Heisenberg-Fisher thermal uncertainty measure

TL;DR

This paper explores the relationship between Wehrl entropy, Fisher's information measure, and thermodynamic entropy in the quantum harmonic oscillator, revealing new insights and uncertainty relations using Fisher information.

Contribution

It establishes a novel connection between Fisher information measures and thermodynamic quantities in the quantum harmonic oscillator, extending the understanding of entropy and uncertainty relations.

Findings

Fisher information relates to the excited spectrum's contribution to mean energy

A Fisher measure corresponds to the canonical partition function minus ground state energy

New Fisher-related uncertainty relations are proposed

Abstract

With the help of the coherent states' basis we establish an interesting connection among i) the so-called Wehrl entropy, ii) Fisher's information measure $I$, and iii) the canonical ensemble entropy for the one-dimensional quantum harmonic oscillator (HO). We show that the contribution of the excited HO spectrum to the mean thermal energy is given by $I$, while the pertinent canonical partition function is given by another Fisher measure: the so-called shift invariant one, minus the HO's ground state energy. Our findings should be of interest in view of the fact that it has been shown that the Legendre transform structure of thermodynamics can be replicated without any change if one replaces the Boltzmann-Gibbs-Shannon entropy by Fisher's information measure [{\it Physical Review E} {\bf 60}, 48 (1999)]. New Fisher-related uncertainty relations are also advanced.