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Different quantum f -divergences and the reversibility of quantum operations

Hiai, F., Milán Mosonyi

Published Jan 1, 2017• 77 citations
OpenAlex

Abstract

The concept of classical $f$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard $f$-divergences), Matsumoto's maximal $f$-divergences, measured $f$-divergences, and sandwiched and $\alpha$-$z$-R\'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum $f$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum $f$-divergence by a quantum operation. In particular, we compare the standard and the maximal $f$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum $f$-divergences are strictly different for non-commuting operators unless $f$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard $f$-divergences. We also study the monotonicity of the $\alpha$-$z$-R\'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\'enyi divergence invariant, and determine domains of the parameters $\alpha,z$ where monotonicity holds, and where the preservation of the $\alpha$-$z$-R\'enyi divergence implies the reversibility of the quantum operation.

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Different quantum f-divergences and the reversibility of quantum operations

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The concept of classical [Formula: see text]-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz’ quasi-entropies (standard [Formula: see text]-divergences), Matsumoto’s maximal [Formula: see text]-divergences, measured [Formula: see text]-divergences, and sandwiched and [Formula: see text]-[Formula: see text]-Rényi divergences. In this paper, we give a systematic overview of the various concepts of quantum [Formula: see text]-divergences, with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum [Formula: see text]-divergence by a quantum operation. In particular, we compare the standard and the maximal [Formula: see text]-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum [Formula: see text]-divergences are strictly different for non-commuting operators unless [Formula: see text] is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard [Formula: see text]-divergences. We also study the monotonicity of the [Formula: see text]-[Formula: see text]-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters [Formula: see text] where monotonicity holds, and where the preservation of the [Formula: see text]-[Formula: see text]-Rényi divergence implies the reversibility of the quantum operation.

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QUANTUM f-DIVERGENCES AND ERROR CORRECTION

Fumio Hiai, Milán Mosonyi, Dėnes Petz +1 more

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

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Hockey stick fff-divergences

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In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum $f$-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in \cite{BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025}. In particular, we consider non-normalized states and hockey stick $f$-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from \cite{HircheTomamichel_integral,LiuHircheCheng2025} to this setting. Our main results here are the representation of the hockey stick $f$-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{Jencova2023} on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi $α$-divergences coincide with the Petz-type Rényi divergences for $α\in(0,1)$ and with the sandwiched Rényi divergences for $α>1$. Moreover, we give some partial results on the characterization of when different notions of quantum $f$-divergences give the same value on a pair of quantum states.

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Reversibility of stochastic maps via quantum divergences

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Relations between different quantum Rényi divergences

Raban Iten

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.

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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
The concept of classical fff-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard fff-divergences), Matsumoto's maximal fff-divergences, measured fff-divergences, and sandwiched and α\alphaα-zzz-R'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum fff-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum fff-divergence by a quantum operation. In particular, we compare the standard and the maximal fff-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum fff-divergences are strictly different for non-commuting operators unless fff is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard fff-divergences. We also study the monotonicity of the α\alphaα-zzz-R'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R'enyi divergence invariant, and determine domains of the parameters α,z\alpha,zα,z where monotonicity holds, and where the preservation of the α\alphaα-zzz-R'enyi divergence implies the reversibility of the quantum operation.
The concept of classical [Formula: see text]-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz’ quasi-entropies (standard [Formula: see text]-divergences), Matsumoto’s maximal [Formula: see text]-divergences, measured [Formula: see text]-divergences, and sandwiched and [Formula: see text]-[Formula: see text]-Rényi divergences. In this paper, we give a systematic overview of the various concepts of quantum [Formula: see text]-divergences, with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum [Formula: see text]-divergence by a quantum operation. In particular, we compare the standard and the maximal [Formula: see text]-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum [Formula: see text]-divergences are strictly different for non-commuting operators unless [Formula: see text] is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard [Formula: see text]-divergences. We also study the monotonicity of the [Formula: see text]-[Formula: see text]-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters [Formula: see text] where monotonicity holds, and where the preservation of the [Formula: see text]-[Formula: see text]-Rényi divergence implies the reversibility of the quantum operation.
Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.
In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum fff-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in \cite{BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025}. In particular, we consider non-normalized states and hockey stick fff-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from \cite{HircheTomamichel_integral,LiuHircheCheng2025} to this setting. Our main results here are the representation of the hockey stick fff-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{Jencova2023} on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi ααα-divergences coincide with the Petz-type Rényi divergences for α∈(0,1)α\in(0,1)α∈(0,1) and with the sandwiched Rényi divergences for α>1α>1α>1. Moreover, we give some partial results on the characterization of when different notions of quantum fff-divergences give the same value on a pair of quantum states.
We consider the reversibility problem for quantum operations (stochastic maps) in connection with the preservation of various quantum divergences, including two different versions of quantum fff-divergences, sandwiched R\'enyi divergences, and α\alphaα-zzz-R\'enyi relative entropies.
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.
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