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

Physical Review A

Comment on “Quantum reciprocity relations for fluctuations of position and momentum”

Iwo Białynicki‐Birula

Published Oct 28, 2019• 1 citation
Publisher

Abstract

The authors of this paper [M. Pandit et al., Phys. Rev. A 100, 012131 (2019)] proposed an alternative formulation of the uncertainty relation in quantum mechanics in terms of the Lipschitz constants. They evaluated these constants in various cases and decided that the product of the square roots of Lipschitz constants for position and momentum probability distributions is bounded below by $0.3/\ensuremath{\hbar}$. In this comment, I invalidate this claim by giving an example of wave functions for which the value of the product of the Lipschitz constants for position and momentum does not obey this bound and tends to zero.

DOI
10.1103/physreva.100.046101

Related papers

Quantum reciprocity relations for fluctuations of position and momentum

Mahasweta Pandit, Anindita Bera, Aditi Sen +1 more

We propose a trade-off between the Lipschitz constants of the position and momentum probability distributions for arbitrary quantum states. We refer to the trade-off as a quantum reciprocity relation. The Lipschitz constant of a function may be considered to quantify the extent of fluctuations of that function and is, in general, independent of its spread. The spreads of the position and momentum distributions are used to obtain the celebrated Heisenberg quantum uncertainty relations. We find that the product of the Lipschitz constants of position and momentum probability distributions is bounded below by a number that is of the order of the inverse square of the Planck's constant.

arXiv:1803.02221•Physical Review A•Jul 29, 2019•3 citations

Reciprocity Relations for Quantum Systems Based on Fisher Information

M. Portesi, Juan Manuel Pujol, Federico Holik

We study reciprocity relations between fluctuations of the probability distributions corresponding to position and momentum, and other observables, in quantum theory. These kinds of relations have been previously studied in terms of quantifiers based on the Lipschitz constants of the concomitant distributions. However, it turned out that they were not valid for all states. Here, we ask the following question: can those relations be described using other quantifiers? By appealing to the Fisher information, we study reciprocity relations for different families of states. In particular, we look for a connection of this problem with previous works.

•Jan 29, 2023•0 citations

Heisenberg uncertainty relation for position and momentum beyond central potentials

Łukasz Rudnicki

Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.

arXiv:1112.5198•Physical Review A•Feb 8, 2012•11 citations

Variance-Based Uncertainty Relations: A Concise Review of Inequalities Discovered Since 1927

V. V. Dodonov

A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.

Quantum Reports•Aug 5, 2025•4 citations

Uncertainty related to position and momentum localization of a quantum state

Łukasz Rudnicki

This paper presents the uncertainty related to position and momentum localization of a quantum state in terms of entropic uncertainty relations. We slightly improve the inequality given in [Phys. Rev. A 74, 052101 (2006)] and introduce a new entropic measure with corresponding uncertainty relation.

arXiv:1010.3269•Oct 15, 2010•7 citations

Uncertainty Relations: Curiosities and Inconsistencies

Krzysztof Urbanowski

Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.

arXiv:2010.08339•Symmetry•Oct 6, 2020•2 citations
The authors of this paper [M. Pandit et al., Phys. Rev. A 100, 012131 (2019)] proposed an alternative formulation of the uncertainty relation in quantum mechanics in terms of the Lipschitz constants. They evaluated these constants in various cases and decided that the product of the square roots of Lipschitz constants for position and momentum probability distributions is bounded below by 0.3/\ensuremathℏ0.3/\ensuremath{\hbar}0.3/\ensuremathℏ. In this comment, I invalidate this claim by giving an example of wave functions for which the value of the product of the Lipschitz constants for position and momentum does not obey this bound and tends to zero.
We propose a trade-off between the Lipschitz constants of the position and momentum probability distributions for arbitrary quantum states. We refer to the trade-off as a quantum reciprocity relation. The Lipschitz constant of a function may be considered to quantify the extent of fluctuations of that function and is, in general, independent of its spread. The spreads of the position and momentum distributions are used to obtain the celebrated Heisenberg quantum uncertainty relations. We find that the product of the Lipschitz constants of position and momentum probability distributions is bounded below by a number that is of the order of the inverse square of the Planck's constant.
We study reciprocity relations between fluctuations of the probability distributions corresponding to position and momentum, and other observables, in quantum theory. These kinds of relations have been previously studied in terms of quantifiers based on the Lipschitz constants of the concomitant distributions. However, it turned out that they were not valid for all states. Here, we ask the following question: can those relations be described using other quantifiers? By appealing to the Fisher information, we study reciprocity relations for different families of states. In particular, we look for a connection of this problem with previous works.
Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.
A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.
Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.