Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras
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Quantum Physics arXiv:2604.23292 (quant-ph) [Submitted on 25 Apr 2026] Title:Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras Authors:Koichi Yamagata View a PDF of the paper titled Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras, by Koichi Yamagata View PDF HTML (experimental) Abstract:We develop a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras. In contrast to the conventional formulation, which is based on families of states, complex completely positive coarse-grainings, and Radon-Nikodym cocycles associated with faithful reference states, our framework allows models consisting of general self-adjoint operators, including derivatives of states. Within this framework, square-root likelihood ratios and symmetric logarithmic derivatives arise naturally as fundamental self-adjoint likelihood-type objects. This makes it possible to treat ordinary quantum statistical models and local quantum statistical structures within a unified setting. We introduce sufficient real positive maps and show that sufficient complex *-subalgebras, sufficient real *-subalgebras, and sufficient real Jordan algebras correspond respectively to complex completely positive maps, real completely positive maps, and real positive maps. We characterize minimal sufficient real *-subalgebras by the likelihood-ratio set together with rho-modular invariance, and show that the real Jordan algebra generated by the likelihood-ratio set and the projected reference state is the minimal sufficient real Jordan algebra. We also obtain Koashi-Imoto type decompositions for sufficient real *-subalgebras and sufficient real Jordan algebras. Our formulation admits degenerate reference states and separates the likelihood-ratio aspect of sufficiency from its genuinely quantum modular aspect. These results suggest that real Jordan structure provides a natural framework for the statistical aspect of quantum theory beyond the conventional complex *-algebraic setting. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.23292 [quant-ph] (or arXiv:2604.23292v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.23292 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Koichi Yamagata [view email] [v1] Sat, 25 Apr 2026 13:13:15 UTC (24 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras, by Koichi YamagataView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 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?)