Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Chain rules for a mutual information based on Rényi zero-relative entropy

Yuan Zhai, Bo Yang, Zhengjun Xi

Published Jul 10, 2023• 1 citation
Publisher

Abstract

Quantum R\'enyi relative entropies play a significant part in characterizing operational tasks in quantum information theory. In this paper, we first give some fundamental properties of the zero-relative entropy, and we find that the geometric R\'enyi relative entropy reduces to the zero-relative entropy in the limit case. Then, we define a new mutual information via the zero-relative entropy, and it is the so-called zero-mutual information. In particular, the zero-mutual information is a lower bound of the Petz-R\'enyi and geometric generalized mutual information. We establish the chain rules for the unsmoothed and smoothed versions of the zero-mutual information. As an application, we discuss generalized mutual information with the Petz-R\'enyi, sandwiched R\'enyi, and geometric R\'enyi types, our result gives a uniform chain rule inequality for quantum generalized mutual information.

DOI
10.1103/physreva.108.012413

Related papers

On quantum R\'enyi entropies: a new definition, some properties and several conjectures

Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr +2 more

The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation. We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.

•Jun 13, 2013•7 citations

Rényi divergence inequalities via interpolation, with applications to generalised entropic uncertainty relations

Alexander Clyde McKinlay

We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.

arXiv:2106.10415•Jun 19, 2021•0 citations

Log-Euclidean Rényi Conditional Mutual Information

Roberto Rubboli, Amir Arqand, Mark M. Wilde

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 $α\to 1$. 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.

quant-pharXiv:2609.34699•Sep 29, 2026•0 citations

On quantum Rényi entropies: A new generalization and some properties

Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr +2 more

The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.

arXiv:1306.3142•Journal of Mathematical Physics•Dec 1, 2013•724 citations

Decomposition rules for quantum Rényi mutual information with an application to information exclusion relations

Alexander McKinlay, Marco Tomamichel

We prove decomposition rules for quantum Rényi mutual information, generalizing the relation I(A : B) = H(A) − H(A|B) to inequalities between Rényi mutual information and Rényi entropy of different orders. The proof uses Beigi’s generalization of Reisz–Thorin interpolation to operator norms [J. Math. Phys. 54(12), 122202 (2013)] and a variation of the argument developed by Dupuis [J. Math. Phys. 56, 022203 (2015)], which was used to show chain rules for conditional Rényi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for Rényi mutual information, generalizing the original relation by Hall [Phys. Rev. Lett. 74, 3307 (1995)].

arXiv:1912.06277•Journal of Mathematical Physics•Jul 1, 2020•6 citations

Rényi–Holevo inequality from α-z-Rényi relative entropies

Diego G. Bussandri, Grzegorz Rajchel-Mieldzioć, Pedro W. Lamberti +1 more

We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.

arXiv:2309.04539•International Journal of Quantum Information•Apr 30, 2024•2 citations
Quantum R'enyi relative entropies play a significant part in characterizing operational tasks in quantum information theory. In this paper, we first give some fundamental properties of the zero-relative entropy, and we find that the geometric R'enyi relative entropy reduces to the zero-relative entropy in the limit case. Then, we define a new mutual information via the zero-relative entropy, and it is the so-called zero-mutual information. In particular, the zero-mutual information is a lower bound of the Petz-R'enyi and geometric generalized mutual information. We establish the chain rules for the unsmoothed and smoothed versions of the zero-mutual information. As an application, we discuss generalized mutual information with the Petz-R'enyi, sandwiched R'enyi, and geometric R'enyi types, our result gives a uniform chain rule inequality for quantum generalized mutual information.
The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation. We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.
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.
The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.
We prove decomposition rules for quantum Rényi mutual information, generalizing the relation I(A : B) = H(A) − H(A|B) to inequalities between Rényi mutual information and Rényi entropy of different orders. The proof uses Beigi’s generalization of Reisz–Thorin interpolation to operator norms [J. Math. Phys. 54(12), 122202 (2013)] and a variation of the argument developed by Dupuis [J. Math. Phys. 56, 022203 (2015)], which was used to show chain rules for conditional Rényi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for Rényi mutual information, generalizing the original relation by Hall [Phys. Rev. Lett. 74, 3307 (1995)].
We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.