Commutativity from a single Bargmann invariant equality
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Quantum Physics arXiv:2605.07405 (quant-ph) [Submitted on 8 May 2026] Title:Commutativity from a single Bargmann invariant equality Authors:Rafael Wagner, Ernesto F. Galvão View a PDF of the paper titled Commutativity from a single Bargmann invariant equality, by Rafael Wagner and Ernesto F. Galv\~ao View PDF HTML (experimental) Abstract:Noncommutativity of states and observables is a fundamental signature of quantum theory, and a minimal requirement for nonclassicality. We provide a universal necessary and sufficient condition for pairwise commutativity of quantum states $\rho_1$ and $\rho_2$: they commute if and only if $\mathrm{tr}(\rho_1^2\rho_2^2) = \mathrm{tr}(\rho_1 \rho_2 \rho_1 \rho_2)$. For qubits the identity simplifies to an equality between polynomials of purities and of the two-state overlap $\mathrm{tr}(\rho_1\rho_2)$. These multivariate traces (known as Bargmann invariants) are directly measurable, allowing commutativity tests that bypass full state tomography. We point out possible applications to the analysis of POVM simulability and partial photonic distinguishability. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2605.07405 [quant-ph] (or arXiv:2605.07405v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.07405 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Rafael Wagner [view email] [v1] Fri, 8 May 2026 08:01:15 UTC (98 KB) Full-text links: Access Paper: View a PDF of the paper titled Commutativity from a single Bargmann invariant equality, by Rafael Wagner and Ernesto F. Galv\~aoView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 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?)