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arXiv:2609.31350

Sharp Data-Processing Region for Quantum Conditional Rényi Entropies

Milad M. Goodarzi, Yucheng Bao, Marco Tomamichel

quant-phSubmitted Sep 25, 2026• Announced Sep 28, 2026• 0 citations
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Abstract

We study the three-parameter family of quantum conditional Rényi entropies $H_{α,z}^λ$ introduced in \textit{Commun. Math. Phys.} 407, 180 (2026). We prove that the previously established data-processing region is sharp. We then characterize the club-sandwiched family through a classical coarse-graining principle: within the full data-processing region, universal monotonicity under deterministic coarse-graining of the classical conditioned register holds if and only if $z = α$. Finally, we establish rigidity of entropic duality by showing that any universal duality relation for finite-dimensional tripartite pure states must satisfy the known parameter relations. Thus, the data-processing region, the club-sandwiched subfamily, and the duality transformation are each uniquely characterized by their respective universal properties.

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We study the three-parameter family of quantum conditional Rényi entropies Hα,zλH_{α,z}^λHα,zλ​ introduced in \textit{Commun. Math. Phys.} 407, 180 (2026). We prove that the previously established data-processing region is sharp. We then characterize the club-sandwiched family through a classical coarse-graining principle: within the full data-processing region, universal monotonicity under deterministic coarse-graining of the classical conditioned register holds if and only if z=αz = αz=α. Finally, we establish rigidity of entropic duality by showing that any universal duality relation for finite-dimensional tripartite pure states must satisfy the known parameter relations. Thus, the data-processing region, the club-sandwiched subfamily, and the duality transformation are each uniquely characterized by their respective universal properties.
The α\alpha α -sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for α≥1/2\alpha \ge 1/2α≥1/2 . In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the α\alpha α -sandwiched Rényi divergence for all α≥1/2\alpha \ge 1/2α≥1/2 . For the range α∈[1/2,1)\alpha \in [1/2,1)α∈[1/2,1) , our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.
The α–z Rényi relative entropies are a two-parameter family of Rényi relative entropies that are quantum generalizations of the classical α-Rényi relative entropies. In the work [Adv. Math. 365, 107053 (2020)], we decided the full range of (α, z) for which the data processing inequality (DPI) is valid. In this paper, we give algebraic conditions for the equality in DPI. For the full range of parameters (α, z), we give necessary conditions and sufficient conditions. For most parameters, we give equivalent conditions. This generalizes and strengthens the results of Leditzky et al. [Lett. Math. Phys. 107, 61–80 (2017)].
There are no universally accepted definitions of Rényi conditional entropy and Rényi mutual information, although motivated by different applications, several definitions have been proposed in the literature. In this paper, we consider a family of two-parameter Rényi conditional entropy and a family of two-parameter Rényi mutual information. By performing a change of variables for the parameters, the two-parameter Rényi conditional entropy we study coincides precisely with the definition introduced by Hayashi and Tan [IEEE Trans. Inf. Theory, 2016], and it also emerges naturally as the classical specialization of the three-parameter quantum Rényi conditional entropy recently put forward by Rubboli, Goodarzi, and Tomamichel [arXiv:2410.21976 (2024)]. We establish several fundamental properties of the two-parameter Rényi conditional entropy, including monotonicity with respect to the parameters and variational expression. The associated two-parameter Rényi mutual information considered in this paper is new and it unifies three commonly used variants of Rényi mutual information. For this quantity, we prove several important properties, including the non-negativity, additivity, data processing inequality, monotonicity with respect to the parameters, variational expression, as well as convexity and concavity. Finally, we demonstrate that these two-parameter Rényi information quantities can be used to characterize the strong converse exponents in privacy amplification and soft covering problems under Rényi divergence of order α∈(0,∞)α\in (0, \infty)α∈(0,∞).
Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here, we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Rényi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Rényi entropies and derive a new entropic uncertainty relation.
Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. This generalizes the well-known duality relation H(A|B)+H(A|C) = 0 for tripartite pure states to Rényi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Rényi entropies.
Abstract The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f -divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz–Rényi relative entropies. In this paper, I introduce the optimized quantum f -divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz’s original approach. Interestingly, the sandwiched Rényi relative entropies are particular examples of the optimized f -divergence. Thus, one benefit of this paper is that there is now a single, unified approach for establishing the data processing inequality for both the Petz–Rényi and sandwiched Rényi relative entropies, for the full range of parameters for which it is known to hold. This paper discusses other aspects of the optimized f -divergence, such as the classical case, the classical-quantum case, and how to construct optimized f -information measures.