Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths

• URL: http://arxiv.org/abs/2512.13046v1
• Date: Mon, 15 Dec 2025 07:13:28 GMT
• Title: Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths
• Authors: You-Wei Ding, Yen Chin Ong, Hao Xu,
• Abstract summary: Abbott et al. predicted that quantum paths exhibit a universal Hausdorff dimension that transitions from $d=2$ to $d=1$ as the momentum of the particle increases.<n>We investigate how quantum measurements alter the fractal geometry of quantum particle paths.
• Score: 8.80493159481596
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In a seminal paper, Abbott et al. analyzed the relationship between a particle's trajectory and the resolution of position measurements performed by an observer at fixed time intervals. They predicted that quantum paths exhibit a universal Hausdorff dimension that transitions from $d=2$ to $d=1$ as the momentum of the particle increases. However, although measurements were assumed to occur at intervals of time, the calculations only involved evaluating the expectation value of operators for the free evolution of wave function within a single interval, with no actual physical measurements performed. In this work we investigate how quantum measurements alter the fractal geometry of quantum particle paths. By modelling sequential measurements using Gaussian wave packets for both the particle and the apparatus, we reveal that the dynamics of the measurement change the roughness of the path and shift the emergent Hausdorff dimension towards a lower value in nonselective evolution. For selective evolution, feedback control forces must be introduced to counteract stochastic wave function collapse, stabilising trajectories and enabling dimensionality to be tuned. When the contribution of the measurement approaches zero, our result reduces to that of Abbott et al. Our work can thus be regarded as a more realistic formulation of their approach, and it connects theoretical quantum fractality with measurement physics, quantifying how detectors reshape spacetime statistics at quantum scales.

Related papers

• Work Statistics and Quantum Trajectories: No-Click Limit and non-Hermitian Hamiltonians [50.24983453990065]
  We present a framework for quantum work statistics in continuously monitored quantum systems.<n>Our approach naturally incorporates non-Hermitian dynamics arising from quantum jump processes.<n>We illustrate our theoretical framework by analyzing a one-dimensional transverse-field Ising model under local spin monitoring.
  arXiv  Detail & Related papers  (2025-04-15T23:21:58Z)
• Action formalism for geometric phases from self-closing quantum trajectories [55.2480439325792]
  We study the geometric phase of a subset of self-closing trajectories induced by a continuous Gaussian measurement of a single qubit system. We show that the geometric phase of the most likely trajectories undergoes a topological transition for self-closing trajectories as a function of the measurement strength parameter.
  arXiv  Detail & Related papers  (2023-12-22T15:20:02Z)
• Topological insulator and quantum memory [0.0]
  Uncertainty relations define the universal accuracy limit of quantum measurements. Relatively recently, it was discovered that quantum correlations and quantum memory might reduce the uncertainty of quantum measurements.
  arXiv  Detail & Related papers  (2023-06-20T17:10:44Z)
• On Repeated Measurements of a Quantum Particle in a Harmonic Potential [0.0]
  We study evolution of a quantum particle in a harmonic potential whose position and momentum are repeatedly monitored. We show how classical trajectories emerge in course of observation and study in detail dispersion of position and momentum of the particle.
  arXiv  Detail & Related papers  (2023-06-14T08:18:38Z)
• Quantifying measurement-induced quantum-to-classical crossover using an open-system entanglement measure [49.1574468325115]
  We study the entanglement of a single particle under continuous measurements. We find that the entanglement at intermediate time scales shows the same qualitative behavior as a function of the measurement strength.
  arXiv  Detail & Related papers  (2023-04-06T09:45:11Z)
• Evolution of many-body systems under ancilla quantum measurements [58.720142291102135]
  We study the concept of implementing quantum measurements by coupling a many-body lattice system to an ancillary degree of freedom. We find evidence of a disentangling-entangling measurement-induced transition as was previously observed in more abstract models.
  arXiv  Detail & Related papers  (2023-03-13T13:06:40Z)
• Geometric phases along quantum trajectories [58.720142291102135]
  We study the distribution function of geometric phases in monitored quantum systems. For the single trajectory exhibiting no quantum jumps, a topological transition in the phase acquired after a cycle. For the same parameters, the density matrix does not show any interference.
  arXiv  Detail & Related papers  (2023-01-10T22:05:18Z)
• Quantum Back-action Limits in Dispersively Measured Bose-Einstein Condensates [0.0]
  We theoretically and experimentally characterize quantum back-action in atomic Bose-Einstein condensates interacting with a far-from resonant laser beam. We experimentally quantify the resulting wavefunction change in terms of the contrast of a Ramsey interferometer. This result is a necessary precursor for achieving true quantum back-action limited measurements of quantum gases.
  arXiv  Detail & Related papers  (2022-09-09T17:04:36Z)
• Measurement induced quantum walks [0.0]
  We investigate a quantum walk on a graph with classical and quantum mechanical properties. For a quantum walk on a line we show that in our system the first detection probability decays classically like $(texttime)-3/2$.
  arXiv  Detail & Related papers  (2021-08-30T08:11:24Z)
• Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit [0.0]
  The time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol. For a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian.
  arXiv  Detail & Related papers  (2020-12-02T13:29:22Z)
• The role of boundary conditions in quantum computations of scattering observables [58.720142291102135]
  Quantum computing may offer the opportunity to simulate strongly-interacting field theories, such as quantum chromodynamics, with physical time evolution. As with present-day calculations, quantum computation strategies still require the restriction to a finite system size. We quantify the volume effects for various $1+1$D Minkowski-signature quantities and show that these can be a significant source of systematic uncertainty.
  arXiv  Detail & Related papers  (2020-07-01T17:43:11Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.