Quantum Measurement as a Canonical Boundary-Channel Projection

This paper proves a finite-dimensional structural theorem identifying canonical projective quantum measurement as a boundary-channel projection. For a finite projection-valued measurement context P = {Π₁, …, Πₘ}, the central sectorwise phase action Uθ = ∑ⱼ eiθⱼΠⱼ has normalized Haar average E_P(X) = ∫ Uθ X Uθ* dθ = ∑ⱼ Πⱼ X Πⱼ. Its trace-dual state-side map is the Lüders nonselective channel Δ_P(ρ) = ∑ⱼ Πⱼ ρ Πⱼ. The theorem characterizes this Lüders, or minimal-disturbance, channel: a projection-valued measurement fixes the outcome weights, but in degenerate sectors it does not by itself determine every compatible instrument. The additional structure used here is the sectorwise phase symmetry together with preservation of the full fixed block algebra. The fixed observable algebra is 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 weights representing the restricted state on Z_out, and selective measurement is conditioning on the corresponding boundary record. Except for the trivial one-sector context, the Lüders projection is idempotent, noninvertible, and nonmultiplicative on the full matrix algebra; it therefore defines a boundary-channel operation rather than full reversible Heisenberg–Schrödinger collapse. The results are interpretation-neutral and isolate the operator-theoretic structure that any finite projective measurement account using the canonical Lüders update must preserve. License note: Distributed under CC BY-NC-ND 4.0.