Causal Rigidity of Born-Type Probability Rules in Infinite-Dimensional Operational Theories
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Quantum Physics arXiv:2602.09056 (quant-ph) [Submitted on 7 Feb 2026] Title:Causal Rigidity of Born-Type Probability Rules in Infinite-Dimensional Operational Theories Authors:Enso O.
Torres Alegre View a PDF of the paper titled Causal Rigidity of Born-Type Probability Rules in Infinite-Dimensional Operational Theories, by Enso O.
Torres Alegre View PDF HTML (experimental) Abstract:We establish an operational rigidity result for a broad class of probability rules in infinite-dimensional settings, applicable under normality and steering assumptions. Starting from a topological generalization of generalized probabilistic theories, we consider probability assignments defined as functions of an operational transition probability between pure states. We show that under three operationally motivated requirements: no superluminal signaling, availability of normal steering via purification in a sigma additive sense, and sigma affinity of probabilities under countable preparation mixtures, any admissible rule within this class must reduce to the identity. In particular, nonlinear deviations generically enable operational signaling distinctions in steering scenarios, while continuity combined with sigma affinity excludes non affine alternatives. This identifies a unique causal fixed point. Within this class of probability rules, the Born rule emerges as the only assignment compatible with no signaling in operational theories admitting normal steering. We connect the operational result to standard infinite-dimensional quantum mechanics through the normal state space of von Neumann algebras and the GNS representation, recovering the conventional Born rule for projective and generalized measurements. We discuss the scope of the assumptions and implications for proposed post quantum modifications in continuous variable and quantum field theoretic regimes. Comments: Subjects: Quantum Physics (quant-ph) MSC classes: 81P05, 81P10, 46L05, 46N50 81P05, 81P10, 46L05, 46N50 81P05, 81P10, 46L05, 46N50 81P05, 81P10, 46L05, 46N50 Cite as: arXiv:2602.09056 [quant-ph] (or arXiv:2602.09056v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.09056 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Enso Onill Torres Alegre [view email] [v1] Sat, 7 Feb 2026 11:04:12 UTC (15 KB) Full-text links: Access Paper: View a PDF of the paper titled Causal Rigidity of Born-Type Probability Rules in Infinite-Dimensional Operational Theories, by Enso O. Torres AlegreView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-02 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?)