Quantum statistical information contained in a semi-classical Fisher--Husimi measure

TL;DR

This paper explores the differences between quantum and semi-classical statistical descriptions using a Fisher information measure based on Husimi distributions, revealing quantum features and refining entropy bounds.

Contribution

It introduces a semi-classical Fisher measure that captures quantum information and refines Lieb's entropy bound, linking delocalization and purity in quantum states.

Findings

Refined Lieb bound for Wehrl entropies

Discovered thermodynamic-like relations involving delocalization

Developed Fisher-based thermal uncertainty relations

Abstract

We study here the difference between quantum statistical treatments and semi-classical ones, using as the main research tool a semi-classical, shift-invariant Fisher information measure built up with Husimi distributions. Its semi-classical character notwithstanding, this measure also contains information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.