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:1205.4595

Classical-Information Theoretical View of Physical Measurements and Generalized Uncertainty Relations

Y. Kurihara

Published May 21, 2012• 0 citations
arXivHTMLPDF

Abstract

General characterization of physical measurements is discussed within the framework of a classical information theory. Uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems controlled under general kinds of equations of motion. We have treated only a mean-square error of the Gauss(-like) distributions in this report. Lessening of the Kennard-Robertson type uncertainties due to boundary conditions are quantitatively discussed using the information entropy.

DOI
10.48550/arxiv.1205.4595

Related papers

Classical information theoretic view of physical measurements and generalized uncertainty relations

Yoshimasa Kurihara

General characterizations of physical measurements are discussed within the framework of the classical information theory. The uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems governed by a Sturm--Liouville type of equation of motion. In the first step, the reduction of Kennard--Robertson type uncertainties due to boundary conditions with a mean-square error is discussed quantitatively with reference to the information entropy. Several concrete examples of generalized uncertainty relations are given. Then, by considering disturbance effects, a universally valid uncertainty relation is investigated for the generalized equation of motion with a certain boundary condition. Necessary conditions for violation (reduction) of the Heisenberg-type uncertainty relation are discussed in detail. The reduction of the generalized uncertainty relation due to the boundary condition is then tested experimentally by re-analyzing data for measured electron densities in a hydrogen molecule encapsulated in a fullerene C60 cage.

arXiv:1212.3880•Journal of theoretical and applied physics•Jan 1, 2013•5 citations

Irreversibility and Information in Mechanical Systems

Arthur Hobson

Concepts from information theory are used to investigate the relation between measurement and irreversibility in finite, closed classical systems having known Hamiltonians. The effects of measurement upon the predictions of the observer are found for the cases of a single measurement, two simultaneous measurements, and two measurements at different times. The physical basis of coarse graining and the significance of the increase in the coarse-grained entropy are discussed. An evolution equation is obtained describing the probability distribution of the measured quantities, and it is shown that this equation is non-Markoffian. Despite this non-Markoffian behavior, the distribution may behave irreversibly and may approach equilibrium. Thus, irreversible behavior is obtained without continual coarse graining. The condition for irreversible behavior is given for the case of a single measurement. The thermodynamic entropy is defined for general nonequilibrium situations.

The Journal of Chemical Physics•Aug 15, 1966•47 citations

Quantitative description of uncertainty andentropic uncertainty relation

Shiqi Zhang, Hua-Tong Yang

Uncertainty is a fundamental characteristic of quantum system. The degree of uncertainty of an observable has long been investigated by the standard deviation of the observable. In recent years, however, by analyzing some special examples, researchers have found that the Shannon entropy of the measurement outcomes of an observable is more suitable to quantify its uncertainty. Formally, Shannon entropy is a special limit of a more general Rényi entropy. In this paper, we discuss the problem of how to predict the measurement outcome of an observable by the existing measurement results of the observable, and how to quantitatively describe the uncertainty of the observable from the perspective of the repeatable probability of the measurement results of this observable in an unknown state. We will argue that if the same observable of different systems in the same state is repeatedly and independently measured many times, then the probability of obtaining an identical measurement result is a decaying function of the number of measurements of obtaining the same result, and the decay rate of the repeatable probability for obtaining the same measurement results and the repeatable number of measurements can represent the degree of uncertainty of the observable in this state. It means that the greater the uncertainty of an observable, the faster the repeatable probability decays with the number of repeatable measurements; conversely, the smaller the uncertainty, the slower the repeatable probability decays with the number of repeatable measurements. This observation enables us to give the Shannon entropy and the Rényi entropy of an observable uniformly by the functional relation between the repeatable probability and the number of repeatable measurements. We show that the Shannon entropy and the Rényi entropy can be formally regarded as the “decay index” of the repeatable probability with the number of repeatable measurements. In this way we also define a generalized Rényi entropy by the repeatable probability for consecutively observing identical results of an observable, and therefore we give a proof of the Maassen-Uffink type entropic uncertainty relation by using this generalized Rényi entropy. This method of defining entropy shows that entropic uncertainty relation is a quantitative limitation for the decay rate of the total probability for obtaining identical measurement results when we simultaneously measure two observables many times.

Acta Physica Sinica•Jan 1, 2023•2 citations

A Universal Formulation of Uncertainty Relation for Errors under Local Representability

Jaeha Lee

A universal formulation of uncertainty relations for quantum measurements is presented with additional focus on the representability of quantum observables by classical observables over a given state. Owing to the simplicity and operational tangibility of the framework, the resultant general relations admit natural operational interpretations and characterisations, and are thus also experimentally verifiable. In view of the universal formulation, Heisenberg's philosophy of the uncertainty principle is also revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. In fact, the relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the Arthurs-Kelly-Goodman, Ozawa, and Watanabe-Sagawa-Ueda relations for quantum measurements. The Schr{ö}dinger relation (hence the standard Kennard-Robertson relation as its trivial corollary as well) is also shown to be a special case when the measurement is non-informative.

arXiv:2203.08197•Mar 15, 2022•0 citations

Renyi and Tsallis formulations of noise-disturbance trade-off relations

Alexey E. Rastegin

We address an information-theoretic approach to noise and disturbance in quantum measurements. Properties of corresponding probability distributions are characterized by means of both the R´enyi and Tsallis entropies. Related information-theoretic measures of noise and disturbance are introduced. These definitions are based on the concept of conditional entropy. To motivate introduced measures, some important properties of the conditional R´enyi and Tsallis entropies are discussed. There exist several formulations of entropic uncertainty relations for a pair of observables. Trade-off relations for noise and disturbance are derived on the base of known formulations of such a kind.

arXiv:1406.0054•Quantum Information and Computation•Mar 1, 2016•3 citations

A universal formulation of uncertainty relation for errors

Jaeha Lee, Izumi Tsutsui

We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.

