Isotropic random walks and Brownian diffusion on complex projective space

• URL: http://arxiv.org/abs/2606.11438v1
• Date: Tue, 09 Jun 2026 20:49:10 GMT
• Title: Isotropic random walks and Brownian diffusion on complex projective space
• Abstract summary: We show that isotropic random walks on the complex projective space provide a canonical and analytically tractable-geometric framework for the exploration of quantum-state space.<n>The approach combines harmonic analysis on compact rank-one symmetric spaces with pure-state evolution and yields explicit analytical expressions for transition kernels, fidelity statistics, and geometric observables associated with the Fubini--Study metric.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that isotropic random walks on the complex projective space provide a canonical and analytically tractable stochastic-geometric framework for the exploration of quantum-state space. The approach combines harmonic analysis on compact rank-one symmetric spaces with stochastic pure-state evolution and yields explicit analytical expressions for transition kernels, fidelity statistics, and geometric observables associated with the Fubini--Study metric. In particular, the framework provides a solvable reference model for isotropic depolarization and Haar equilibration, reproducing Haar-random fidelity statistics and the invariant measure on projective Hilbert space without specifying a microscopic Lindblad generator. In the short-time regime, the stochastic evolution converges to Brownian diffusion generated by the Fubini--Study Laplace--Beltrami operator, while the long-time limit exhibits concentration-of-measure behaviour characteristic of high-dimensional random quantum states. We further derive analytical and asymptotic results for the first-passage-time problem, including closed-form expressions in the Brownian limit for the mean first passage time and the long-time tail of the first-passage-time distribution. For high-fidelity target states, the mean first passage time exhibits a strong dimension-dependent divergence originating from the concentration properties of the Fubini--Study geometry.

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