On a Conjecture regarding Fisher Information

Abstract

Fisher’s information measureIplays a very important role in diverse areas of theoretical physics. The associated measuresIxandIp, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The productIxIphas been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimensionIxIp≥4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifiesIxIp→0fort→∞.

Heisenberg-like and Fisher-information-based uncertainty relations forN-electrond-dimensional systems

“…Then we use some results of the previous section to obtain a mathematical formulation of the positionmomentum uncertainty principle for these systems. The resulting expressions extend and generalize various similar conjectures and inequalities in the sense already discussed in the first section [43][44][45][46][47].…”

“…Nowadays it remains a strongly controversial problem [43][44][45][46][47][48][49]. First, it was conjectured [45] in 2000 that the position-momentum Fisher information product had the lower bound I 1 (ρ)I 1 (γ ) 4 for one-dimensional quantum systems with the position and momentum densities ρ(x) = | (x)| 2 and γ (p) = | (p)| 2 , (p) being the Fourier transform of (x).…”

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

“…However, this is not the most universal uncertainty relation expressible as a lower bound to the product of the Fisher measures and because the Fisher product can be made arbitrarily small [ 137 ]. See also [ 138 ], where this problem has been discussed for pure and mixed states.…”

Relative Fisher information for Morse potential and isotropic quantum oscillators

“…The behavior of the relative Fisher information when evaluated on time-dependent solutions of the Schrödinger equation for some quantum systems may exhibit a distinct property. Since the time evolution of the product of the Fisher information in coordinate and momentum spaces under the Schrödinger equation converges with zero as time goes to infinity (e.g., [23]), this inquiry would be interesting as a future work.…”