Abstract

We study here the difference between quantum statistical treatments and semiclassical ones, using as the main research tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant 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.

Highlights

  • A quarter of century before Shannon, R.A

  • We briefly review basic notions about i) Fisher’s information measure and ii) coherent states and Husimi distributions for a system’s thermal state. The nucleus of this communication is developed in Section III: appropriately employing the semi-classical, shift-invariant Fisher measure so as to uncover the rather surprising amount of purely quantum information that it carries

  • The semi-classical Wehrl entropy W is a useful measure of localization in phase-space [8, 21]. It is built up using coherent states |z [8, 16, 22] and constitutes a powerful tool in statistical physics

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Summary

INTRODUCTION

A quarter of century before Shannon, R.A. Fisher advanced a method to measure the information content of continuous, rather than digital inputs, using not the binary computer codes but rather the statistical distribution of classical probability theory [1, 2]. We will show that these differences can be neatly expressed entirely in terms of a semi-classical, shift-invariant Fisher measure This measure helps to refine the socalled Lieb–bound [8] and connects this refinement with a Fisher-description of delocalization in phase-space, that, can be visualized as information loss. The nucleus of this communication is developed in Section III: appropriately employing the semi-classical, shift-invariant Fisher measure so as to uncover the rather surprising amount of purely quantum information that it carries.

Fisher’s information measure
Semi-classical purity
SEMI-CLASSICAL FISHER’S MEASURE
MaxEnt approach
Delocalization revisited
Second moment of the Husimi distribution
Purity and delocalization
THERMODYNAMICS-LIKE RELATIONS
DEGREES OF PURITY’S RELATIONS
VIII. CONCLUSIONS
GENERALIZATIONS
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