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arXiv:2609.23946

A constructive violation of additivity of minimum output von Neumann entropy

Laura Shou, Alexey V. Gorshkov

quant-phmath-phSubmitted Sep 20, 2026• Announced Sep 22, 2026• 0 citations
v1 · Sep 20, 2026v2 · Sep 24, 2026
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Abstract

We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel also gives a violation of additivity for the minimum output Rényi-$p$ entropy for any $1\le p\le\infty$.

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13 pages

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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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Non-Additivity of Minimum Output p-Reˊnyi\mathbf{R\acute{e}nyi}Reˊnyi Entropy
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