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

Quantum q-divergence

Sumiyoshi Abe

Physica A Statistical Mechanics and its ApplicationsJul 6, 200423 citations

Test-Measured Rényi Divergences

Milán Mosonyi, Fumio Hiai

One possibility of defining a quantum Rényi$\alpha $-divergence of two quantum states is to optimize the classical Rényi$ \alpha $-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi$ \alpha $-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi$ \alpha $-divergence coincides with the sandwiched Rényi$ \alpha $-divergence when$ \alpha >1 $. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi$ \alpha $-divergence for$ n $copies might require a number of measurement outcomes that diverges in$ n $, in general). In view of this, it seems natural to expect the same when$ \alpha < 1 $; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi$ \alpha $-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi$ \alpha $-divergence is not even a quantum extension of the classical Rényi divergence when$ \alpha < 1 $, in sharp contrast to the$ \alpha >1$case.One possibility of defining a quantum Rényi$\alpha $-divergence of two quantum states is to optimize the classical Rényi$\alpha $-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi$\alpha $-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi$\alpha $-divergence coincides with the sandwiched Rényi$\alpha $-divergence when$\alpha >1$. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi$\alpha $-divergence for$n$copies might require a number of measurement outcomes that diverges in$n$, in general). In view of this, it seems natural to expect the same when$\alpha < 1$; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi$\alpha $-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi$\alpha $-divergence is not even a quantum extension of the classical Rényi divergence when$\alpha < 1$, in sharp contrast to the$\alpha >1$case.

arXiv:2201.05477IEEE Transactions on Information TheorySep 26, 202214 citations

The

\alpha \to 1

Limit of the Sharp Quantum Rényi Divergence

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.

One possibility of defining a quantum Rényi$\alpha

-divergence of two quantum states is to optimize the classical Rényi

\alpha

-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi

\alpha

-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi

\alpha

-divergence coincides with the sandwiched Rényi

\alpha

-divergence when

\alpha >1

. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi

\alpha

-divergence for

copies might require a number of measurement outcomes that diverges in

, in general). In view of this, it seems natural to expect the same when

\alpha < 1

; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi

\alpha

-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi

\alpha

-divergence is not even a quantum extension of the classical Rényi divergence when

\alpha < 1

, in sharp contrast to the

\alpha >1$case.