Isotropic random walks and Brownian diffusion on complex projective space

arXiv:2606.11438v2 Announce Type: replace 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.