Branch-Coherence Transfer in Projected GHZ Resources

This work formulates a tensor-transfer principle for two-branch Schmidt and GHZ-type quantum resources under projection, filtering, and endpoint reduction. The central message is that coherence-sector tensor components can survive projection as endpoint phase visibility, while diagonal branch-axis components generally do not transfer as lower-rank tensor entries. Projection does not simply delete tensor indices; it performs a protocol-dependent contraction followed by normalization. For two-qubit Schmidt states, the singlet-like phase kernel is recovered only in the maximally entangled case. Away from maximal entanglement, a Schmidt-axis obstruction remains, showing that branch coherence and branch-axis structure are distinct tensor sectors. For GHZ-type resources, the distinction becomes especially sharp. The longitudinal parity coefficient dN = ⟨Z⊗N⟩ belongs to the full rank-N resource tensor and is not generally inherited as a two-terminal endpoint coefficient. Under a successful completely positive trace-nonincreasing operation on the intermediate qubits, the endpoint pair is controlled instead by the filtered branch coherence Γ = Tr E(|0M⟩⟨1M|). If A and B are filtered branch weights and the input amplitudes are cos η and sin η, the endpoint visibility is Veff = 2 cos η sin η |Γ| / (cos²η A + sin²η B). The correct transfer law is therefore T⁽ᴺ⁾zz⋯z ↛ T⁽²⁾zz, Γ → Veff. This identifies Γ, not raw multipartite longitudinal parity, as the operationally relevant endpoint quantity. The same normalized branch coherence determines the main two-terminal figures of merit: Veff = C(ρAB) = 2𝒩(ρAB), S⁽²⁾max = 2√(1 + Veff²), and Fmax = (2 + Veff) / 3. The work also derives the engineering form Veff = q · 2r / (1 + r²), separating q, the quality of surviving branch coherence, from r, the amplitude-balance ratio between filtered branches. Thus high endpoint entanglement requires both coherence preservation and branch balance. The principle also extends to dephased two-branch resources: dephasing suppresses Γ and therefore reduces Veff, concurrence, negativity, CHSH strength, and teleportation fidelity, but it does not restore any transfer of the diagonal longitudinal coefficient into the endpoint tensor. All results are finite-dimensional quantum-mechanical identities: no hidden variables, no modified Born rule, no new Bell inequality, and no new topological invariant are introduced. The core lesson is simple: parity does not transfer; filtered coherence does. Precision Entanglement: The Architecture of Branch-Coherence Transfer (.pdf)