Quantum channels preserving sigma-additivity on diagonal von Neumann algebras and Ulam measurable cardinals

[2604.25854] Quantum channels preserving sigma-additivity on diagonal von Neumann algebras and Ulam measurable cardinals Skip to main content Search Submit Donate Log in Quantum Physics arXiv:2604.25854 (quant-ph) [Submitted on 28 Apr 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Quantum channels preserving sigma-additivity on diagonal von Neumann algebras and Ulam measurable cardinals Authors: S. V. Dzhenzher View PDF HTML (experimental) Abstract: This paper investigates the interplay between the properties of quantum states on the Hilbert space \(\ell_2(\kappa)\) and the set-theoretic nature of the cardinal $\kappa$. We focus on the existence of singular $\sigma$-additive states~--- functionals whose induced measures are $\sigma$-additive yet vanish on singletons. While the existence of such states is known to be equivalent to the Ulam measurability of $\kappa$, their structural and dynamical properties remain largely unexplored. We prove that any $\sigma$-additive state on the diagonal algebra is representable as a Pettis integral over a singular $\sigma$-additive measure, extending the classical representation theory to the non-normal sector. Furthermore, we construct a class of quantum channels using $\sigma$-complete ultrafilters that map normal states to singular $\sigma$-additive states, effectively <<archiving>> information into the singular part of the state space. Comments: 8 pages, 1 figure; the introduction is expanded; the title is clarified Subjects: Quantum Physics (quant-ph) ; Functional Analysis (math.FA) Cite as: arXiv:2604.25854 [quant-ph] (or arXiv:2604.25854v3 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.25854 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sviatoslav Dzhenzher [ view email ] [v1] Tue, 28 Apr 2026 16:57:17 UTC (20 KB) [v2] Thu, 14 May 2026 07:37:14 UTC (21 KB) [v3] Thu, 27 Aug 2026 08:54:39 UTC (27 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2026-04 Change to browse by: math math.FA References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation 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? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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