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

arXiv:2012.08327

Relations between different quantum Rényi divergences

Raban Iten

Published Dec 12, 2020• 1 citation
arXivHTMLPDF

Abstract

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 $\bar{D}_α$ and the minimal quantum Rényi divergence $\widetilde{D}_α$. Moreover, the maximum quantum Rényi divergence $\widehat{D}_α$ is of particular mathematical interest. In this Master thesis, we investigate relations between these divergences and their applications in quantum information theory. Our main result is a reverse Araki-Lieb-Thirring inequality that implies a new relation between the minimal and the Petz divergence, namely that $α\bar{D}_α(ρ\| σ) \leqslant \widetilde{D}_α(ρ\| σ)$ for $α\in [0,1]$ 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. In addition, we provide a new proof of the inequality $\widetilde{D}_{1}(ρ\| σ) \leqslant \widehat{D}_{1}(ρ\| σ)\, ,$ based on the Araki-Lieb-Thirring inequality. This leads to an elegant proof of the logarithmic form of the reverse Golden-Thompson inequality.

DOI
10.48550/arxiv.2012.08327

Related papers

Pretty good measures in quantum information theory

Raban Iten, Joseph M. Renes, David Sutter

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.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2017•1 citation

Pretty Good Measures in Quantum Information Theory

Raban Iten, Joseph M. Renes, David Sutter

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 D̃α. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely, αD̅α(Q∥σ) ≤ D̃α(Q∥σ) for α ∈[0,1] and where Q 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. We also find a new necessary and sufficient condition for optimality of the pretty good measurement and singlet fraction.

arXiv:1608.08229•IEEE Transactions on Information Theory•Dec 14, 2016•25 citations

Optimized Quantum F-Divergences

Mark M. Wilde

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-Renyi relative entropies. In this contribution, 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 Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252

arXiv:2103.16797•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2018•1 citation

Optimized Quantum F-Divergences

Mark M. Wilde

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-Renyi relative entropies. In this contribution, 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 Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252

Civil War Book Review•Aug 15, 2018•1 citation

Relative entropies and their use in quantum information theory

Felix Leditzky

This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the $α$-sandwiched Rényi divergence ($α$-SRD). We first investigate the limit $α\to 0$ to determine whether this recovers the well-known $0$-Rényi relative divergence. We then prove various new results for entropic quantities derived from the $α$-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the $α$-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the $α$-SRD.

arXiv:1611.08802•Nov 27, 2016•18 citations

Optimized quantum f -divergences and data processing

Mark M. Wilde

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.

arXiv:1710.10252•Journal of Physics A Mathematical and Theoretical•Jul 25, 2018•47 citations
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 Dˉα\bar{D}_αDˉα​ and the minimal quantum Rényi divergence D~α\widetilde{D}_αDα​. Moreover, the maximum quantum Rényi divergence D^α\widehat{D}_αDα​ is of particular mathematical interest. In this Master thesis, we investigate relations between these divergences and their applications in quantum information theory. Our main result is a reverse Araki-Lieb-Thirring inequality that implies a new relation between the minimal and the Petz divergence, namely that αDˉα(ρ∥σ)⩽D~α(ρ∥σ)α\bar{D}_α(ρ\| σ) \leqslant \widetilde{D}_α(ρ\| σ)αDˉα​(ρ∥σ)⩽Dα​(ρ∥σ) for α∈[0,1]α\in [0,1]α∈[0,1] and where ρρρ and σσσ 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. In addition, we provide a new proof of the inequality D~1(ρ∥σ)⩽D^1(ρ∥σ) ,\widetilde{D}_{1}(ρ\| σ) \leqslant \widehat{D}_{1}(ρ\| σ)\, ,D1​(ρ∥σ)⩽D1​(ρ∥σ), based on the Araki-Lieb-Thirring inequality. This leads to an elegant proof of the logarithmic form of the reverse Golden-Thompson inequality.
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.
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 D̃α. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely, αD̅α(Q∥σ) ≤ D̃α(Q∥σ) for α ∈[0,1] and where Q 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. We also find a new necessary and sufficient condition for optimality of the pretty good measurement and singlet fraction.
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-Renyi relative entropies. In this contribution, 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 Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252
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-Renyi relative entropies. In this contribution, 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 Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252
This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the ααα-sandwiched Rényi divergence (ααα-SRD). We first investigate the limit α→0α\to 0α→0 to determine whether this recovers the well-known 000-Rényi relative divergence. We then prove various new results for entropic quantities derived from the ααα-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the ααα-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the ααα-SRD.
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.