Exact Information Contraction and Nonvanishing Divergence for Finite Quantum Channels

We compare three quantum information contraction coefficients and determine when contraction close to its optimum can retain nonvanishing input divergence. Using the exact complete coefficient and complex inverse-pencil inequality of SR-012, we prove that conditional relative-entropy contraction, conditional mutual-information contraction, and complete equal-reference contraction have the same coefficient for every finite quantum channel. For a depolarizing qubit with parameter −1/3 ≤ λ ≤ 1, the ordinary mutual-information coefficient is 2λ²/(1+λ), whereas the complete and conditional coefficients are 3λ²/(1+2λ). These differ strictly except at λ = 0, 1; at half-depolarization the values are 1/3 and 3/8. For a general channel with input dimension at least two, let P project onto the largest output subspace whose tensor product with the input space lies in the Choi support, and put b = ‖N*(I−P)‖op. A maximizing sequence with input divergence bounded away from zero exists exactly when b = ηc. In this case every positive limiting divergence is possible with reference dimension at most the square of the input dimension. Otherwise an explicit nonlinear deficit is uniform over all finite references, including growing ones. The proof gives an eighteenth-power trace-distance deficit in general and a fourth-power bound for faithful Choi matrices. Product criteria and logarithmic copy bounds follow. The mutual-information lower family converges pointwise with vanishing input information; the positive-divergence construction has vanishing trace distance. Neither construction establishes a fixed-error or capacity statement, and the resource constants are not claimed optimal. A visual guide records the mathematical routes, joint proof dependencies, and provenance of the results.