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

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.