AI 中文总结
本文针对CHLMW提出的显式量子信道对，填补了其正阶Rényi可加性违背严格认证的空白，给出0<p≤1/22的严格区间及可验证证书，是该对信道首个可计算机验证的正阶端点。
AI 中文摘要
Cubitt、Harrow、Leung、Montanaro和Winter（CHLMW）展示了一对显式量子信道，其最小输出Rényi熵在零阶时是非加性的，并报道了在接近零的正阶处的数值违背。他们的论文指出，半定规划论证为同一例子产生了一个严格的正阶区间，但未给出端点或可验证的证书；Leung、Lovitz和Wu的最新论文记录称，对于这对信道“未获得严格端点”。我们针对第2节中固定的已公布信道的保迹归一化填补了这一空白。小型有理见证矩阵证明，任一信道的每个输出的所有特征值都在301/100000和2/3之间；单个显式纠缠输入具有秩为8的精确有理联合输出谱；两个独立的初等区间论证将这三个事实转化为严格可加性违背的证明，即对于每个实阶0<p≤1/22，有S_p^min(N_R⊗N_\ar{S}) < S_p^min(N_R)+S_p^min(N_\ar{S})。验证的每一步都简化为整数比较，完整证书是几个小型有理矩阵，读者可通过简短程序或对于任意单个阶手动检查。据我们所知，与Leung、Lovitz和Wu的评估一致，这是这对显式信道的第一个已公布的、可计算机验证的正阶端点证书。我们不主张该现象或特征值下限机制的新颖性，二者均归功于CHLMW，也不主张该端点的最优性。
英文摘要
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.
Comments8 pages; v2: corrected proof of the envelope lemma, expanded literature discussion; verification artifact v1.2 archived at doi:10.5281/zenodo.22097554
