Exact certification of a positive-order R\'enyi additivity violation for an explicit channel pair

Abstract:Cubitt, Harrow, Leung, Montanaro, and Winter (CHLMW) exhibited an explicit pair of quantum channels whose minimum output Rényi entropy is nonadditive at order zero, and reported numerical violations at positive orders close to zero. Their paper states that a semidefinite-programming argument yields a rigorous positive-order interval for the same example, without printing the endpoint or a verifiable certificate; a recent paper by Leung, Lovitz, and Wu records that for this pair "no rigorous endpoint was obtained". We close that gap for the trace-preserving normalization of the printed pair fixed in Section 2. Small rational witness matrices prove that every output of either channel has all eigenvalues between $301/100000$ and $2/3$; a single explicit entangled input has an exact rational joint output spectrum of rank eight; and two independent elementary interval arguments turn these three facts into a proof of strict additivity violation, $S_p^{\min}(\mathrm{N}_R\otimes\mathrm{N}_{\bar S}) < S_p^{\min}(\mathrm{N}_R)+S_p^{\min}(\mathrm{N}_{\bar S}),$ for every real order $0<p\le 1/22$. Every step of the verification reduces to comparisons of integers, and the complete certificate is a few small rational matrices that a reader can check with a short program -- or, for any single order, by hand. To our knowledge, consistent with the assessment of Leung, Lovitz, and Wu, this is the first printed, computer-verifiable certified positive-order endpoint for this explicit pair. We claim no novelty for the phenomenon or for the eigenvalue-floor mechanism, both due to CHLMW, and no optimality of the endpoint.

6 pages; verification artifact archived at doi:https://doi.org/10.5281/zenodo.21968558

arXiv:2608.17376 [quant-ph]

(or arXiv:2608.17376v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.17376 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.