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⟩ is the two-outcome projector
{∣ψ⟩⟨ψ∣,I−∣ψ⟩⟨ψ∣}, which is copy-optimal but often experimentally unrealistic or outside the intended measurement model. In this work, we introduce BM-Cert, a single-copy verification protocol for the
n-qubit Greenberger--Horne--Zeilinger (GHZ) 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
νBM(n)=1−O(1/n), so our depth-2 protocol already 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 enough to achieve asymptotically ideal projective certification. The same Bell-matching outcomes also yield BM-Fid, an unbiased estimator of the GHZ fidelity whose leading Hoeffding coefficient in the sample complexity tends to the ideal value achieved by direct projection. For the open-boundary linear nearest neighbor setting, we further introduce Brick-Cert, a disjoint 2-local certification protocol whose spectral gap
4/5 is optimal within that restricted architecture.
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.
Abstract The objective of Quantum State Verification (QSV) is to evaluate whether an unknown quantum state is sufficiently similar to a predetermined target state at a certain confidence level. The required number of identical and independent copies, N, of an unknown state for such a task is inversely proportional to the spectral gap of a verification operator. Stabilizer states form a foundational class of quantum states, extensively utilized in quantum computing. Defined through a group of stabilizer operators constructed from Pauli matrices in qubit systems, their structural properties not only simplify the mathematical representation and simulation of quantum states but also make them particularly suitable for verification protocols that rely on local measurements. Restricting the verification operator to be those that can be achieved by local measurements, previous researches had determined an upper bound for the spectral gap to be 2/3 in the cases of qubit stabilizer states. In this research, we show a similar upper bound of d/(d+1) in the cases of graph states of d-level systems by a technique from [1] but adapted to d-level graph states. In particular, we show that d-level graph states are maximally entangled across any bipartition of the graph, which allows us to reduce the problem to the case of a bipartite maximally entangled d-level systems for which the maximal spectral gap is known.