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

WORLD SCIENTIFIC eBooks

Asymptotics of Quantum Relative Entropy from a Representation Theoretical Viewpoint

Masahito Hayashi

Published Feb 1, 2005• 0 citations
Publisher

Abstract

In this paper it was proved that the quantum relative entropy D(σ‖ρ) can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of POVMs. The sequence of POVMs depends on ρ, but is independent of the choice of σ. 1
DOI
10.1142/9789812563071_0006

Related papers

Asymptotics of quantum relative entropy from a representation theoretical viewpoint

Masahito Hayashi

In this paper it is proved that the quantum relative entropy D (ρ||σ) can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive-operator-valued measures (POVMs). The sequence of POVMs depends on σ, but is independent of the choice of ρ.

arXiv:quant-ph/9704040•Journal of Physics A Mathematical and General•Apr 10, 2001•47 citations

Relative entropy via distribution of observables

George Androulakis, Tiju Cherian John

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.

arXiv:2203.01964•Infinite Dimensional Analysis Quantum Probability and Related Topics•Aug 12, 2023•5 citations

Further results on the relative entropy

Matthew J. Donald

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.

Mathematical Proceedings of the Cambridge Philosophical Society•Mar 1, 1987•48 citations

Kullback-Leibler divergence between quantum distributions, and its upper-bound

Vincenzo Bonnici

This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution $U$ which maximizes the KL divergence from a given distribution $P$ under the assumption that $P$ and $U$ have been generated by distributing a given discrete quantity, a quantum. Quantum distributions naturally represent a wide range of probability distributions that are used in practical applications. Moreover, such a class of distributions can be obtained as an approximation of any probability distribution. The retrieving of an upper-bound for the entropic divergence is here shown to be possible under the condition that the compared distributions are quantum distributions over the same quantum value, thus they become comparable. Thus, entropic divergence acquires a more powerful meaning when it is applied to comparable distributions. This aspect should be taken into account in future developments of divergences. The theoretical findings are used for proposing a notion of normalized KL divergence that is empirically shown to behave differently from already known measures.

arXiv:2008.05932•Aug 13, 2020•3 citations

A Bayesian characterization of relative entropy

John C. Baez, Tobias Fritz

We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence.We use a number of interesting categories related to probability theory.In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function f : X → Y together with a stochastic right inverse s : Y → X.The function f can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement.Given this data we can define the entropy of the probability distribution on X relative to the 'prior' given by pushing the probability distribution on Y forwards along s.We say that s is 'optimal' if these distributions agree.We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid [0, ∞] which vanishes when s is optimal must be a scalar multiple of this relative entropy.Our proof is independent of all earlier characterizations, but inspired by the work of Petz.We thank Ryszard Kostecki and Rob Spekkens for discussions

arXiv:1402.3067•Theory and applications of categories•Jan 1, 2014•42 citations

A limit of the quantum Rényi divergence

Nilanjana Datta, Felix Leditzky

Recently, an interesting quantity called the quantum Rényi divergence (or 'sandwiched' Rényi relative entropy) was defined for pairs of positive semi-definite operators ρ and σ. It depends on a parameter α and acts as a parent quantity for other relative entropies which have important operational significance in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Rényi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Rényi entropy is obtainable from the quantum Rényi divergence only if ρ and σ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived—the quantum Rényi divergence with parameter α ⩾ 1/2, and the α-relative Rényi entropy with α ∈ [0, 1).

arXiv:1308.5961•Journal of Physics A Mathematical and Theoretical•Jan 8, 2014•47 citations
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.
Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.
This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution UUU which maximizes the KL divergence from a given distribution PPP under the assumption that PPP and UUU have been generated by distributing a given discrete quantity, a quantum. Quantum distributions naturally represent a wide range of probability distributions that are used in practical applications. Moreover, such a class of distributions can be obtained as an approximation of any probability distribution. The retrieving of an upper-bound for the entropic divergence is here shown to be possible under the condition that the compared distributions are quantum distributions over the same quantum value, thus they become comparable. Thus, entropic divergence acquires a more powerful meaning when it is applied to comparable distributions. This aspect should be taken into account in future developments of divergences. The theoretical findings are used for proposing a notion of normalized KL divergence that is empirically shown to behave differently from already known measures.
We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence.We use a number of interesting categories related to probability theory.In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function f : X → Y together with a stochastic right inverse s : Y → X.The function f can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement.Given this data we can define the entropy of the probability distribution on X relative to the 'prior' given by pushing the probability distribution on Y forwards along s.We say that s is 'optimal' if these distributions agree.We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid [0, ∞] which vanishes when s is optimal must be a scalar multiple of this relative entropy.Our proof is independent of all earlier characterizations, but inspired by the work of Petz.We thank Ryszard Kostecki and Rob Spekkens for discussions
Recently, an interesting quantity called the quantum Rényi divergence (or 'sandwiched' Rényi relative entropy) was defined for pairs of positive semi-definite operators ρ and σ. It depends on a parameter α and acts as a parent quantity for other relative entropies which have important operational significance in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Rényi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Rényi entropy is obtainable from the quantum Rényi divergence only if ρ and σ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived—the quantum Rényi divergence with parameter α ⩾ 1/2, and the α-relative Rényi entropy with α ∈ [0, 1).