Isotropic random walks and Brownian diffusion on complex projective space
Abstract
We study isotropic random walks on complex projective space in the context of stochastic state-space exploration in open quantum systems. Using results from harmonic analysis on compact rank-one symmetric spaces, first we obtain explicit transition kernels and characterise the evolution of fidelity and geometric observables. Then, we establish the correspondence between the small-step limit and Fubini-Study Brownian motion, and also show that the long-time limit reproduces Haar-random state statistics. At the level of the density matrix, the ensemble evolution formally coincides with an isotropic depolarizing channel at all times. Our main results concern first-passage statistics to high-fidelity target states. In the Brownian limit, we derive a closed-form expression for the mean first-passage time and an explicit spectral representation of the first-passage-time distribution, with decay rates determined by the eigenvalues of the killed Fubini-Study Laplace-Beltrami operator. The resulting asymptotics reveal a strong dependence on target fidelity and Hilbert-space dimension, demonstrating the rapidly increasing difficulty of reaching high-fidelity targets through unbiased isotropic exploration. The model is purely geometric and does not incorporate microscopic Hamiltonians or Lindblad dynamics. Nevertheless, the results suggest that it offers a natural starting point for stochastic models of open quantum systems that incorporate Hamiltonian and dissipative effects, with potential applications to stochastic quantum state preparation and control.
Related
- Asymptotic Behaviour for Isotropic Pearson Random Walks
- Approximation of Random Differential Equations Driven by Physical Brownian Motion with Fast Oscillating Noise
- An Inverse Random Source Problem for the Moore-Gibson-Thompson Equation Driven by Fractional Brownian Motion
- Geometric influence on the limiting behavior of diffusion processes in one-sided Brownian environments on disconnected fractal sets
- Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications
BibTeX
@misc{toth2026isotropic,
title = {{Isotropic random walks and Brownian diffusion on complex projective space}},
author = {Gyula I. Tóth},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.11438},
eprint = {2606.11438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11438},
}
RIS
TY - UNPB
TI - Isotropic random walks and Brownian diffusion on complex projective space
AU - Gyula I. Tóth
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2606.11438
UR - https://arxiv.org/abs/2606.11438
AB - We study isotropic random walks on complex projective space in the context of stochastic state-space exploration in open quantum systems. Using results from harmonic analysis on compact rank-one symmetric spaces, first we obtain explicit transition kernels and characterise the evolution of fidelity and geometric observables. Then, we establish the correspondence between the small-step limit and Fubini-Study Brownian motion, and also show that the long-time limit reproduces Haar-random state statistics. At the level of the density matrix, the ensemble evolution formally coincides with an isotropic depolarizing channel at all times. Our main results concern first-passage statistics to high-fidelity target states. In the Brownian limit, we derive a closed-form expression for the mean first-passage time and an explicit spectral representation of the first-passage-time distribution, with decay rates determined by the eigenvalues of the killed Fubini-Study Laplace-Beltrami operator. The resulting asymptotics reveal a strong dependence on target fidelity and Hilbert-space dimension, demonstrating the rapidly increasing difficulty of reaching high-fidelity targets through unbiased isotropic exploration. The model is purely geometric and does not incorporate microscopic Hamiltonians or Lindblad dynamics. Nevertheless, the results suggest that it offers a natural starting point for stochastic models of open quantum systems that incorporate Hamiltonian and dissipative effects, with potential applications to stochastic quantum state preparation and control.
ER -