Third-Order Local Randomized Measurements for Finite-size Entanglement Certification

Quantum Physics [Submitted on 14 Apr 2026] Third-Order Local Randomized Measurements for Finite-size Entanglement Certification Giovanni Scala, Gniewomir Sarbicki 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 Submission history From: Giovanni Scala [view email] [v1] Tue, 14 Apr 2026 18:00:02 UTC (30 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)