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arXiv:2608.31081

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

Guocheng Zhen, Chengkai Zhu, Ranyiliu Chen, Xin Wang

quant-phSubmitted Aug 31, 2026• Announced Sep 1, 2026• 0 citations
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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 $η>0$ satisfying $\log K>2(3+η)^2$, a deterministic polynomial-time algorithm, for every sufficiently large target size $N$, outputs $K$ permutations on $N'=N+o_{K,η}(N)$ points. Restricting their permutation matrices to the nontrivial standard representation yields real orthogonal Stinespring blocks and a channel $Φ_{N'}:M_{N'-1}(\mathbb{C})\to M_K(\mathbb{C})$ 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.

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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≥2K\ge 2K≥2 and rational η>0η>0η>0 satisfying log⁡K>2(3+η)2\log K>2(3+η)^2logK>2(3+η)2, a deterministic polynomial-time algorithm, for every sufficiently large target size NNN, outputs KKK permutations on N′=N+oK,η(N)N'=N+o_{K,η}(N)N′=N+oK,η​(N) points. Restricting their permutation matrices to the nontrivial standard representation yields real orthogonal Stinespring blocks and a channel ΦN′:MN′−1(C)→MK(C)Φ_{N'}:M_{N'-1}(\mathbb{C})\to M_K(\mathbb{C})ΦN′​:MN′−1​(C)→MK​(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 333 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.
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We prove that minimum-output Rényi-entropy additivity can fail by almost one bit simultaneously at every nonnegative order. For every ε∈(0,log⁡2)\varepsilon\in(0,\log2)ε∈(0,log2), there exists a finite-dimensional quantum channel with a real Stinespring isometry such that the same maximally entangled input witnesses a tensor-square entropy gap of at least log⁡2−ε\log2-\varepsilonlog2−ε for all p∈[0,∞]p\in[0,\infty]p∈[0,∞]. The output dimension can be chosen to be O(ε−3)O(\varepsilon^{-3})O(ε−3) as ε↓0\varepsilon\downarrow0ε↓0. The construction uses direct products of free groups: tensorized Haagerup estimates control the one-copy outputs, while commutation between distinct factors forces exact Bell-branch collisions at two copies. Strong convergence gives both an existential realization through finite-dimensional representations of right-angled Artin groups followed by realification, and a Haar-orthogonal model whose success probability tends to one as the matrix dimension grows. We also determine the exact Bell quotient, prove asymptotically sharp regular-radius bounds, and show that the cubic output-dimension scale is optimal within the present purity--rank certificate.
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Counterexamples to additivity of minimum output ppp-Rényi entropy of quantum channels for all p≥0p\ge 0p≥0
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