Longitudinal GHZ Axis Coefficients Do Not Transfer under Two-Terminal Projection

This work clarifies a basic but often overlooked distinction in GHZ-based projection and repeater-type protocols: the correlation tensor of the original N-partite resource is not the same object as the effective two-terminal tensor obtained after projection, filtering, or postselection. The central result is an exact non-transfer theorem for the longitudinal GHZ coefficient. For the GHZ-type state |G(N)η⟩ = cos η |0⟩⊗N + sin η |1⟩⊗N, the resource tensor separates into a longitudinal branch-axis sector with coefficient dN(η) = cos²η + (−1)N sin²η and an equatorial phase-kernel sector with visibility V = sin 2η. Equivalently, the resource correlation has the form EN(a1, …, aN) = dN ∏j aj,z + V Re ∏j(aj,x − i aj,y). This decomposition shows that the GHZ resource contains two structurally different tensor sectors: the longitudinal coefficient dN = ⟨Z⊗N⟩ is an N-body resource component, while the equatorial term measures branch coherence. The main point is that dN does not generally transfer to the endpoint coefficient T(2)zz. After any successful operation on the intermediate qubits, the normalized endpoint state remains supported in span{|00⟩, |11⟩} before endpoint bit-flip feedforward. Therefore the two-terminal longitudinal coefficient is fixed as T(2)zz = ⟨Z ⊗ Z⟩eff = 1, independently of the original value of dN. For odd N, however, the resource coefficient is dN = cos 2η, which is generically not equal to 1. Thus the correct tensor rule is T(N)zz…z ↛ T(2)zz. The endpoint tensor is not obtained by deleting tensor indices; it is produced by a protocol-dependent contraction with the successful measurement or filtering operation, followed by normalization. The quantity that does transfer operationally is the filtered branch coherence Γ. For a general successful completely positive trace-nonincreasing operation on the intermediate subsystem, with branch weights A, B, coherence Γ, and success probability p = cos²η A + sin²η B, the endpoint phase visibility is Veff = 2 cos η sin η |Γ| / (cos²η A + sin²η B). This same normalized coherence controls the endpoint entanglement and Bell strength: Veff = C(ρAB) = 2 N(ρAB) and S(2)max = 2 √(1 + Veff²). In the ideal coherent balanced case, Veff = sin 2η, so the transverse phase visibility is preserved, but this still does not imply transfer of the longitudinal coefficient dN. A sharp diagnostic appears already for N = 3 and η = π/4. The full GHZ resource has T(3)zzz = 0, while an X-basis projection of the middle qubit produces, up to a known phase, a two-qubit Bell endpoint state with T(2)zz = 1. This directly demonstrates that a vanishing three-body longitudinal resource correlation can coexist with a saturated two-terminal longitudinal endpoint correlation. The correct operational summary is therefore dN ↛ T(2)zz, Γ → T(2)⊥: the longitudinal GHZ coefficient remains a property of the original N-partite resource, whereas the filtered branch coherence determines the two-terminal visibility, entanglement, and CHSH strength. .