Third-Order Local Randomized Measurements for Finite-size Entanglement Certification
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Quantum Physics arXiv:2604.13165 (quant-ph) [Submitted on 14 Apr 2026] Title:Third-Order Local Randomized Measurements for Finite-size Entanglement Certification Authors:Giovanni Scala, Gniewomir Sarbicki View a PDF of the paper titled Third-Order Local Randomized Measurements for Finite-size Entanglement Certification, by Giovanni Scala and Gniewomir Sarbicki View PDF HTML (experimental) Abstract:Randomized measurements access nonlinear functionals without full tomography, yet turning third-order local single-copy data into a strong entanglement test remains difficult. We convert the reduction criterion into an experimentally measurable separability criterion by testing it on squared affine combinations of the identity, the local marginals, and the state itself. This yields a $4\times4$ matrix $\bar{\mathfrak{M}}(\rho)$ built from experimentally accessible second- and third-order local invariants. Entanglement is certified when its minimum eigenvalue $\mathcal{E}_4(\rho)$ becomes negative. We prove that all separable states satisfy $\bar{\mathfrak{M}}(\rho)\succeq0$, and that the sign of $\mathcal{E}_4(\rho)$ can be inferred from single-copy randomized measurements with dimension-independent sample complexity. For isotropic states on $d\times d$, the second-order purity criterion detects entanglement only for $p\sim d^{-1/2}$, whereas our third-order witness reaches $p\sim 2/d$, close to the separability threshold $p\sim 1/d$. A complementary nonisotropic benchmark shows that the affine marginal directions become essential once the local states are not maximally mixed. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.13165 [quant-ph] (or arXiv:2604.13165v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.13165 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Giovanni Scala [view email] [v1] Tue, 14 Apr 2026 18:00:02 UTC (30 KB) Full-text links: Access Paper: View a PDF of the paper titled Third-Order Local Randomized Measurements for Finite-size Entanglement Certification, by Giovanni Scala and Gniewomir SarbickiView 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?)