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.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.

quant-pharXiv:2609.17517Sep 16, 20260 citations

Maps preserving entropies of the arithmetic mean of density operators

Junliang Ge, Wenhua Qian, Wenming Wu +1 more

Physics Letters AJul 24, 20260 citations

The Probability Law for Generic Density Operators

Matteo Campanella, David Jou, Maria Stella Mongiovı̀

UNIPA Springer seriesJan 1, 20200 citations

The Density Operator and the Boltzmann Distribution

Optical science and engineeringMay 23, 20050 citations

Thermodynamic extension of density-functional theory. III. Zero-temperature limit of the ensemble spin-density functional theory

Robert Balawender

In this work, the zero-temperature limit of the thermodynamic spin-density functional theory is investigated. The coarse-grained approach to the equilibrium density operator is used to describe the equilibrium state. The characteristic functions of a macrostate are introduced and their zero-temperature limits are investigated. A detailed discussion of the spin-grand-canonical ensemble in the entropy and energy representations is performed. The maps between the state function variables at 0K limit are rigorously studied for both representations. In the spin-canonical ensemble, the energy surface and the discontinuity pattern are investigated. Finally, based on the maps between the state function variables at 0K limit, the Hohenberg-Kohn theorem for the systems with non-integer electron and spin numbers at zero-temperature limit is formulated.In this work, the zero-temperature limit of the thermodynamic spin-density functional theory is investigated. The coarse-grained approach to the equilibrium density operator is used to describe the equilibrium state. The characteristic functions of a macrostate are introduced and their zero-temperature limits are investigated. A detailed discussion of the spin-grand-canonical ensemble in the entropy and energy representations is performed. The maps between the state function variables at 0K limit are rigorously studied for both representations. In the spin-canonical ensemble, the energy surface and the discontinuity pattern are investigated. Finally, based on the maps between the state function variables at 0K limit, the Hohenberg-Kohn theorem for the systems with non-integer electron and spin numbers at zero-temperature limit is formulated.

arXiv:1212.1367Dec 6, 20123 citations

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.