Vistas of Algebraic Probability: Quantum Computation and Information
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Quantum Physics arXiv:2602.04351 (quant-ph) [Submitted on 4 Feb 2026] Title:Vistas of Algebraic Probability: Quantum Computation and Information Authors:Antonio Falcó, Hermann G. Matthies View a PDF of the paper titled Vistas of Algebraic Probability: Quantum Computation and Information, by Antonio Falc\'o and Hermann G. Matthies View PDF Abstract:Kolmogorov's foundation of probability takes measure spaces, $\sigma$-algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum effects, and more generally for \emph{quantum-like} situations. A broader formulation is provided by an algebraic viewpoint: one starts from an algebra of random variables equipped with a distinguished linear functional -- the \emph{state} -- interpreted as expectation. In this sense, the approach can also be viewed as a modern reading of ideas already implicit in early probability (e.g., the Bernoullis), while its contemporary form has been developed and used extensively in quantum physics. The algebraic framework accommodates both classical and quantum-like behaviours, yet it remains underused in classical probability and uncertainty quantification, where it can nevertheless open new perspectives and clarify structural features. Although the language carries a physics flavor, the subject is purely probabilistic. The key distinction between classical and quantum-like behaviour is \emph{commutativity}: its failure produces the characteristic effects of quantum-like situations. The rise of quantum computing is a prominent setting in which such behaviour may become relevant even for practitioners in computational science. Here we focus on the purely algebraic core of the approach. By restricting attention to finite-dimensional algebras, we avoid many analytical subtleties while retaining the main ideas, their classical limit, and their applicability to quantum-like models and quantum computation. Subjects: Quantum Physics (quant-ph); Probability (math.PR) MSC classes: 46L53, 47L90, 46K10, 81R05, 81S25, 46N50, 46L89, 81P68, 60A99 Cite as: arXiv:2602.04351 [quant-ph] (or arXiv:2602.04351v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.04351 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Antonio Falcó [view email] [v1] Wed, 4 Feb 2026 09:19:11 UTC (198 KB) Full-text links: Access Paper: View a PDF of the paper titled Vistas of Algebraic Probability: Quantum Computation and Information, by Antonio Falc\'o and Hermann G. MatthiesView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-02 Change to browse by: math math.PR 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?)