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

Reciprocity Relations for Quantum Systems Based on Fisher Information

M. Portesi, Juan Manuel Pujol, Federico Holik

Published Jan 29, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.3390/psf2022005044

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

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

Iwo Białynicki‐Birula

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.

Physical Review A•Oct 28, 2019•1 citation

Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials

E. Romera, Pablo Sánchez-Moreno, J. S. Dehesa

An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.

Journal of Mathematical Physics•Oct 1, 2006•93 citations

The Fisher Information: Properties and Physico-Chemical Applications

The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.

BENTHAM SCIENCE PUBLISHERS eBooks•Dec 11, 2013•1 citation

Relations between parameters of a decoherent system and Fisher information

B. Roy Frieden

Fisher information (FI) is normally developed and applied within the context of classical mathematical statistics. Here we show that it alternatively arises as physical properties of a generally decoherent system. Two kinds of Fisher information are treated: the classical FI $I[p]$ and the FI channel capacity 〈I〉. These are shown to be linearly related to the partition function Z, the coherence function $\ensuremath{\rho}{(x,x}^{\ensuremath{'}}),$ the two-point coherence function $\ensuremath{\tau}(\ensuremath{\Delta}x),$ the entropy of fluctuation $H(\ensuremath{\Delta}x),$ and the cross entropy ${H}_{\mathrm{KL}}(\ensuremath{\rho}{/r}_{\ensuremath{\Delta}})$ between the system probability density function $\ensuremath{\rho}(x,x)$ and its shifted version $\ensuremath{\rho}(x+\ensuremath{\Delta}x,x+\ensuremath{\Delta}x).$

Physical Review A•Aug 12, 2002•10 citations

Quantum Fisher information and its dynamical nature

Matteo Scandi, Paolo Abiuso, Jacopo Surace +1 more

The importance of the Fisher information metrics and its quantum generalisations is testified by the number of applications that this has in very different fields, ranging from hypothesis testing to metrology, passing through thermodynamics. Still, from the rich range of possible quantum Fisher informations, only a handful are typically used and studied. This review aims at collecting a number of results scattered in the literature and provide a cohesive treatment to people who begin the study of Fisher information and to those who are already working on it to have a more organic understanding of the topic. Moreover, we complement the review with new results about the relation between Fisher information and physical evolutions. Extending previous works, we show that dynamical properties such as (complete) positivity, Markovianity, detailed balance, retrodictive power of evolution maps can be characterised in terms of their relation with respect to the Fisher information metrics. These results show a fact that was partially overseen in the literature, namely the inherently dynamical nature of Fisher information.

arXiv:2304.14984•Reports on Progress in Physics•Jun 13, 2025•23 citations
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.
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.
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.
An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
Fisher information (FI) is normally developed and applied within the context of classical mathematical statistics. Here we show that it alternatively arises as physical properties of a generally decoherent system. Two kinds of Fisher information are treated: the classical FI I[p]I[p]I[p] and the FI channel capacity 〈I〉. These are shown to be linearly related to the partition function Z, the coherence function $\ensuremath{\rho}{(x,x}^{\ensuremath{'}}),$ the two-point coherence function $\ensuremath{\tau}(\ensuremath{\Delta}x),$ the entropy of fluctuation $H(\ensuremath{\Delta}x),$ and the cross entropy ${H}_{\mathrm{KL}}(\ensuremath{\rho}{/r}_{\ensuremath{\Delta}})$ between the system probability density function $\ensuremath{\rho}(x,x)$ and its shifted version $\ensuremath{\rho}(x+\ensuremath{\Delta}x,x+\ensuremath{\Delta}x).$
The importance of the Fisher information metrics and its quantum generalisations is testified by the number of applications that this has in very different fields, ranging from hypothesis testing to metrology, passing through thermodynamics. Still, from the rich range of possible quantum Fisher informations, only a handful are typically used and studied. This review aims at collecting a number of results scattered in the literature and provide a cohesive treatment to people who begin the study of Fisher information and to those who are already working on it to have a more organic understanding of the topic. Moreover, we complement the review with new results about the relation between Fisher information and physical evolutions. Extending previous works, we show that dynamical properties such as (complete) positivity, Markovianity, detailed balance, retrodictive power of evolution maps can be characterised in terms of their relation with respect to the Fisher information metrics. These results show a fact that was partially overseen in the literature, namely the inherently dynamical nature of Fisher information.