Isotropic random walks and Brownian diffusion on complex projective space
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Quantum Physics arXiv:2606.11438 (quant-ph) [Submitted on 9 Jun 2026] Title:Isotropic random walks and Brownian diffusion on complex projective space Authors:Gyula I. Tóth View a PDF of the paper titled Isotropic random walks and Brownian diffusion on complex projective space, by Gyula I. T\'oth View PDF HTML (experimental) 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. Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) Cite as: arXiv:2606.11438 [quant-ph] (or arXiv:2606.11438v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2606.11438 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Gyula Toth [view email] [v1] Tue, 9 Jun 2026 20:49:10 UTC (22 KB) Full-text links: Access Paper: View a PDF of the paper titled Isotropic random walks and Brownian diffusion on complex projective space, by Gyula I. T\'othView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-06 Change to browse by: math math-ph math.MP References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)