Computable Multibranch Transfer Kernels for Endpoint Quantum Usefulness

This work introduces a finite-dimensional operational framework for analysing multibranch quantum transfer when only the endpoint interference data are relevant. Instead of treating the full parent state as the primary object, the paper identifies the correct computable quantity: the branch-transfer kernel K_br, a positive contraction satisfying 0 ≤ K_br ≤ I. This kernel captures how distinguishability, decoherence, and branch-overlap structure are transferred into the observable endpoint state. The central idea is simple but powerful: given branch amplitudes collected in D_a and a branch-transfer kernel K, the observable endpoint state is determined by R = D_a K D_a† / p, where p = Tr(D_a K D_a†). This provides a compact normal form for multibranch interference experiments, quantum-channel reductions, and operational reconstructions in finite dimension. The framework is not proposed as a new physical dynamics; it is a computable representation theorem for how endpoint-visible coherence is encoded. A key result is the converse realization theorem: every admissible positive contraction K can be realised by an appropriate environment or transfer construction. This makes the formalism operationally complete: K is not merely a fitting matrix, but a valid effective object that can arise from an underlying quantum process. The paper also clarifies an important limitation of pairwise visibility data: for d ≥ 3, knowing all pairwise overlaps is generally insufficient to reconstruct the full multibranch structure. Higher-order consistency is encoded in cycle phases, such as Φ_ijk = arg(K_ij K_jk K_ki), which capture genuinely multibranch interference information. The manuscript also provides a practical computational layer. It includes Python routines for checking positivity, estimating diagonal feasibility, constructing transfer kernels, and testing whether endpoint data admit a valid kernel representation. This turns the theoretical result into a usable finite-dimensional tool for quantum information, interferometry, decoherence modelling, and operational state reconstruction. The main contribution of the work is therefore the identification of K_br as the natural endpoint object: not the full parent tensor, not merely pairwise visibilities, but a positive semidefinite transfer kernel carrying both magnitude and phase consistency. In this sense, the paper provides a clean bridge between abstract quantum coherence and computable experimental data. It is a compact mathematical framework for asking: what part of a multibranch quantum process is actually visible at the endpoint, and how can it be represented, checked, and reconstructed?