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Non-Gaussian Noise: A Minimal Structural Model Based on Spread–Contraction Dynamics

ab_ab

Published May 24, 2026• 0 citations
Publisher

Abstract

This work investigates the structural origin of non‑Gaussian quantum noise by introducing two universal degrees of freedom: spread and contraction. These two components appear broadly in quantum optics, measurement theory, and statistical physics, and they characterize how a distribution acquires smoothness, nonlocality, sharpness, or irreversibility. Using a minimal 1+1 dimensional dynamical model, we show that the competition between spread and contraction determines the shape of the distribution intensity p(x). When the ratio between the two crosses a critical value, the system undergoes a continuous yet abrupt transition through three regimes: Gaussian → non‑Gaussian → two‑component. The model reproduces key signatures of non‑Gaussianity—such as changes in kurtosis, skewness, and the emergence of two‑component spectra—without introducing additional parameters. This framework provides a unified structural perspective on phenomena including the disappearance of interference fringes, sudden contraction in weak measurements, and non‑Gaussian noise observed in optical state tomography and decoherence experiments. The spread–contraction plane serves as a general classification map for understanding how non‑Gaussian noise arises from the balance between smoothing and localization mechanisms, independent of any specific physical theory.

DOI
10.5281/zenodo.20362883

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This work investigates the structural origin of non‑Gaussian quantum noise by introducing two universal degrees of freedom: spread and contraction. These two components appear broadly in quantum optics, measurement theory, and statistical physics, and they characterize how a distribution acquires smoothness, nonlocality, sharpness, or irreversibility. Using a minimal 1+1 dimensional dynamical model, we show that the competition between spread and contraction determines the shape of the distribution intensity p(x). When the ratio between the two crosses a critical value, the system undergoes a continuous yet abrupt transition through three regimes: Gaussian → non‑Gaussian → two‑component. The model reproduces key signatures of non‑Gaussianity—such as changes in kurtosis, skewness, and the emergence of two‑component spectra—without introducing additional parameters. This framework provides a unified structural perspective on phenomena including the disappearance of interference fringes, sudden contraction in weak measurements, and non‑Gaussian noise observed in optical state tomography and decoherence experiments. The spread–contraction plane serves as a general classification map for understanding how non‑Gaussian noise arises from the balance between smoothing and localization mechanisms, independent of any specific physical theory.
This paper studies the evolution of non-Gaussian phase fluctuations in stochastic and quantum systems. We construct an observable based on second and fourth cumulants and show that it approaches a constant value under broad physical conditions.The main result is the emergence of a universal invariant:Y(t) = σ⁴(t) / (κ(t) · t)We show that this quantity converges to a constant λ under short-memory, finite-cumulant dynamics. This connects stochastic processes, decoherence physics, and renormalization flow into a unified structure. In many physical systems, noise is commonly approximated as Gaussian. This assumption simplifies analysis but hides higher-order statistical structure.However, in real quantum systems (e.g., superconducting qubits, optical interferometers, spin environments), deviations from Gaussianity are observed due to:environmental memory effectsrare event contributionsinteraction-induced correlationsThe question addressed here is:Does non-Gaussianity decay in a universal way under time evolution?To answer this, we define a measurable invariant based on cumulants.
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This work presents a new theoretical and numerical framework for understanding the sharp quantum–classical crossover by introducing two internal structural degrees of freedom—the local spread S(x) and contraction C(x)—derived from the gradient and curvature of a probability distribution. These quantities form a two‑dimensional Spread–Contraction (SC) phase diagram that characterizes the transition between quantum‑dominated spreading and classical localization. A minimal covariant action motivated by information geometry is constructed, from which the SC Ordinary Differential Equation (SC‑ODE) is obtained via variational principles. The SC‑ODE integrates spreading and contracting contributions in a minimal form and provides a structural mechanism for the steep drop in interference visibility observed in experiments. To investigate the role of spatial structure, the study introduces a conservative SC Partial Differential Equation (SC‑PDE) incorporating degenerate mobility and backward diffusion. Numerical experiments under extreme initial conditions—where micro‑noise is superimposed on a uniform field—reveal a three‑phase structure consisting of stable, critical, and collapse regimes. This behavior contrasts with the always‑stable SC‑ODE and demonstrates that spatial degrees of freedom are essential for generating critical phenomena. A comparison with the Cahn–Hilliard equation shows that, within the scope of the numerical experiments, the SC‑PDE preserves non‑negativity and boundedness even under conditions where the Cahn–Hilliard model fails. This robustness arises from the automatic suppression of backward diffusion near the vacuum due to the degenerate mobility structure. Overall, this work provides a unified structural framework for describing the sharpness of the quantum–classical crossover and offers a numerically stable PDE formulation that may be relevant for surface evolution, thin‑film processes, and nonlinear field dynamics.
Author's Note (August 24, 2026): This exploratory preprint is currently being re-evaluated as part of a broader methodological audit of the subsequent research program. In particular, the structural interpretation of the reported results and its relationship to later abstract transformation models are being reassessed. This notice does not indicate that the numerical results reported in this preprint have been shown to be incorrect; rather, their interpretation and scope are under review. Any necessary corrections or revisions will be documented transparently in a subsequent version. This exploratory preprint examines whether representative quantum noise channels can be interpreted as distinct modes of structural reconfiguration in observable quantum correlation distributions. Using Bell-state-based simulations, the study compares depolarizing noise, phase damping, and amplitude damping across Z, X, and Y measurement bases. The results suggest that noise-channel type and measurement basis jointly determine whether the observed distribution tends toward preservation, uniformization, fixation, or observation-dependent deformation.
Description This work introduces a structural framework for understanding the quantum–classical crossover using two internal quantities of the Thickness Structure Hypothesis (TSH): the spread deviation Δf and the shrink tension γT. These quantities characterize the competing tendencies of quantum spreading and classical or gravitational contraction. By organizing physical behaviors on the Δf–γT phase diagram, the framework provides a unified structural classification of quantum, classical, and gravitational regimes. A key result is that the sharp disappearance of quantum interference can be interpreted as a critical phenomenon. Near the boundary between the Stable (quantum) and Composite (classical) phases, the interference visibility follows a Landau‑type scaling law with critical exponent β = 1/2. This suggests that the abrupt switching observed in the quantum–classical transition arises from structural changes in Δf and γT rather than from external decoherence alone. Within this framework, wave‑packet collapse and the loss of interference visibility correspond to motion toward larger γT in the structural parameter space. The Δf–γT phase diagram thus provides a general, theory‑independent viewpoint for describing interference, localization, and the onset of classical behavior. The approach offers a structural mechanism for the sharpness of the quantum–classical crossover and connects quantum, classical, and gravitational behaviors within a single unified diagram.