On a conjecture regarding Fisher information

Abstract

Fisher's information measure I plays a very important role in diverse areas of theoretical physics. The associated measures I<SUB>x</SUB> and I<SUB>p</SUB>, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The product I<SUB>x</SUB>I<SUB>p</SUB> has 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 dimension I<SUB>x</SUB>I<SUB>p</SUB> ≥ 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 verifies I<SUB>x</SUB>I<SUB>p</SUB> → 0 for t → ∞.