Quantum Measurement Channels from Universal Carrier Descent

This paper proves a finite-dimensional carrier-descent theorem for canonical projective quantum measurement channels. Under explicit universal-carrier, Ward-admissibility, and finite projective-sector hypotheses, each nonzero sector carries a compact U(1) phase carrier, and a finite family of orthogonal sectors determines a central sectorwise phase torus. Haar averaging over this compact carrier gives the conditional expectation E_P(X) = ∑ⱼ Πⱼ X Πⱼ, whose trace-dual state map is the Lüders nonselective measurement channel Δ_P(ρ) = ∑ⱼ Πⱼ ρ Πⱼ. The fixed observable algebra is the block algebra B_fix = {X : [X, Πⱼ] = 0 for all j}, while the classical outcome algebra is its center Z_out = span{Π₁, …, Πₘ}. The Born weights pⱼ(ρ) = Tr(Πⱼρ) are the unique coefficients representing the restricted state on Z_out, and selective measurement is conditioning on the corresponding boundary record. Except in the trivial one-sector case, the Lüders projection is idempotent, noninvertible, and nonmultiplicative on the full matrix algebra; hence the associated boundary-channel datum is a boundary-channel projection rather than full reversible Heisenberg–Schrödinger collapse, i.e. full-algebra unitary equivalence. The paper proves that canonical finite projective measurement is the endpoint of a carrier-to-phase-to-Haar boundary-channel descent: phase selection determines Haar projection, Haar projection determines the Lüders channel, central restriction determines the Born weights, and the collapse criterion certifies the obstruction to full reversible Heisenberg–Schrödinger collapse in every nontrivial projective context. License note: Distributed under CC BY-NC-ND 4.0.