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arXiv:2609.08635

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

Giovanni Scala, Corrado Cosimo Mattiacci, Dorota Pietryka, Gniewomir Sarbicki

quant-phSubmitted Sep 8, 2026• Announced Sep 9, 2026• 0 citations
v1 · Sep 8, 2026v2 · Sep 11, 2026
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Abstract

Certifying entanglement in high-dimensional systems usually requires full state tomography, whose cost grows rapidly with the system dimension. Local randomized measurements offer a scalable alternative, but existing tests based on second-order correlations access only limited information about the state. Here, we derive a finite-size entanglement certificate that extends local randomized measurements to third order. The additional third-order information reveals entanglement that remains undetected at second order, while a dimension-independent concentration bound provides rigorous control of finite-sample errors. Our result opens a practical route to extracting stronger entanglement information from experimental platforms without the dimension-dependent overhead of state tomography.

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Certifying entanglement in high-dimensional systems usually requires full state tomography, whose cost grows rapidly with the system dimension. Local randomized measurements offer a scalable alternative, but existing tests based on second-order correlations access only limited information about the state. Here, we derive a finite-size entanglement certificate that extends local randomized measurements to third order. The additional third-order information reveals entanglement that remains undetected at second order, while a dimension-independent concentration bound provides rigorous control of finite-sample errors. Our result opens a practical route to extracting stronger entanglement information from experimental platforms without the dimension-dependent overhead of state tomography.
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 4×44\times44×4 matrix Mˉ(ρ)\bar{\mathfrak{M}}(ρ)Mˉ(ρ) built from experimentally accessible second- and third-order local invariants. Entanglement is certified when its minimum eigenvalue E4(ρ)\mathcal{E}_4(ρ)E4​(ρ) becomes negative. We prove that all separable states satisfy Mˉ(ρ)⪰0\bar{\mathfrak{M}}(ρ)\succeq0Mˉ(ρ)⪰0, and that the sign of E4(ρ)\mathcal{E}_4(ρ)E4​(ρ) can be inferred from single-copy randomized measurements with dimension-independent sample complexity. For isotropic states on d×dd\times dd×d, the second-order purity criterion detects entanglement only for p∼d−1/2p\sim d^{-1/2}p∼d−1/2, whereas our third-order witness reaches p∼2/dp\sim 2/dp∼2/d, close to the separability threshold p∼1/dp\sim 1/dp∼1/d. A complementary nonisotropic benchmark shows that the affine marginal directions become essential once the local states are not maximally mixed.
Abstract We compute all third-order local invariants accessible via randomized measurements and employ them to derive separability criteria. The reconstruction of the invariants yields experimentally accessible entanglement criteria for multipartite states with arbitrary local dimensions. The results show that third-order invariants capture inter-subsystem correlations beyond second-order spectral criteria within more feasible entanglement detection protocols than full tomography. As an example, for Werner states in d = 3, the entanglement is detected for p > 1 2 for the second-order correlations and it is improved to p > 1 10 3 at the third-order.
Certifying genuine multipartite entanglement in quantum systems can require a number of measurements that grows exponentially with the system size. Here we introduce a criterion based on correlation-tensor subsector lengths restricted to local measurement planes and show that it can be evaluated using partially randomized measurements without an explicit exponential dependence on the number of qubits. We derive the corresponding bounds for kkk-separable states and illustrate the criterion using representative families of multipartite entangled states. Finally, we demonstrate the practical applicability of the method on an ion-trap quantum computer by certifying genuine five-partite entanglement.
High-dimensional entangled states offer higher information capacity and stronger resilience to noise compared with two-dimensional systems. However, the large number of modes and sensitivity to random rotations complicate experimental entanglement certification. Here, we experimentally certify three-dimensional entanglement in a five-dimensional two-photon state using 800 Haar-random measurements implemented via a ten-plane programmable light converter. We further demonstrate the robustness of this approach against random rotations, certifying high-dimensional entanglement despite arbitrary phase randomization of the optical modes. This method, which requires no common reference frame between parties, opens the door for high-dimensional entanglement distribution through long-range random links.
High-dimensional entanglement certification: bounding relative entropy of entanglement in 2d+12d+12d+1 experiment-friendly measurements
Entanglement -- the coherent correlations between parties in a joint quantum system -- is well-understood and quantifiable in the two-dimensional, two-party case. Higher (>2)-dimensional entangled systems hold promise in extending the capabilities of various quantum information applications. Despite the utility of such systems, methods for quantifying high-dimensional entanglement are more limited and experimentally challenging. We review entanglement certification approaches and the large number of -- often difficult -- measurements required to apply them. We present a novel certification method whose measurement requirements scale linearly with subsystem dimension (scaling with the square-root of the system dimension) and which requires only a single complex measurement. The certification method places a lower-bound on the relative entropy of entanglement of any maximally correlated state thereby certifying system entanglement. A lower bound is also shown for any maximally correlated state in the presence of noise -- the expected experimental case. We discuss experimental realization of all required measurements.
In recent work Phys. Rev. X 16, 031057 (2026), we proposed a self-consistent approach to entanglement certification based on generalized noncontextuality. It requires no prior characterization of the measurement devices and, given access to all local measurements, can certify every entangled state in a Bell circuit. Here, we study the complexity of this protocol in a finite experiment. At fixed local dimension ddd, we show that the optimal number of distinct local effects needed to reach a trace-distance \textit{entanglement resolution} γγγ is Θd(γ−(d−1))Θ_d(γ^{-(d-1)})Θd​(γ−(d−1)). This scaling is necessary even when the measurements are tailored to the target state. Independent Haar-random projective measurements instead require Θd(γ−(d−1)log⁡(1/γ))Θ_d(γ^{-(d-1)}\log(1/γ))Θd​(γ−(d−1)log(1/γ)) effects. Finally, we show that, once the exact operational identities are known, Θd(γ−2log⁡(1/δ))Θ_d(γ^{-2}\log(1/δ))Θd​(γ−2log(1/δ)) copies of the target state are necessary and sufficient for certification with error probability at most δδδ. Determining the optimal sample complexity when those identities must instead be inferred from the same finite target-state data remains open.