Closed-Form Arbitrary-Gap Rényi Nonadditivity Above Any Positive Order via Direct Products of Free Groups and High-Girth Graphs
Abstract
We give a closed-form finite-dimensional construction of quantum channels with real algebraic entries exhibiting arbitrarily large violations of minimum-output Rényi entropy additivity uniformly above any prescribed positive Rényi order, using a different finite realization based on direct products of free groups and fixed-field high-girth graphs. More precisely, for every p0>0 and g>0, we construct one channel and one Bell input for which the additivity gap is greater than g simultaneously for all p∈[p0,∞]. With r=⌈2g⌉, the construction uses 15r output qubits and O(r230r+r2(1+p0−1)) input qubits. Compared with the stated parameters of Shou and Gorshkov \cite{ShouGorshkov2026}, our construction uses fewer output qubits for every target gap and has a smaller asymptotic input-qubit bound as the gap grows with the positive cutoff fixed. At the von Neumann order, a separate specialization gives an explicit channel with gap greater than 1/1024 using $37608898$ input qubits and $8$ output qubits.
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BibTeX
@misc{zhen2026closedform,
title = {{Closed-Form Arbitrary-Gap Rényi Nonadditivity Above Any Positive Order via Direct Products of Free Groups and High-Girth Graphs}},
author = {Guocheng Zhen and Ranyiliu Chen and Xin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06533},
eprint = {2610.06533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06533},
}
RIS
TY - UNPB
TI - Closed-Form Arbitrary-Gap Rényi Nonadditivity Above Any Positive Order via Direct Products of Free Groups and High-Girth Graphs
AU - Guocheng Zhen
AU - Ranyiliu Chen
AU - Xin Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06533
UR - https://arxiv.org/abs/2610.06533
AB - We give a closed-form finite-dimensional construction of quantum channels with real algebraic entries exhibiting arbitrarily large violations of minimum-output Rényi entropy additivity uniformly above any prescribed positive Rényi order, using a different finite realization based on direct products of free groups and fixed-field high-girth graphs. More precisely, for every p_0>0 and g>0, we construct one channel and one Bell input for which the additivity gap is greater than g simultaneously for all p∈[p_0,∞]. With r=⌈2g⌉, the construction uses 15r output qubits and O(r2^30r+r^2(1+p_0^−1)) input qubits. Compared with the stated parameters of Shou and Gorshkov \cite{ShouGorshkov2026}, our construction uses fewer output qubits for every target gap and has a smaller asymptotic input-qubit bound as the gap grows with the positive cutoff fixed. At the von Neumann order, a separate specialization gives an explicit channel with gap greater than 1/1024 using $37608898$ input qubits and $8$ output qubits.
ER -