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

UNCERTAINTY RELATIONS AND ERROR RELATIONS IN OPTICAL MEASUREMENTS OPTOPHYSICAL MEASUREMENTS

Vadim I. Smirnov

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

We analyze extremal measurement error relations for time and frequency parameters of optical pulses, which are compared with the uncertainty relation. It is shown that the product of the mean-square measurement error limit for a single photon corresponds to the minimum value of ˙/2 in the quantum uncertainty relation for time-energy (coordinate-momentum) and decreases with deviation from a Gaussian pulse shape, for which the information entropy is maximal. The error relation is in the form of the uncertainty relation with the inequality reversed.

Related papers

Position-momentum entropic uncertainty relation and complementarity in single-slit and double-slit experiments

Jorge Sánchez-Ruiz

The exact analytical values of the position and momentum information entropies are calculated for the single-slit and double-slit diffraction experiments. In both cases, the product of the exponentials of the entropies is strictly greater than the lower bound $\ensuremath{\pi}e\ensuremath{\Elzxh}$ given by the optimal entropic uncertainty relation for position and momentum, which implies that the single-slit and double-slit configurations are not minimum-uncertainty states. The results obtained show that the position-momentum entropic uncertainty relation provides a rigorous quantitative expression for the uncertainty principle in these experiments, unlike the Heisenberg inequality for standard deviations. However, it is also shown that, in the double-slit experiment, wave-particle duality cannot be derived from the entropic uncertainty relation.

Physical Review A•Mar 1, 1998•18 citations

Uncertainty relation and the measurement error–perturbation relation

Yurii I. Vorontsov

The origins and physical consequences of the traditionally used relation between the position measurement error and the momentum perturbation, Δm2xΔp2p ≥ 2/4, are discussed. It is demonstrated that the corresponding increase in the momentum variance for the aposteriori state occurs only in some special cases. The product of Δm2A and Δp2B is shown to essentially differ from the one given by the uncertainty relation if the commutator is an operator. The error quantum limits for the joint homodyne measurement of quadrature amplitudes for an optical mode are found. It is shown that similar results can be obtained if the quadratures of a harmonic oscillator are estimated by means of continuous position measurement.

Physics-Uspekhi•Oct 31, 2005•7 citations

REMARKS ON TIME-ENERGY UNCERTAINTY RELATIONS

ROMEO BRUNETTI, Klaus Fredenhagen

Using a recent construction of observables characterizing the time of occurence of an effect in quantum theory, we present a rigorous derivation of the standard time-energy uncertainty relation. In addition, we prove an uncertainty relation for time measurements alone.

arXiv:quant-ph/0207048•Reviews in Mathematical Physics•Jul 1, 2002•18 citations

Energy–time and frequency–time uncertainty relations: exact inequalities

V. V. Dodonov, Alexandre V. Dodonov

We give a short review of known exact inequalities that can be interpreted as ‘energy–time’ and ‘frequency–time’ uncertainty relations. In particular we discuss a precise form of signals minimizing the physical frequency–time uncertainty product. Also, we calculate the ‘stationarity time’ for mixed Gaussian states of a quantum harmonic oscillator, showing explicitly that pure quantum states are ‘more fragile’ than mixed ones with the same value of the energy dispersion. The problems of quantum evolution speed limits, time operators and measurements of energy and time are briefly discussed, too.

arXiv:1504.00862•Physica Scripta•Jun 1, 2015•50 citations

Uncertainty relations: a small zoo of remarkable inequalities discovered since 1927

V. V. Dodonov

A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.

quant-pharXiv:2603.07293•Revista Brasileira de Ensino de Física•Mar 10, 2026•1 citation

Lower bounds for the time-bandwidth product of a single-photon pulse

Valentin Averchenko, D A Reiß, Сыч Денис Васильевич +1 more

Abstract Two complementary properties, such as frequency and duration of a pulse, cannot be precisely measured simultaneously. The lower bound on the time-bandwidth product, known also as uncertainty relation, plays a very important role in quantum theory. In this work, we consider single-photon pulses with arbitrary temporal and spectral profiles, and derive a lower bound for the single-photon time-bandwidth product. We investigate how this bound depends on the modal purity, the single-mode fidelity, and the effective number of modes. Our results can be used to solve the inverse problem, namely, to estimate characteristics of the single-photon pulse based on the time-bandwidth product obtained from temporal and spectral measurements of the pulse.

Physica Scripta•Nov 20, 2019•5 citations
We analyze extremal measurement error relations for time and frequency parameters of optical pulses, which are compared with the uncertainty relation. It is shown that the product of the mean-square measurement error limit for a single photon corresponds to the minimum value of ˙/2 in the quantum uncertainty relation for time-energy (coordinate-momentum) and decreases with deviation from a Gaussian pulse shape, for which the information entropy is maximal. The error relation is in the form of the uncertainty relation with the inequality reversed.
The exact analytical values of the position and momentum information entropies are calculated for the single-slit and double-slit diffraction experiments. In both cases, the product of the exponentials of the entropies is strictly greater than the lower bound $\ensuremath{\pi}e\ensuremath{\Elzxh}$ given by the optimal entropic uncertainty relation for position and momentum, which implies that the single-slit and double-slit configurations are not minimum-uncertainty states. The results obtained show that the position-momentum entropic uncertainty relation provides a rigorous quantitative expression for the uncertainty principle in these experiments, unlike the Heisenberg inequality for standard deviations. However, it is also shown that, in the double-slit experiment, wave-particle duality cannot be derived from the entropic uncertainty relation.
The origins and physical consequences of the traditionally used relation between the position measurement error and the momentum perturbation, Δm2xΔp2p ≥ 2/4, are discussed. It is demonstrated that the corresponding increase in the momentum variance for the aposteriori state occurs only in some special cases. The product of Δm2A and Δp2B is shown to essentially differ from the one given by the uncertainty relation if the commutator is an operator. The error quantum limits for the joint homodyne measurement of quadrature amplitudes for an optical mode are found. It is shown that similar results can be obtained if the quadratures of a harmonic oscillator are estimated by means of continuous position measurement.
We give a short review of known exact inequalities that can be interpreted as ‘energy–time’ and ‘frequency–time’ uncertainty relations. In particular we discuss a precise form of signals minimizing the physical frequency–time uncertainty product. Also, we calculate the ‘stationarity time’ for mixed Gaussian states of a quantum harmonic oscillator, showing explicitly that pure quantum states are ‘more fragile’ than mixed ones with the same value of the energy dispersion. The problems of quantum evolution speed limits, time operators and measurements of energy and time are briefly discussed, too.
A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.
Abstract Two complementary properties, such as frequency and duration of a pulse, cannot be precisely measured simultaneously. The lower bound on the time-bandwidth product, known also as uncertainty relation, plays a very important role in quantum theory. In this work, we consider single-photon pulses with arbitrary temporal and spectral profiles, and derive a lower bound for the single-photon time-bandwidth product. We investigate how this bound depends on the modal purity, the single-mode fidelity, and the effective number of modes. Our results can be used to solve the inverse problem, namely, to estimate characteristics of the single-photon pulse based on the time-bandwidth product obtained from temporal and spectral measurements of the pulse.