We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz–Rényi [Formula: see text]-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.
We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz–Rényi [Formula: see text]-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.
Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.
Chapter 6 builds upon the foundation of divergences from Chapter 5, advancing into entropies and relative entropies with an axiomatic approach, and the inclusion of the additivity axiom. The chapter delves into the classical and quantum relative entropies, establishing their core properties and revealing the significance of the KL-divergence introduced in Chapter 5, notably characterized by asymptotic continuity. Quantum relative entropies are addressed as generalizations of classical ones, with a focus on the conditions necessary for these measures in the quantum framework. Several variants of relative entropies are discussed, including Renyi relative entropies and their extensions to quantum domain such as the Petz quantum Renyi divergence, minimal quantum Renyi divergence, and the maximal quantum Renyi divergence. This discourse underlines the relevance of continuity and its relation to faithfulness in relative entropies. The concept of entropy is portrayed as a measure with a broad spectrum of interpretations and applications across fields, from thermodynamics and information theory to cosmology and economics.
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
We revisit the monotonicity of relative entropy under the action of quantum channels, a foundational result in quantum information theory. Among the several available proofs, we focus on those by Petz and Uhlmann, which we reformulate within a unified, finite-dimensional operator-theoretic framework. In the first part, we examine Petz's strategy, identify a subtle flaw in his original use of Jensen's contractive operator inequality, and point out how it was corrected to restore the validity of his line of reasoning. In the second part, we develop Uhlmann's approach, which is based on interpolations of positive sesquilinear forms and applies automatically to non-invertible density operators. By comparing these two approaches, we highlight their complementary strengths: Petz's method is more direct and clear; Uhlmann's method is more abstract and general. Our treatment aims to clarify the mathematical structure underlying the monotonicity of relative entropy and to make these proofs more accessible to a broader audience interested in both the foundations and applications of quantum information theory.
Relative entropies play important roles in classical and quantum information theory. In this paper, we discuss the sandwiched Rényi relative entropy for [Formula: see text] on [Formula: see text] (the cone of positive trace-class operators acting on an infinite-dimensional complex Hilbert space [Formula: see text]) and characterize all surjective maps preserving the sandwiched Rényi relative entropy on [Formula: see text]. Such transformations have the form [Formula: see text] for each [Formula: see text], where [Formula: see text] and [Formula: see text] is either a unitary or an anti-unitary operator on [Formula: see text]. Particularly, all surjective maps preserving sandwiched Rényi relative entropy on [Formula: see text] (the set of all quantum states on [Formula: see text]) are necessarily implemented by either a unitary or an anti-unitary operator.