coherence kernel, (iii) a transverse Born model with frequency-resolved bath integrals and population-coherence coupling, and (iv) a non-rotating-wave quantum Rabi model whose memory kernel has no closed form. Numerical experiments demonstrate that Pade captures nontrivial temporal structures such as oscillatory memory, algebraic tails, and phase-sensitive coherence transfer, and that the learned models generalize across ensembles of physically admissible initial states. We perform a parametrization-invariant sensitivity analysis and show that the trajectories are insensitive to the unrecoverable parts of the kernel, so the learned models stay predictive despite severe ill-conditioning in kernel recovery. These results together illustrate that data-driven rational approximation provides an effective route to identifying non-Markovian kernels of practical relevance in quantum technologies.
The Jaynes-Cummings (JC) model, introduced in 1963 and central to cavity quantum electrodynamics, describes a two-level system coupled to a single bosonic mode under the rotating-wave approximation. When the mode is projected out via the Nakajima-Zwanzig (NZ) formalism, the memory-kernel generator QLQ is manifestly non-Hermitian -- yet we prove analytically that its spectrum is strictly real at every coupling and every finite truncation, for both vacuum and thermal baths. For the vacuum bath the characteristic polynomial factorises completely; the nonzero eigenvalues reproduce the JC dressed-state ladder for n >= 2, while the lowest mode is suppressed by exactly sqrt(2) relative to the bare-Hamiltonian prediction. For any thermal Gibbs state, the squared generator reduces to an asymmetric rank-one perturbation symmetrised by a closed-form diagonal metric, with eigenvalues guaranteed real by Cauchy interlacing. We construct a positive-definite metric eta_osc intertwining QLQ with its adjoint on the nonzero spectral subspace, proving hidden pseudo-Hermiticity. A classification theorem extends these results to the full U(1)-conserving single-photon-exchange class with arbitrary complex couplings. The phase boundary under counter-rotating deformation is mapped analytically at resonance and numerically across the coupling-truncation plane, revealing a weak-coupling protected wedge, re-entrant real-spectrum bubbles, and N-universal plateaus organised by a three-family band catalog with closed-form level spacings. The full phase structure is proved well-defined in the N -> infinity thermodynamic limit. The sqrt(2) suppression provides a platform-independent experimental falsification target spanning seven orders of magnitude in coupling strength.
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.
This monograph delivers a rigorous, non-perturbative closed-form analytical solution to the generalized non-Markovian Lindblad-Nakajima-Zwanzig master equation governing open quantum-classical systems coupled to arbitrary non-equilibrium thermal baths. By lifting the dissipative Liouville-von Neumann evolution onto an extended non-Hermitian supersymmetric (SUSY) fibre bundle equipped with a Cartan-Darboux connection, the environmental memory kernel is mapped onto the characteristic flow of an exceptional point foliation. We prove two foundational theorems: (1) Global Closed-Form Integrability, reducing non-local integro-differential dynamics to exact meromorphic Wilson holonomy loops with O(N^3) deterministic complexity, and (2) Strict Complete Positivity and Trace Preservation (CPTP) across all finite and asymptotic temporal regimes, definitively overcoming the unphysical negative-probability breakdown during informational backflow intervals. The formulation includes a production-certified, zero-heap embedded ISO C99 deterministic kernel executing in 1.85 microseconds on standard bare-metal controllers, establishing a new foundation for decoherence-resilient autonomous cybernetic computing.