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arXiv:2604.13165

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

Giovanni Scala, Gniewomir Sarbicki

quant-phSubmitted Apr 14, 2026• Announced Apr 16, 2026• 0 citations
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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 $\bar{\mathfrak{M}}(ρ)$ built from experimentally accessible second- and third-order local invariants. Entanglement is certified when its minimum eigenvalue $\mathcal{E}_4(ρ)$ becomes negative. We prove that all separable states satisfy $\bar{\mathfrak{M}}(ρ)\succeq0$, and that the sign of $\mathcal{E}_4(ρ)$ 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.

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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.
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.
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.
The certification of quantum systems is essential for emerging quantum technologies, particularly in quantum communication, networks, and distributed computing, where maintaining a common reference frame across distant nodes poses significant challenges. Reference frame independent approaches, such as randomized measurement schemes, offer a promising route by reducing experimental demands while granting access to basis-independent quantities, including entanglement. However, the efficiency of such schemes in measuring such local invariants has remained unclear. In this work, we determine the minimal number of measurement settings required to access all two-qubit invariants. We further demonstrate that entanglement certification necessarily involves the most demanding invariants, establishing it as a maximally difficult task. Our results reveal a fundamental hierarchy among invariants, with direct implications for experimental feasibility and theoretical understanding of quantum certification. Finally, we extend our analysis beyond bipartite systems by applying it to the Kempe invariant in three-qubit systems, improving known measurement protocols and providing a first step toward uncovering similar hierarchies in higher dimensions.
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.
For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.