On a Conjecture regarding Fisher Information

Fisher’s information measure plays a very important role in diverse areas of theoretical physics. The associated measures and , as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The product 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 . 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 for .