Deterministic Minimum-Output-Entropy Nonadditivity via Haagerup's Inequality and Near-Free Permutation Representations

Abstract: We give a deterministic realization of the finite-dimensional quadratic certificate underlying Collins's mixed-unitary proof of minimum-output-entropy nonadditivity. For every fixed integer $K\ge 2$ and rational $η&gt;0$ satisfying $\log K&gt;2(3+η)<sup>2$, a deterministic polynomial-time algorithm, for every sufficiently large target size $N$, outputs $K$ permutations on $N&#39;=N+o_{K,η}(N)$ points. Restricting their permutation matrices to the nontrivial standard representation yields real orthogonal Stinespring blocks and a channel $Φ<em>{N&#39;}:M</em>{N&#39;-1}(\mathbb{C})\to M_K(\mathbb{C})$ such that [ 2H_{\min}(Φ{N'}) -H{\min}(Φ_{N'}^{\otimes 2}) \ge \frac{\log K}{K} -2\log\left(1+\frac{(3+η)^2}{K}\right) >0. ] The construction combines Haagerup's length-two inequality with the simultaneous deterministic spectral approximation of O'Donnell and Wu. We further show that the constant $3$ is asymptotically sharp on the relevant Hermitian zero-diagonal coefficient class and that the finite spectral transfer is nearly saturated, thereby isolating the finer geometry of the full output body as the natural next level of refinement beyond the scalar-radius method. Finally, a standard covariant extension converts the same deterministic entropy gap exactly into self-tensor superadditivity of the one-shot Holevo quantity.