Analytic Maximal Violation of Extended MABK Inequalities for Generalized GHZ States

• URL: http://arxiv.org/abs/2609.02045v1
• Date: Wed, 02 Sep 2026 03:19:58 GMT
• Title: Analytic Maximal Violation of Extended MABK Inequalities for Generalized GHZ States
• Abstract summary: We analytically characterize the maximal quantum violation of the extended Mermin-Ardehali-Belinskii-Klyshko family of inequalities by $n$-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states.<n>We construct two complementary measurement strategies that saturate the bound for EMABK throughout the entire parameter range.
• Score: 9.699517824851613
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We analytically characterize the maximal quantum violation of the extended Mermin-Ardehali-Belinskii-Klyshko (EMABK) family of inequalities by $n$-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states. We develop a correlation-tensor approach in which the Bell value is expressed as the Frobenius inner product of the quantum correlation tensor and an effective coefficient tensor. A rank constraint on the latter, together with the von Neumann trace inequality, yields an upper bound governed by the two largest singular values of the reshaped correlation-tensor. For generalized GHZ states, we determine the complete singular spectrum and obtain a piecewise analytic upper bound. We then construct two complementary measurement strategies that saturate the bound for EMABK throughout the entire parameter range. The first is a purely MABK strategy in which all measurement directions lie in the equatorial plane of the Bloch sphere. The second is a hybrid strategy that recursively combines lower-order MABK anti-diagonal operators with the fully-$σ_z$ tensor product. The resulting piecewise analytic expression recovers the standard MABK Tsirelson bound $2^{(n-1)/2}$ in the maximally entangled limit and approaches the classical local-hidden-variable bound in the product-state limits. It further shows that every entangled generalized GHZ state exhibits a strict quantum-classical separation under the EMABK inequality, thereby eliminating the nonviolation region of the standard MABK inequality in the partially entangled regime.

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