Additivity of barycentric Rényi divergences holds only for minimal case
Regularized barycentric Rényi divergences
Information Theory
Summary
Quantum Rényi divergences help compare quantum states but can be tricky to define when combining multiple states. The paper studies a special way called barycentric Rényi divergences to combine these measures. The authors showed that only one particular version, called the minimal barycentric Rényi divergence, behaves nicely when combining quantum systems. This clarifies which divergence is unique in maintaining a key mathematical property called additivity.
What this means in practice
- •For quantum information developers: Design quantum communication protocols relying on additive quantum divergence measures ensured by the minimal barycentric Rényi divergence.
- •For quantum computing engineers: Improve quantum error correction and resource estimation by leveraging the unique additivity property of the minimal barycentric Rényi divergence in system modeling.
A theory result. No direct application yet.
Authors
Milán Mosonyi
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 $D^{q_x}$. 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) admits an explicit expression. Here we settle the problem of additivity by showing that for any choice of additive and monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on strictly positive inputs. This in turn implies that the only additive barycentric Rényi divergence is the minimal one.