Physics Letters A•Oct 5, 2024•3 citations
General characterization of physical measurements is discussed within the framework of a classical information theory. Uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems controlled under general kinds of equations of motion. We have treated only a mean-square error of the Gauss(-like) distributions in this report. Lessening of the Kennard-Robertson type uncertainties due to boundary conditions are quantitatively discussed using the information entropy.
General characterizations of physical measurements are discussed within the framework of the classical information theory. The uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems governed by a Sturm--Liouville type of equation of motion. In the first step, the reduction of Kennard--Robertson type uncertainties due to boundary conditions with a mean-square error is discussed quantitatively with reference to the information entropy. Several concrete examples of generalized uncertainty relations are given. Then, by considering disturbance effects, a universally valid uncertainty relation is investigated for the generalized equation of motion with a certain boundary condition. Necessary conditions for violation (reduction) of the Heisenberg-type uncertainty relation are discussed in detail. The reduction of the generalized uncertainty relation due to the boundary condition is then tested experimentally by re-analyzing data for measured electron densities in a hydrogen molecule encapsulated in a fullerene C60 cage.
Concepts from information theory are used to investigate the relation between measurement and irreversibility in finite, closed classical systems having known Hamiltonians. The effects of measurement upon the predictions of the observer are found for the cases of a single measurement, two simultaneous measurements, and two measurements at different times. The physical basis of coarse graining and the significance of the increase in the coarse-grained entropy are discussed. An evolution equation is obtained describing the probability distribution of the measured quantities, and it is shown that this equation is non-Markoffian. Despite this non-Markoffian behavior, the distribution may behave irreversibly and may approach equilibrium. Thus, irreversible behavior is obtained without continual coarse graining. The condition for irreversible behavior is given for the case of a single measurement. The thermodynamic entropy is defined for general nonequilibrium situations.
Uncertainty is a fundamental characteristic of quantum system. The degree of uncertainty of an observable has long been investigated by the standard deviation of the observable. In recent years, however, by analyzing some special examples, researchers have found that the Shannon entropy of the measurement outcomes of an observable is more suitable to quantify its uncertainty. Formally, Shannon entropy is a special limit of a more general Rényi entropy. In this paper, we discuss the problem of how to predict the measurement outcome of an observable by the existing measurement results of the observable, and how to quantitatively describe the uncertainty of the observable from the perspective of the repeatable probability of the measurement results of this observable in an unknown state. We will argue that if the same observable of different systems in the same state is repeatedly and independently measured many times, then the probability of obtaining an identical measurement result is a decaying function of the number of measurements of obtaining the same result, and the decay rate of the repeatable probability for obtaining the same measurement results and the repeatable number of measurements can represent the degree of uncertainty of the observable in this state. It means that the greater the uncertainty of an observable, the faster the repeatable probability decays with the number of repeatable measurements; conversely, the smaller the uncertainty, the slower the repeatable probability decays with the number of repeatable measurements. This observation enables us to give the Shannon entropy and the Rényi entropy of an observable uniformly by the functional relation between the repeatable probability and the number of repeatable measurements. We show that the Shannon entropy and the Rényi entropy can be formally regarded as the “decay index” of the repeatable probability with the number of repeatable measurements. In this way we also define a generalized Rényi entropy by the repeatable probability for consecutively observing identical results of an observable, and therefore we give a proof of the Maassen-Uffink type entropic uncertainty relation by using this generalized Rényi entropy. This method of defining entropy shows that entropic uncertainty relation is a quantitative limitation for the decay rate of the total probability for obtaining identical measurement results when we simultaneously measure two observables many times.
A universal formulation of uncertainty relations for quantum measurements is presented with additional focus on the representability of quantum observables by classical observables over a given state. Owing to the simplicity and operational tangibility of the framework, the resultant general relations admit natural operational interpretations and characterisations, and are thus also experimentally verifiable. In view of the universal formulation, Heisenberg's philosophy of the uncertainty principle is also revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. In fact, the relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the Arthurs-Kelly-Goodman, Ozawa, and Watanabe-Sagawa-Ueda relations for quantum measurements. The Schr{ö}dinger relation (hence the standard Kennard-Robertson relation as its trivial corollary as well) is also shown to be a special case when the measurement is non-informative.
We address an information-theoretic approach to noise and disturbance in quantum measurements. Properties of corresponding probability distributions are characterized by means of both the R´enyi and Tsallis entropies. Related information-theoretic measures of noise and disturbance are introduced. These definitions are based on the concept of conditional entropy. To motivate introduced measures, some important properties of the conditional R´enyi and Tsallis entropies are discussed. There exist several formulations of entropic uncertainty relations for a pair of observables. Trade-off relations for noise and disturbance are derived on the base of known formulations of such a kind.
We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.