Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality
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Quantum Physics arXiv:2602.12334 (quant-ph) [Submitted on 12 Feb 2026] Title:Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality Authors:Jacopo Surace View a PDF of the paper titled Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality, by Jacopo Surace View PDF HTML (experimental) Abstract:Quasi-probabilities appear across diverse areas of physics, but their conceptual foundations remain unclear: they are often treated merely as computational tools, and operations like conditioning and Bayes' theorem become ambiguous. We address both issues by developing a principled framework that derives quasi-probabilities and their conditional calculus from structural consistency requirements on how statements are valued across different universes of discourse, understood as finite Boolean algebras of statements.We begin with a universal valuation that assigns definite (possibly complex) values to all statements. The central concept is Syntactic Locality: every universe can be embedded within a larger ambient one, and the universal valuation must behave coherently under such embeddings and restrictions. From a set of structural principles, we prove a representation theorem showing that every admissible valuation can be re-expressed as a finitely additive measure on mutually exclusive statements, mirroring the usual probability sum rule. We call such additive representatives pre-probabilities. This representation is unique up to an additive regraduation freedom. When this freedom can be fixed canonically, pre-probabilities reduce to finite quasi-probabilities, thereby elevating quasi-probability theory from a computational device to a uniquely determined additive representation of universal valuations. Classical finite probabilities arise as the subclass of quasi-probabilities stable under relativisation, i.e., closed under restriction to sub-universes. Finally, the same framework enables us to define a coherent theory of conditionals, yielding a well-defined generalized Bayes' theorem applicable to both pre-probabilities and quasi-probabilities. We conclude by discussing additional regularity conditions, including the role of rational versus irrational probabilities in this setting. Comments: Subjects: Quantum Physics (quant-ph); Logic (math.LO); Probability (math.PR); Statistics Theory (math.ST) Cite as: arXiv:2602.12334 [quant-ph] (or arXiv:2602.12334v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.12334 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Jacopo Surace [view email] [v1] Thu, 12 Feb 2026 19:00:08 UTC (878 KB) Full-text links: Access Paper: View a PDF of the paper titled Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality, by Jacopo SuraceView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-02 Change to browse by: math math.LO math.PR math.ST stat stat.TH 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?)