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Journal of Physics A Mathematical and Theoretical

A quantum uncertainty relation based on Fisher's information

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

Published Jan 10, 2011• 55 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/44/6/065301

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Abstract With special attention to the role of information theory in physical sciences we present analytical results for the coordinate- and momentum-space Fisher information of some important one-dimensional quantum systems which differ in spacing of their energy levels. The studies envisaged allow us to relate the coordinate-space information ( I ρ ) with the familiar energy levels of the quantum system. The corresponding momentum-space information ( I γ ) does not obey such a simple relationship with the energy spectrum. Our results for the product ( I ρ I γ ) depend quadratically on the principal quantum number n and satisfy an appropriate uncertainty relation derived by Dehesa et al (2007 J. Phys. A: Math. Theor. 40 1845 )

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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.
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The coordinate-space wave function ψ(x)ψ(x)ψ(x) of quasi-one-dimensional atoms is defined in the x≥0x\geq 0x≥0 region only. This poses a typical problem to write a physically acceptable momentum-space wave function ϕ(p)ϕ(p)ϕ(p) from the Fourier transform of ψ(x)ψ(x)ψ(x). We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function ϕ(p)ϕ(p)ϕ(p) as a function of ppp and confirm that ϕi(p)ϕ_i(p)ϕi​(p) (the imaginary part of ϕ(p)ϕ(p)ϕ(p)) represents the correct momentum-space wave function. We make use of the results for ψ(x)ψ(x)ψ(x) and ϕi(p)ϕ_i(p)ϕi​(p) to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.
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.
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Abstract With special attention to the role of information theory in physical sciences we present analytical results for the coordinate- and momentum-space Fisher information of some important one-dimensional quantum systems which differ in spacing of their energy levels. The studies envisaged allow us to relate the coordinate-space information ( I ρ ) with the familiar energy levels of the quantum system. The corresponding momentum-space information ( I γ ) does not obey such a simple relationship with the energy spectrum. Our results for the product ( I ρ I γ ) depend quadratically on the principal quantum number n and satisfy an appropriate uncertainty relation derived by Dehesa et al (2007 J. Phys. A: Math. Theor. 40 1845 )