发表机构
Budapest University of Technology and Economics（布达佩斯科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了对于任何可加且单调的量子相对熵，正则化重心Rényi散度在严格正输入上与最小重心Rényi散度一致，从而解决了可加性问题，并表明唯一可加的重心Rényi散度是最小散度。
AI 中文摘要
重心Rényi散度在[Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024]中被引入，作为标准Kubo-Ando构造的替代方案，用于定义多元量子Rényi散度。它们通过变分表达式定义，并依赖于有限集合的量子相对熵$D^{q_x}$。当所有相对熵在CPTP映射下是单调的时，相应的重心Rényi散度也是单调的；当所有相对熵是可加的时，相应的重心Rényi散度在张量积下是次可加的。可加性此前仅在所有$D^{q_x}$都选为Umegaki相对熵的情况下被建立，这也是重心Rényi散度（称为最小散度）具有显式表达式的唯一情况。在这里，我们通过证明对于任何可加且单调的量子相对熵的选择，正则化重心Rényi散度在严格正输入上与最小重心Rényi散度一致，从而解决了可加性问题。这进而意味着唯一可加的重心Rényi散度是最小散度。
英文摘要
Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies $D^{q_x}$. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all $D^{q_x}$ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the $D^{q_x}$ are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.
Comments21 pages. v4: Improved presentation, slightly stronger results
