Exact Complete Relative-Entropy Contraction of Finite Quantum Channels

We determine the complete equal-reference Umegaki relative-entropy contraction coefficient of every finite-dimensional quantum channel. For its unnormalized Choi matrix J, let H be the largest output subspace with ℂⁿ ⊗ H ⊆ supp J, and let M(ρ) be the compression to H of Tr_A[(ρᵀ ⊗ I)J⁺]. The coefficient is 1 − min_ρ Tr_H M(ρ)⁻¹, with the minimum set to zero when H = {0}. A Kraus cancellation inequality gives the upper bound, and a weighted trace-zero operator basis gives matching faithful pairs with both marginals equal and reference dimension n². The complementary coefficient has a reciprocal radius formula, a finite semidefinite representation, and an exact product law. Strict contraction is equivalent to a nonzero completely positive replacer minorant. The formula also gives the exact coefficient of depolarizing channels throughout their physical parameter range. The general matching family has vanishing input divergence; its logarithmic rate is pointwise for a fixed faithful input density, with separate finite-accuracy bounds for optimization and conditioning. A different construction retains any prescribed positive limiting divergence when the coefficient is one. This cannot hold universally: a complete nonlinear bound for binary symmetric channels forces every maximizing sequence to lose input divergence, regardless of reference dimension. The positive construction still has vanishing trace distance and exponentially small regularization; no fixed-error or capacity conclusion follows. A visual guide records the mathematical routes, proof dependencies, and provenance of the result.