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arXiv:2511.02297

Two-Parameter Rényi Information Quantities with Applications to Privacy Amplification and Soft Covering

Li, Shi-Bing, Ke Li, Yu, Lei

Published Nov 4, 2025• 0 citations
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Abstract

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 $α\in (0, \infty)$.

DOI
10.48550/arxiv.2511.02297

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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,∞).
We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders α∈[12,∞)α\in[\frac{1}{2},\infty)α∈[21​,∞). For α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1), the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for α∈[1,∞)α\in[1,\infty)α∈[1,∞), it is characterized by the order-ααα sandwiched R{é}nyi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched R{é}nyi divergence for α∈(2,∞)α\in(2,\infty)α∈(2,∞), expressed in terms of the corresponding order-ααα sandwiched R{é}nyi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the KKK-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1).
In this paper, we introduce a novel mixed-order Rényi divergence and investigate its fundamental properties. Using this divergence, we define a family of mixed-order order-two Rényi mutual information and Rényi conditional entropy. We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched Rényi divergence with order α∈[2,∞)α\in[2,\infty)α∈[2,∞). The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order Rényi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.
Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→1α\to 1α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.
In this work, maximal ααα-leakage is introduced to quantify how much a quantum adversary can learn about any sensitive information of data upon observing its disturbed version via a quantum privacy mechanism. We first show that an adversary's maximal expected ααα-gain using optimal measurement is characterized by measured conditional Rényi entropy. This can be viewed as a parametric generalization of König et al.'s famous guessing probability formula [IEEE Trans. Inf. Theory, 55(9), 2009]. Then, we prove that the ααα-leakage and maximal ααα-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured Rényi capacity, respectively. Various properties of maximal ααα-leakage, such as data processing inequality and composition property are established as well. Moreover, we show that regularized ααα-leakage and regularized maximal ααα-leakage for identical and independent quantum privacy mechanisms coincide with ααα-tilted sandwiched Rényi information and sandwiched Rényi capacity, respectively.
Maximal α\alphaα - Leakage for Quantum Privacy Mechanisms and Operational Meaning of Measured Renyi Capacity
In this work, maximal a-leakage is introduced to quantify how much a quantum adversary can learn about any sensitive information of data upon observing its disturbed versionvia a quantum privacy mechanism. We first show that an adversary's maximal expected a-gain using optimal measurement is characterized by measured conditional Renyi entropy. This can be viewed as a parametric generalization of Konig et al.'s famous guessing probability formula [IEEE Trans. Inf. Theory, 55(9), 2009]. Then, we prove that the a-leakage and maximalα\alphaα-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured Renyi capacity, respectively. Various properties of maximal a-leakage, such as data processing inequality and composition property are established as well. Moreover, we show that regularized a-leakage and regularized maximal a-leakage for identical and independent quantum privacy mechanisms coincide with a-tilted sandwiched Renyi information and sandwiched Renyi capacity, respectively. A full version of this paper is accessible at: https://arxiv.org/pdf/2403.14450 [1].