On a Conjecture regarding Fisher Information

Abstract EN

Fisher’s information measure I plays a very important role in diverse areas of theoretical physics.

The associated measures I x and I p , as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations.

The product I x I p 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 x I p ≥ 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 x I p → 0 for t → ∞ .