Description:A unified geometric analysis of GHZ–Mermin experiments connecting the binary parity obstruction, classical correlation polytope, independent attenuation cone, and finite-sample information geometry. The paper derives an exact multiplicative criterion for attenuation compatibility and a sharp tradeoff between Mermin violation and attenuation-model failure, W ≤ 3 + e⁻δ_A, then develops Fisher/KL diagnostics for statistically separating nonclassicality from noise-model mismatch. Keywords:GHZ–Mermin inequality; Bell nonlocality; quantum contextuality; parity obstruction; incidence geometry; attenuation cone; noise models; correlation polytopes; information geometry; Kullback–Leibler divergence; Fisher information; experimental design; quantum foundations; multipartite entanglement.
Description:A unified geometric analysis of GHZ–Mermin experiments connecting the binary parity obstruction, classical correlation polytope, independent attenuation cone, and finite-sample information geometry. The paper derives an exact multiplicative criterion for attenuation compatibility and a sharp tradeoff between Mermin violation and attenuation-model failure, W ≤ 3 + e⁻δ_A, then develops Fisher/KL diagnostics for statistically separating nonclassicality from noise-model mismatch. Keywords:GHZ–Mermin inequality; Bell nonlocality; quantum contextuality; parity obstruction; incidence geometry; attenuation cone; noise models; correlation polytopes; information geometry; Kullback–Leibler divergence; Fisher information; experimental design; quantum foundations; multipartite entanglement.
Description:A geometric analysis of GHZ–Mermin experiments that separates deterministic parity obstruction, classical Bell constraints, and independent attenuation mechanisms using the incidence structure of K4. The paper derives an exact edge-cone compatibility criterion, proves uniqueness of attenuation failure, establishes the sharp tradeoff W≤3+e−δA, extends the construction to Kn, and develops a coarse-grained KL/Fisher framework for statistical separation and experimental design. Keywords:GHZ–Mermin experiments; Bell nonlocality; incidence geometry; edge cones; parity obstruction; attenuation models; Mermin inequality; contextuality; Kullback–Leibler divergence; Fisher information; experimental design; statistical separation; complete graphs.
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\MP_\beta$ obtained from the Mermin scenario, parametrized by a function β on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\MP_0$ and $\MP_1$ depending on the parity of β. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\MP_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\MP_1$ can be seen as a nonlocal toy version of Λ-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the 2-qubit case, we provide a decomposition of the Λ-polytope using $\MP_1$, whose vertices are classified, and the nonsignaling polytope of the (2,3,2) Bell scenario, whose vertices are well-known.
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes MPβ obtained from the Mermin scenario, parametrized by a function β on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes MP0 and MP1 depending on the parity of β. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of MP0 turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. MP1 can be seen as a nonlocal toy version of Λ-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the 2-qubit case, we provide a decomposition of the Λ-polytope using MP1, whose vertices are classified, and the nonsignaling polytope of the (2,3,2) Bell scenario, whose vertices are well-known.
In multipartite Bell scenarios, we study the nonlocality robustness of the Greenberger-Horne-Zeilinger (GHZ) state. When each party performs planar measurements forming a regular polygon, we exploit the symmetry of the resulting correlation tensor to drastically accelerate the computation of (i) a Bell inequality via Frank-Wolfe algorithms and (ii) the corresponding local bound. The Bell inequalities obtained are facets of the symmetrized local polytope and they give the best-known upper bounds on the nonlocality robustness of the GHZ state for three to ten parties. Moreover, for four measurements per party, we generalize our facets and hence show, for any number of parties, an improvement on Mermin's inequality in terms of noise robustness. We also compute the detection efficiency of our inequalities and show that some give rise to the activation of nonlocality in star networks, a property that was only shown with an infinite number of measurements.
Bell’s theorem constrains locally causal or factorizable hidden-variable completions of quantum correlations, while GHZ and Mermin extend the incompatibility with context-independent local assignments into multipartite systems [2,3,6,7]. A preceding paper in this program proposed a geometric interpretation of bipartite entanglement in which the source is represented as a nonseparable bivector relation rather than as a preassigned table of local vector outcomes [1]. The present paper develops the higher-order extension of that construction by replacing the heuristic progression from to with an explicit relational exterior representation. For qubits, each subsystem is assigned a local Bloch space , and the Pauli coefficient hierarchy is represented in the restricted cross-system exterior space whose real dimension is . Pauli correlation terms supported on distinct subsystems map to relational exterior -forms. The bipartite singlet correlation becomes while the phase-sensitive GHZ correlation becomes obtained by contracting a full-support relational form with the corresponding measurement blade. The GHZ family then yields a closure result. Every proper reduced density operator of is independent of , while suitable full--party measurements recover that phase. Consequently, the future disclosure map does not factor through the proper-marginal quotient. We call this a closure obstruction and define a future-relative closure rank as the minimum relational order required by a sufficient representation. For the phase-sensitive GHZ future, . Relational grade is kept distinct from connected-correlation order, quantum entanglement, Bell violation, and closure rank. Classical higher-order correlations can possess high closure rank, while GHZ and W states can share tripartite relational order yet belong to different multipartite entanglement classes [11,12]. The framework therefore does not identify higher exterior grade with greater quantumness. Instead, relational grade records correlation support, separability theory records quantum decomposability, and closure theory asks which relational distinctions must survive for a specified future. Keywords: Bell theorem; GHZ states; multipartite entanglement; relational geometry; exterior algebra; bivectors; trivectors; closure mathematics; future sufficiency; quantum foundations; Mermin inequality; quantum information.
Bell’s theorem constrains locally causal or factorizable hidden-variable completions of quantum correlations, while GHZ and Mermin extend the incompatibility with context-independent local assignments into multipartite systems [2,3,6,7]. A preceding paper in this program proposed a geometric interpretation of bipartite entanglement in which the source is represented as a nonseparable bivector relation rather than as a preassigned table of local vector outcomes [1]. The present paper develops the higher-order extension of that construction by replacing the heuristic progression from to with an explicit relational exterior representation. For qubits, each subsystem is assigned a local Bloch space , and the Pauli coefficient hierarchy is represented in the restricted cross-system exterior space whose real dimension is . Pauli correlation terms supported on distinct subsystems map to relational exterior -forms. The bipartite singlet correlation becomes while the phase-sensitive GHZ correlation becomes obtained by contracting a full-support relational form with the corresponding measurement blade. The GHZ family then yields a closure result. Every proper reduced density operator of is independent of , while suitable full--party measurements recover that phase. Consequently, the future disclosure map does not factor through the proper-marginal quotient. We call this a closure obstruction and define a future-relative closure rank as the minimum relational order required by a sufficient representation. For the phase-sensitive GHZ future, . Relational grade is kept distinct from connected-correlation order, quantum entanglement, Bell violation, and closure rank. Classical higher-order correlations can possess high closure rank, while GHZ and W states can share tripartite relational order yet belong to different multipartite entanglement classes [11,12]. The framework therefore does not identify higher exterior grade with greater quantumness. Instead, relational grade records correlation support, separability theory records quantum decomposability, and closure theory asks which relational distinctions must survive for a specified future. Keywords: Bell theorem; GHZ states; multipartite entanglement; relational geometry; exterior algebra; bivectors; trivectors; closure mathematics; future sufficiency; quantum foundations; Mermin inequality; quantum information.