Regularized barycentric Rényi divergences
Abstract
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 Dqx. 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 Dqx 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 Dqx are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.
Related
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BibTeX
@misc{mosonyi2026regularized,
title = {{Regularized barycentric Rényi divergences}},
author = {Milán Mosonyi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17517},
eprint = {2609.17517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17517},
}
RIS
TY - UNPB
TI - Regularized barycentric Rényi divergences
AU - Milán Mosonyi
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.17517
UR - https://arxiv.org/abs/2609.17517
AB - 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.
ER -