Exact Complete Relative-Entropy Contraction of Finite Classical-to-Quantum Channels

We determine the complete equal-reference Umegaki relative-entropy contraction coefficient of every finite preparation channel after input dephasing. For prepared states ρ₁, …, ρₙ, let H be the intersection of their supports and let Aᵢ be the compression of ρᵢ⁺ to H. If H is nonzero, the coefficient is 1 − min over p in Δₙ of Tr_H[(Σᵢ pᵢAᵢ)⁻¹]; it equals one when H is zero. A noncommutative metric inequality gives the upper bound, and a weighted operator-simplex construction gives matching faithful classical–quantum input pairs with equal classical and reference marginals. Reference dimension n suffices for the supremum. The complementary coefficient is the reciprocal of an exponential left order-two Petz–Rényi radius and is multiplicative under Cartesian products. Faithful binary-input qubit-output channels can consequently have complete coefficient tending to one while their ordinary coefficient tends to zero, with reference dimension two sufficient throughout. An exact rational example has complete coefficient 9/41. The matching input family converges pointwise, has input relative entropy tending to zero, and can converge logarithmically slowly; it provides no fixed-error or capacity conclusion. A visual guide records the mathematical routes, proof dependencies, and provenance of the result.