Local Balancing and Branch Decoherence in Projected GHZ Resources

This work develops an operational diagnostic calculus for projected GHZ resources, showing that endpoint usefulness is controlled not by raw multipartite longitudinal parity, but by the filtered branch coherence transferred through the intermediate subsystem. For a two-branch GHZ-type resource c|0⟩⊗ᴺ + s|1⟩⊗ᴺ, any successful completely positive trace-nonincreasing operation on the middle subsystem defines a positive branch-transfer kernel K_E = [[A, Γ], [Γ, B]] ≥ 0*, where A and B are filtered branch weights and Γ is the filtered branch coherence. The endpoint visibility is V_eff = 2cs|Γ| / (c²A + s²B). The main result is the exact factorization V_eff = q · 2r/(1 + r²) = q sech(ln r), with q = |Γ|/√(AB) and r = c√A/(s√B). Here q measures the normalized coherence quality of the transferred branches, while r measures their population imbalance. This separates two different mechanisms of endpoint degradation: branch imbalance is locally balanceable, whereas branch decoherence is not corrected by branch-preserving local balancing. The paper proves that branch-preserving local filters leave q = |Γ|/√(AB) invariant whenever both filtered branches remain present, but can probabilistically set r = 1 and remove the imbalance penalty. At the balanced optimum, the filtered endpoint concurrence reaches C_max = q, with conditional balancing success probability P_bal = 1 − |m|, where m = ⟨Z ⊗ I⟩ is the endpoint population imbalance. Thus, local filtering can concentrate the usable endpoint resource only up to the transferred branch-coherence quality q. The same separation can be read directly from endpoint tomography. From the observables T_xx⁽²⁾, T_xy⁽²⁾, and m = ⟨Z ⊗ I⟩, one obtains V_eff = √((T_xx⁽²⁾)² + (T_xy⁽²⁾)²), r = √((1 + m)/(1 − m)), and q = V_eff/√(1 − m²) on the open two-branch sector. These relations allow endpoint measurements to distinguish a correctable imbalance problem from genuine branch decoherence. Several operational figures of merit are then controlled by the same visibility: C(ρ_AB) = V_eff, N(ρ_AB) = V_eff/2, F_tel^max = (2 + V_eff)/3, and S_max⁽²⁾ = 2√(1 + V_eff²). In practical terms, if r ≠ 1 the endpoint branches should be rebalanced; if q < 1 the loss is branch decoherence; and if q = 1 with r = 1, the endpoint state is Bell-equivalent. The results are finite-dimensional quantum-mechanical identities and introduce no modified dynamics, hidden variables, or new Bell inequality; their contribution is an exact operational separation of endpoint degradation in projected GHZ and repeater-motivated two-branch resources.