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:math-ph/0701062 · Journal of Mathematical Physics

Uncertainty principle and quantum Fisher information. II.

Paolo Gibilisco, Daniele Imparato, Tommaso Isola

Published Jul 1, 2007• 80 citations
arXivHTMLPDF

Abstract

Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances Varρ(A)Varρ(B) if the observables A,B are not compatible, namely, if the commutator [A,B] is not zero. In this paper, we prove an uncertainty principle in Schrödinger form where the bound for the product of variances Varρ(A)Varρ(B) depends on the area spanned by the commutators i[ρ,A] and i[ρ,B] with respect to an arbitrary quantum version of the Fisher information.

DOI
10.1063/1.2748210

Related papers

Uncertainty principle and quantum Fisher information. II

Gibilisco, P., Daniele Imparato, Isola, T.

Heisenberg and Schr{\"o}dinger uncertainty principles give lower bounds for the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$, in a state $\rho$, if the observables $A,B$ are not compatible, namely if the commutator $[A,B]$ is not zero. In this paper we prove an uncertainty principle in Schr{\"o}dinger form where the bound for the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$ depends on the area spanned by the commutators $[\rho,A]$ and $[\rho,B]$ with respect to an arbitrary quantum version of the Fisher information.

•Jan 1, 2007•88 citations

Uncertainty principle with quantum Fisher information

Attila Andai

In this paper we prove a lower bound for the determinant of the covariance matrix of quantum mechanical observables, which was conjectured by Gibilisco et al. and has the interpretation of uncertainty. The lower bound is given in terms of the commutator of the state and the observables and quantum Fisher information (generated by an operator monotone function).

arXiv:0707.1147•Journal of Mathematical Physics•Jan 1, 2008•26 citations

On some states minimizing uncertainty relations: A new look at these relations

K. Urbanowski

The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.

arXiv:2411.08131•Modern Physics Letters A•Mar 25, 2026•0 citations

Variance uncertainty relations without covariances for three and four observables

V. V. Dodonov

Sum and product uncertainty relations, containing variances of three or four observables, but not containing explicitly their covariances, are derived. Their consequences are, in particular, inequalities, giving nonzero lower bounds for the products of two variances in the case of zero mean value of the commutator between the related operators. Explicit examples show that the bounds can be better than the known Robertson-Schr\"odinger one.

arXiv:1711.04037•Physical Review A•Feb 7, 2018•30 citations

A quantum uncertainty relation based on Fisher's information

Pablo Sánchez-Moreno, Angel Ricardo Plastino, J. S. Dehesa

We explore quantum uncertainty relations involving the Fisher information functionals I x and I p evaluated, respectively, on a wavefunction Ψ( x ) defined on a D -dimensional configuration space and the concomitant wavefunction on the conjugate momentum space. We prove that the associated Fisher functionals obey the uncertainty relation I x I p ⩾ 4 D 2 when either Ψ( x ) or is real. On the other hand, there is no lower bound to the above product for arbitrary complex wavefunctions. We give explicit examples of complex wavefunctions not obeying the above bound. In particular, we provide a parametrized wavefunction for which the product I x I p can be made arbitrarily small.

Journal of Physics A Mathematical and Theoretical•Jan 10, 2011•55 citations

Stronger uncertainty relations

Lorenzo Maccone, Arun Kumar Pati

Heisenberg-Robertson's uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two non-compatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.

•Jul 1, 2014•0 citations
Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances Varρ(A)Varρ(B) if the observables A,B are not compatible, namely, if the commutator [A,B] is not zero. In this paper, we prove an uncertainty principle in Schrödinger form where the bound for the product of variances Varρ(A)Varρ(B) depends on the area spanned by the commutators i[ρ,A] and i[ρ,B] with respect to an arbitrary quantum version of the Fisher information.
Heisenberg and Schr{\"o}dinger uncertainty principles give lower bounds for the product of variances Varρ(A)⋅Varρ(B)Var_{\rho}(A)\cdot Var_{\rho}(B)Varρ​(A)⋅Varρ​(B), in a state ρ\rhoρ, if the observables A,BA,BA,B are not compatible, namely if the commutator [A,B][A,B][A,B] is not zero. In this paper we prove an uncertainty principle in Schr{\"o}dinger form where the bound for the product of variances Varρ(A)⋅Varρ(B)Var_{\rho}(A)\cdot Var_{\rho}(B)Varρ​(A)⋅Varρ​(B) depends on the area spanned by the commutators [ρ,A][\rho,A][ρ,A] and [ρ,B][\rho,B][ρ,B] with respect to an arbitrary quantum version of the Fisher information.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.
We explore quantum uncertainty relations involving the Fisher information functionals I x and I p evaluated, respectively, on a wavefunction Ψ( x ) defined on a D -dimensional configuration space and the concomitant wavefunction on the conjugate momentum space. We prove that the associated Fisher functionals obey the uncertainty relation I x I p ⩾ 4 D 2 when either Ψ( x ) or is real. On the other hand, there is no lower bound to the above product for arbitrary complex wavefunctions. We give explicit examples of complex wavefunctions not obeying the above bound. In particular, we provide a parametrized wavefunction for which the product I x I p can be made arbitrarily small.
Heisenberg-Robertson's uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two non-compatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.