Uncertainty Relation for a Single Observable
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Quantum Physics arXiv:2605.26331 (quant-ph) [Submitted on 25 May 2026] Title:Uncertainty Relation for a Single Observable Authors:Haruki Yamashita, Aina Mayumi, Gen Kimura View a PDF of the paper titled Uncertainty Relation for a Single Observable, by Haruki Yamashita and 2 other authors View PDF HTML (experimental) Abstract:Uncertainty relations are usually formulated as trade-off relations between two or more observables. Here we show that the uncertainty of a single observable already has a nontrivial lower bound originating from the noncommutativity between the observable and the quantum state. We prove sharp lower bounds on the variance of a single observable and then sharpen them further by introducing the classical uncertainty of the observable under a fixed state. The optimal coefficient is determined solely by the smallest and largest eigenvalues of the state. Our results include an optimal state-dependent improvement of Luo's Wigner--Yanase-type relation and a direct bound showing that coherence or asymmetry of the state with respect to the observable gives an unavoidable contribution to its uncertainty. For qubits, the sharpened bounds become exact identities, giving a complete decomposition of the variance into classical and noncommutative parts. These single-observable relations also yield improved product-form uncertainty relations for pairs of observables. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) Cite as: arXiv:2605.26331 [quant-ph] (or arXiv:2605.26331v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.26331 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Gen Kimura [view email] [v1] Mon, 25 May 2026 21:07:54 UTC (116 KB) Full-text links: Access Paper: View a PDF of the paper titled Uncertainty Relation for a Single Observable, by Haruki Yamashita and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: math math-ph math.MP 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?)