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Linear and Multilinear Algebra

Rescaled ♮ α -Rényi divergences of all real orders

Yūki Seo

Published Oct 7, 2025• 0 citations
Publisher

Abstract

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).

DOI
10.1080/03081087.2025.2564153

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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).
There are many proposals for defining a quantum version of the Rényi divergence on the differentiable manifold P of positive definite matrices. The geodesic connecting two points on P is a weighted matrix geometric mean, and its tangent vector is the relative operator entropy. It is a natural idea to use the relative operator entropy to describe the difference between two points on P. The quantum version of the classical results of Rényi divergence does not always hold due to its non-commutativity. Due to this difficulty, we use the upper bound and the lower one of the spectrum of positive definite matrices to estimate the order relation between quantum divergences, which is called the Mond Pečarić method in operator theory. In this paper, we show fundamental properties of the ♮α-Rényi divergence of real order α∈R, including monotonicity of order, the relation of max- and min-divergence, data processing inequality, bound estimates, and convexity. We also estimate the difference without the α–z-Rényi divergence and ♮α-Rényi divergence in terms of the generalized Kantorovich constant and the Specht ratio.
Abstract Given two density matrices ρ and σ , there are a number of different expressions that reduce to the α -Rényi relative entropy of ρ with respect to σ in the classical case; i.e. when ρ and σ commute. Only those expressions for which the data processing inequality (DPI) is valid are of potential interest as quantum divergences in quantum information theory. Audenaert and Datta have made a conjecture on the validity of the DPI for an interesting family of quantum generalizations of the α -Rényi relative entropies, the -Rényi relative entropies. They and others have contributed to the partial solution of this conjecture. We review the problem, its context, and the methods that have been used to obtain the results that are known at present, presenting a unified treatment of developments that have unfolded in a number of different papers.
Quantum generalizations of Rényi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Rényi divergence D̅αand the minimal quantum Rényi divergence D̅α. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely that αD̅α(ρ∥σ) ≤ D̅α(ρ∥σ) for α ϵ [0, 1] and where ρ and σ are density operators. This bound suggests defining a “pretty good fidelity”, whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction.
Abstract The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter α>1\alpha >1α>1 α > 1 . The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (α,z)(\alpha ,z)(α,z) ( α , z ) -divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.
Fawzi and Fawzi [Quantum 5, 387 (2021)] recently defined the sharp Rényi divergence, Dα#, for α ∈ (1, ∞), as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit α → 1 of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin–Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences for which we prove various properties, including the data-processing inequality.