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

Asymptotic Attainment for Quantum Relative Entropy

Masahito Hayashi

Published Apr 24, 1997• 2 citations
OpenAlex

Abstract

In this paper it was proved that the quantum relative entropy D(k) can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of measurements. The sequence of measurements depends on , but is independent of the choice of . In classical statistical theory the relative entropy D(pkq) is an information quantity which means the statistical eÆciency in order to distinguish a probability measure p of a measurable space from another probability measure q of the same measurable space. The states correspond to measures on measurable space. When p; q are discrete probabilities, the relative entropy (called also information divergence) introduced by Kullback and Leibler is dened by [1]: D(pkq) := X i p i log p i q i : In general, when p; q are measures on measurable space the relative entropy is dened by: D(pkq) := Z log dq dp (!)p( d!); where dq dp (!) is Radon-Nikodym derivative of q with respect to p. In this paper, we co...

Related papers

Asymptotics of Quantum Relative Entropy from a Representation Theoretical Viewpoint

Masahito Hayashi

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

WORLD SCIENTIFIC eBooks•Feb 1, 2005•0 citations

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

Canonical divergence for measuring classical and quantum complexity

Domenico Felice, Stefano Mancini, Nihat Ay

A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.

•Apr 24, 2019•7 citations

Canonical Divergence for Measuring Classical and Quantum Complexity

Domenico Felice, Stefano Mancini, Nihat Ay

A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both classical and quantum systems. On the simplex of probability measures, it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.

arXiv:1903.09797•Entropy•Apr 24, 2019•9 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

Classical and quantum relative entropy

Patrícia Szokol

The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.

•May 6, 2010•0 citations
In this paper it was proved that the quantum relative entropy D(k) can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of measurements. The sequence of measurements depends on , but is independent of the choice of . In classical statistical theory the relative entropy D(pkq) is an information quantity which means the statistical eÆciency in order to distinguish a probability measure p of a measurable space from another probability measure q of the same measurable space. The states correspond to measures on measurable space. When p; q are discrete probabilities, the relative entropy (called also information divergence) introduced by Kullback and Leibler is dened by [1]: D(pkq) := X i p i log p i q i : In general, when p; q are measures on measurable space the relative entropy is dened by: D(pkq) := Z log dq dp (!)p( d!); where dq dp (!) is Radon-Nikodym derivative of q with respect to p. In this paper, we co...
A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.
A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both classical and quantum systems. On the simplex of probability measures, it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.
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.
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.