On a conjecture regarding Fisher information

TL;DR

This paper disproves a conjecture about a lower bound on the product of Fisher information measures in quantum physics, providing counterexamples using free-particle Schrödinger solutions and proposing a new conjecture.

Contribution

The authors demonstrate that a previously conjectured non-trivial lower bound on Fisher information product does not hold, and introduce a new conjecture for time-dependent solutions.

Findings

Counterexamples for pure states solving the free-particle Schrödinger equation.

Disproof of the conjectured lower bound on Fisher information product.

Proposal of a new conjecture for normalizable solutions.

Abstract

Fisher's information measure plays a very important role in diverse areas of theoretical physics. The associated measures as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The product of them has been conjectured to exhibit a non trivial lower bound in [Phys. Rev. A (2000) 62 012107]. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schr\"odinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schr\"odinger equation. We also give a new conjecture regarding any normalizable time-dependent solution of this equation.