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

European Journal of Physics

Entropic uncertainty relations

Vladimír Majerník, Lukáš Richterek

Published Mar 1, 1997• 72 citations
Publisher

Abstract

The uncertainty principle in quantum physics for non-commuting observables can be quantitatively expressed in various mathematical forms. Well known are common Heisenberg-like uncertainty relations in which the product of dispersions of two non-commuting observables is used. An alternative form of uncertainty relations represents the so-called entropic uncertainty relations in which the sum of entropies of the considered non-commuting observables is employed. The aim of this paper is to show that the entropic uncertainty relations exhibit several interesting mathematical properties and can be taken as an adequate alternative formulation of the uncertainty principle in quantum mechanics. To compare the mathematical properties of the common and entropic uncertainty relations we present some simple quantum systems with two non-commuting observables. Dedicated to Dr B Mamojka on the occasion of his fiftieth birthday. Abstrakt. Princip neurcitosti pro nekomutující veliciny v kvantové fyzice mze být matematicky vyjádren r formou. Dobre známé jsou relace neurcitosti Heisenbergova typu, v nichz se vyuzí vá soucinu disperzí dvou nekomutujících fyzikálních velicin. Alternativou jsou tzv. entropické relace neurcitosti, v nichz vystupuje soucet entropií uvazovaných velicin. Cílem tohoto clánku je ukázat, ze tyto relace mají nekolik z matematického hlediska zajímavých vlastností a mohou být pouzity k alternativnímu vyjádrení principu neurcitosti v kvantové mechanice. Srovnání vlastností Heisengergových a entropických relací neurcitosti je demonstrováno na jednoduchém kvantovém systému s dvema nekomutujícími velicinami.

DOI
10.1088/0143-0807/18/2/005

Related papers

Entropic uncertainty relations for the infinite well

Vladimír Majerník, Lukáš Richterek

We calculate the entropic and standard uncertainty relations for the position and momentum of the infinite potential well as functions of its quantum states. We show that the entropic uncertainty relations express more adequately the uncertainty principle than the standard ones which use the dispersions as measures of the uncertainties for two non-commuting observables.

Journal of Physics A Mathematical and General•Feb 21, 1997•53 citations

Entropic uncertainty relations and the quantum-to-classical transition

Isadora Veeren, Fernando de Melo

Our knowledge of quantum mechanics can satisfactorily describe simple microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measurements must increase in order to keep quantum properties, a desirable feature for large quantum computers.

arXiv:2003.02086•Physical Review A•Aug 10, 2020•11 citations

Entropic uncertainty relations – A survey

Stephanie Wehner, Andreas Winter

Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, rather little is known about entropic uncertainty relations with more than two measurement settings. In this note we review known results and open questions. The uncertainty principle is one of the fundamental ideas of quantum mechanics. Since Heisenberg’s uncertainty relations for canonically conjugate variables, they have been one of the most prominent examples of how quantum mechanics differs from the classical world (Heisenberg, 1927). Uncertainty relations today are probably best known in the form given by (Robertson, 1929), who extended Heisenberg’s result to two arbitrary observables A and B. Robertson’s relation states that if we prepare many copies of the state |ψ〉, and measure each copy individually using either A or B, we have

•Jan 1, 2009•239 citations

A Framework for Uncertainty Relations

Yunlong Xiao

Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.

•Dec 9, 2016•1 citation

Entropic uncertainty relation for successive measurements

Kyunghyun Baek, Wonmin Son

In quantum mechanics, the uncertainty is fundamental concept, so its underlying meaning has been argued intensively due to its weird properties. On this account, there has been various fomulations expressing the uncertainty principle in dierent ways. In this paper, the new entropic uncertainty relations will be suggested showing dierent aspects of uncertainty as we consider successive measurements. Already, Deutsch and Unk suggested entropic uncertainty relations, so our work will be to modify it as focusing on another aspect (error-disurbance) of uncertainry. The uncertainty relations Heisenberg is the rst man who suggested the uncertainty principle. In 1927 [1], he proposed the uncertainty relation,

•Jan 1, 2012•0 citations

Entropic uncertainty relations and the stabilizer formalism

Sönke Niekamp, Matthias Kleinmann, Otfried Gühne

Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen–Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We compare the relative strengths of variance-based and various entropic uncertainty relations for dichotomic anticommuting observables.

arXiv:1103.2316•Journal of Mathematical Physics•Jan 1, 2012•12 citations
The uncertainty principle in quantum physics for non-commuting observables can be quantitatively expressed in various mathematical forms. Well known are common Heisenberg-like uncertainty relations in which the product of dispersions of two non-commuting observables is used. An alternative form of uncertainty relations represents the so-called entropic uncertainty relations in which the sum of entropies of the considered non-commuting observables is employed. The aim of this paper is to show that the entropic uncertainty relations exhibit several interesting mathematical properties and can be taken as an adequate alternative formulation of the uncertainty principle in quantum mechanics. To compare the mathematical properties of the common and entropic uncertainty relations we present some simple quantum systems with two non-commuting observables. Dedicated to Dr B Mamojka on the occasion of his fiftieth birthday. Abstrakt. Princip neurcitosti pro nekomutující veliciny v kvantové fyzice mze být matematicky vyjádren r formou. Dobre známé jsou relace neurcitosti Heisenbergova typu, v nichz se vyuzí vá soucinu disperzí dvou nekomutujících fyzikálních velicin. Alternativou jsou tzv. entropické relace neurcitosti, v nichz vystupuje soucet entropií uvazovaných velicin. Cílem tohoto clánku je ukázat, ze tyto relace mají nekolik z matematického hlediska zajímavých vlastností a mohou být pouzity k alternativnímu vyjádrení principu neurcitosti v kvantové mechanice. Srovnání vlastností Heisengergových a entropických relací neurcitosti je demonstrováno na jednoduchém kvantovém systému s dvema nekomutujícími velicinami.
Our knowledge of quantum mechanics can satisfactorily describe simple microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measurements must increase in order to keep quantum properties, a desirable feature for large quantum computers.
Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, rather little is known about entropic uncertainty relations with more than two measurement settings. In this note we review known results and open questions. The uncertainty principle is one of the fundamental ideas of quantum mechanics. Since Heisenberg’s uncertainty relations for canonically conjugate variables, they have been one of the most prominent examples of how quantum mechanics differs from the classical world (Heisenberg, 1927). Uncertainty relations today are probably best known in the form given by (Robertson, 1929), who extended Heisenberg’s result to two arbitrary observables A and B. Robertson’s relation states that if we prepare many copies of the state |ψ〉, and measure each copy individually using either A or B, we have
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
In quantum mechanics, the uncertainty is fundamental concept, so its underlying meaning has been argued intensively due to its weird properties. On this account, there has been various fomulations expressing the uncertainty principle in dierent ways. In this paper, the new entropic uncertainty relations will be suggested showing dierent aspects of uncertainty as we consider successive measurements. Already, Deutsch and Unk suggested entropic uncertainty relations, so our work will be to modify it as focusing on another aspect (error-disurbance) of uncertainry. The uncertainty relations Heisenberg is the rst man who suggested the uncertainty principle. In 1927 [1], he proposed the uncertainty relation,
Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen–Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We compare the relative strengths of variance-based and various entropic uncertainty relations for dichotomic anticommuting observables.