The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized Rényi divergences between such states, including $(α,z)$ , log-Euclidean, geometric, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $α$ , the regularized Rényi $(α,z)$ -divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized Rényi divergences between such states, including $(α,z)$, log-Euclidean, geometric, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $α$, the regularized Rényi $(α,z)$-divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.

quant-pharXiv:2605.31379Jun 1, 20260 citations

Approximate and Asymptotic Majorization and Divergence

Takahiro Sagawa

SpringerBriefs in mathematical physicsJan 1, 20220 citations

Regularized barycentric Rényi divergences

Milán Mosonyi

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

Rescaled ♮ α -Rényi divergences of all real orders

Yūki Seo

In this paper, inspired by the idea due to Takahashi and Fujiwara, for density matrices ρ,σ, we define the rescaled ♮α-Rényi divergence Dˆα(ρ||σ):=1α(α−1)log⁡Tr[σ♮αρ] of all real orders α∈R of ρ from σ, where σ♮αρ:=σ1/2(σ−1/2ρσ−1/2)ασ1/2 for α∈R, and study its properties such as the positivity, the data processing inequality, continuity on R and jointly convexity. Moreover, we determine the order relation among Dˆα(ρ||σ), the rescaled sandwiched α-Rényi divergence D~α(ρ||σ) and the rescaled α-Rényi divergence Dα(ρ||σ).In this paper, inspired by the idea due to Takahashi and Fujiwara, for density matrices ρ,σ, we define the rescaled ♮α-Rényi divergence Dˆα(ρ||σ):=1α(α−1)log⁡Tr[σ♮αρ] of all real orders α∈R of ρ from σ, where σ♮αρ:=σ1/2(σ−1/2ρσ−1/2)ασ1/2 for α∈R, and study its properties such as the positivity, the data processing inequality, continuity on R and jointly convexity. Moreover, we determine the order relation among Dˆα(ρ||σ), the rescaled sandwiched α-Rényi divergence D~α(ρ||σ) and the rescaled α-Rényi divergence Dα(ρ||σ).

Linear and Multilinear AlgebraOct 7, 20250 citations

Quantum Rényi Divergence

Marco Tomamichel

SpringerBriefs in mathematical physicsOct 8, 20151 citation

(α,z)

, log-Euclidean, geometric, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed

α

, the regularized Rényi

(α,z)

-divergences give different regularized values for different

z

parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.

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.