Disjoint Bell measurements enable near-projective GHZ certification

Abstract

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 $|\psi\rangle$ is the two-outcome projector $\{|\psi\rangle\langle\psi|,\mathsf{I}-|\psi\rangle\langle\psi|\}$, 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 $\nu_\mathrm{BM}(n)=1-O(1/n)$, 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.