Certifying multipartite entangled states is a basic task in quantum information processing, but the achievable copy complexity depends crucially on the measurements available to the verifier. The strongest possible certification measurement for a known pure target state \(|ψ\rangle\) is the two-outcome projector \(\{|ψ\rangle\langleψ|,\mathsf{I}-|ψ\rangle\langleψ|\}\), which is copy-optimal but often experimentally unrealistic or outside the intended measurement model. In this work, we introduce Bell-Matching Certification (BM-Cert), a single-copy verification protocol for the \(n\)-qubit Greenberger--Horne--Zeilinger state using only disjoint two-qubit Bell-basis measurements, together with one single-qubit \(X\)-basis measurement when \(n\) is odd. Surprisingly, a simple combinatorial effect yields perfect completeness and a verification spectral gap \(ν_\mathrm{BM}(n)=1-O(1/n)\), so the protocol approaches the ideal projective verification asymptotically as \(n\) grows. This contrasts with local Pauli GHZ verification, whose optimal spectral gap remains bounded away from \(1\). Thus, allowing only two-qubit entangling measurements on disjoint pairs is already enough to achieve asymptotically ideal projective certification.
Certifying multipartite entangled states is a basic task in quantum information processing, but the achievable copy complexity depends crucially on the measurements available to the verifier. The strongest possible certification measurement for a known pure target state (|ψ\rangle) is the two-outcome projector ({|ψ\rangle\langleψ|,\mathsf{I}-|ψ\rangle\langleψ|}), which is copy-optimal but often experimentally unrealistic or outside the intended measurement model. In this work, we introduce Bell-Matching Certification (BM-Cert), a single-copy verification protocol for the (n)-qubit Greenberger--Horne--Zeilinger state using only disjoint two-qubit Bell-basis measurements, together with one single-qubit (X)-basis measurement when (n) is odd. Surprisingly, a simple combinatorial effect yields perfect completeness and a verification spectral gap (ν_\mathrm{BM}(n)=1-O(1/n)), so the protocol approaches the ideal projective verification asymptotically as (n) grows. This contrasts with local Pauli GHZ verification, whose optimal spectral gap remains bounded away from (1). Thus, allowing only two-qubit entangling measurements on disjoint pairs is already enough to achieve asymptotically ideal projective certification.
Quantum state verification is a statistically reliable way to test whether an experimental device prepares a desired quantum state, without reconstructing the full density matrix. We study nonadaptive, single-copy verification of the W state. Motivated by the recent Bell-matching verification for GHZ states, we introduce an analogous verifier based on disjoint two-qubit measurements. Increasing the allowed measurement locality by just one makes the spectral gap bounded away from zero. Consequently, the copy complexity becomes independent of the number of qubits. The same measurement also yields a fidelity estimation algorithm. By retaining an additional indicator from the outcome, we obtain an unbiased estimator of the W-state fidelity with copy complexity independent of the number of qubits.
A central task in quantum information science is state certification: testing whether an unknown state is ε1-close to a fixed target state, or ε2-far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary n-qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant ε1, they can only positively certify states within ε1=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated.
Here we present robust certification protocols based on few-qubit measurements that apply to all but a O(2−n)-fraction of pure target states. Our first protocol achieves constant robustness, i.e. ε1=Θ(1), using a single O(logn)-qubit measurement along with single-qubit measurements in the Z or X basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in n) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust: ε1=Ω(1/logn). Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.
We construct optimal protocols for verifying qubit and qudit Greenberger-Horne-Zeilinger (GHZ) states using local projective measurements. When the local dimension is a prime, an optimal protocol is constructed from Pauli measurements only. Our protocols provide a highly efficient way to estimate the fidelity and to certify genuine multipartite entanglement. In particular, they enable the certification of genuine multipartite entanglement using only one test when the local dimension is sufficiently large. By virtue of adaptive local projective measurements, we then construct protocols for verifying GHZ-like states that are optimal over all protocols based on one-way communication. The efficiency can be improved further if additional communications are allowed. Finally, we construct optimal protocols for verifying GHZ states and nearly optimal protocols for GHZ-like states in the adversarial scenario.
We introduce a basis-independent algorithm that only uses single-copy measurements to certify the Schmidt number
of 2n-qubit bipartite pure states. Assuming the local system dimension is d=2n, the Schmidt number χ is O(1), and the non-zero Schmidt coefficients are bounded below by Ω(1), our protocol achieves a total sample complexity of O(d2χ2)
with only O(χ4) measurement settings. This work establishes a measurement-setting-efficient approach to high-dimensional entanglement
certification.
Techniques developed for device-independent characterizations allow one to certify certain physical properties of quantum systems without assuming any knowledge of their internal workings. Such a certification, however, often relies on the employment of device-independent witnesses catered for the particular property of interest. In this work, we consider a one-parameter family of multipartite, two-setting, two-outcome Bell inequalities and demonstrate the extent to which they are suited for the device-independent certification of genuine many-body entanglement (and hence the entanglement depth) present in certain well-known multipartite quantum states, including the generalized Greenberger-Horne-Zeilinger (GHZ) states with unbalanced weights, the higher-dimensional generalizations of balanced GHZ states, and the W states. As a by-product of our investigations, we have found that, in contrast with well-established results, provided trivial qubit measurements are allowed, full-correlation Bell inequalities can also be used to demonstrate the nonlocality of weakly entangled unbalanced-weight GHZ states. Besides, we also demonstrate how two-setting, two-outcome Bell inequalities can be constructed, based on the so-called GHZ paradox, to witness the entanglement depth of various graph states, including the ring graph states, the fully connected graph states, and some linear graph states, etc.
Multipartite Bell tests provide a correlation-only route to benchmarking quantum processors, but their application at large scales is hindered by the rapid decay of many-body correlators under noise and exponentially many terms in conventional Bell expressions. Here we address these scalability obstacles by introducing a finite-setting generalized Mermin family of state-tailored Bell inequalities with analytic certification bounds, in which the measurement-setting number m provides an additional certification dimension complementary to the system size n. We show that, for the powers-of-two setting choices considered here, increasing m leaves the ideal normalized multipartite quantum value unchanged while lowering the relevant classical bounds, thereby strengthening the Bell-violation ratios and yielding an improved noise-robustness scaling compared to the standard Mermin inequality. We test this construction experimentally on a programmable superconducting processor by preparing Greenberger-Horne-Zeilinger (GHZ) states of up to 80 qubits. Using randomized sampling for direct Bell-operator estimation, we observe Bell ratios that grow exponentially with system size, certify a nonlocality depth of 14, and show that increasing m strengthens both the Bell ratio and depth certification. All results are obtained solely from measured correlators and analytical bounds, without readout correction, tomography, or model-based mitigation. Generalized Mermin inequalities therefore provide a sharper Bell benchmark for noisy large-scale GHZ states.