arXiv:2609.17517v5 Announce Type: replace-cross Abstract: Barycentric R\'enyi 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 R\'enyi 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 R\'enyi divergences, and when all the relative entropies are additive then the corresponding barycentric R\'enyi 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 R\'enyi 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 R\'enyi divergence coincides with the minimal barycentric R\'enyi 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 R\'enyi divergence is not even weakly additive.