Near-projective GHZ certification from disjoint Bell measurements

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 \(|ψ\rangle\) 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.