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:1402.3067 · Theory and applications of categories

A Bayesian characterization of relative entropy

John C. Baez, Tobias Fritz

Published Jan 1, 2014• 42 citations
arXivHTMLPDF

Abstract

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

DOI
10.70930/tac/myyolhqw

Related papers

On a 2-Relative Entropy

James Fullwood

We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a ‘2-relative entropy’ satisfies natural 2-categorial analogues of convex linearity, vanishing under optimal hypotheses, and lower semicontinuity. While relative entropy is a relative measure of information between probability distributions, we view our construction as a relative measure of information between channels.

arXiv:2112.03582•Entropy•Dec 31, 2021•1 citation

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

A characterization of von Neumann entropy using functors

Kenji Nakahira

Baez, Fritz, and Leinster derived a method for characterizing Shannon entropy in classical systems. In this method, they considered a functor from a certain category to the monoid of non-negative real numbers with addition as a map from measure-preserving functions to non-negative real numbers, and derived Shannon entropy by imposing several simple conditions. We propose a method for characterizing von Neumann entropy by extending their results to quantum systems.

arXiv:2309.10353•Sep 19, 2023•0 citations

On maximization of measured $f$-divergence between a given pair of quantum states

Keiji Matsumoto

This paper deals with maximization of classical $f$-divergence between the distributions of a measurement outputs of a given pair of quantum states. $f$-divergence $D_{f}$ between the probability density functions $p_{1}$ and $p_{2}$ over a discrete set is defined as $D_{f}( p_{1}||p_{2}) :=\sum_{x}p_{2}(x) f\left(p_{1}(x)/p_{2}( x) \right) $. For example, Kullback-Leibler divergence and Renyi type relative entropy are well-known examples with good operational meanings. Thus, finding the maximal value $D_{f}^{\min}$ of measured measured $f$-divergence is also an interesting question. But so far the question is solved only for very restricted example of $f$. \ The purposes of the present paper is to advance the study further, by investigating its properties, rewriting the maximization problem to more tractable form, and giving closed formulas of the quantity in some special cases.

arXiv:1412.3676•Dec 11, 2014•2 citations

The Homological Nature of Entropy

Pierre Baudot, Daniel Bennequin

We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.

Entropy•May 13, 2015•70 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
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
We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a ‘2-relative entropy’ satisfies natural 2-categorial analogues of convex linearity, vanishing under optimal hypotheses, and lower semicontinuity. While relative entropy is a relative measure of information between probability distributions, we view our construction as a relative measure of information between channels.
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.
On maximization of measured fff-divergence between a given pair of quantum states
This paper deals with maximization of classical fff-divergence between the distributions of a measurement outputs of a given pair of quantum states. fff-divergence DfD_{f}Df​ between the probability density functions p1p_{1}p1​ and p2p_{2}p2​ over a discrete set is defined as $D_{f}( p_{1}||p_{2}) :=\sum_{x}p_{2}(x) f\left(p_{1}(x)/p_{2}( x) \right) $. For example, Kullback-Leibler divergence and Renyi type relative entropy are well-known examples with good operational meanings. Thus, finding the maximal value Dfmin⁡D_{f}^{\min}Dfmin​ of measured measured fff-divergence is also an interesting question. But so far the question is solved only for very restricted example of fff. \ The purposes of the present paper is to advance the study further, by investigating its properties, rewriting the maximization problem to more tractable form, and giving closed formulas of the quantity in some special cases.
We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.
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.