Unbounded Holevo additivity gaps in finite dimensions

We establish unbounded two-use Holevo additivity gaps in finite dimensions. For each sufficiently large fixed integer $K$ and all sufficiently large $n$, we construct channels $T_n$ with output dimension $K^n$, input dimension $\exp ({\mathrm{\Theta }}_K(n^2))$, and $$\chi (T_n^{\otimes 2})-2\chi (T_n)\ge n[\frac{{\log }_{2}K}{K}-2{\log }_2(1+9\mathrm{/}K)]-O(1\mathrm{/}n)\mathrm{.}$$ The gap is linear in output qubits, with an explicit quadratic input-qubit cost and an explicit threshold on $n$. We also obtain channels whose single-use Holevo quantity tends to zero while their two-use Holevo information, and hence classical capacity, diverges. The channels arise from structured tensor products of the complementary mixed-unitary channels used in Collins’s free-probabilistic proof. We establish minimum-output-entropy gaps by combining Collins–Youn’s product-group Haagerup inequality with Bordenave–Collins’s quantitative strong-convergence estimates