One-parameter class of uncertainty relations based on entropy power

TL;DR

This paper introduces a new one-parameter family of entropy-based uncertainty relations for conjugate observables, enabling detailed analysis of quantum states' information distribution in infinite-dimensional spaces.

Contribution

It develops an infinite class of higher-order entropy power uncertainty relations, extending traditional bounds and allowing shape determination of quantum information distributions.

Findings

Demonstrates the new relations with superpositions of vacuum and squeezed states

Analyzes heavy-tailed wave functions with Cauchy-type distributions

Shows potential for detailed quantum state characterization

Abstract

We use the concept of entropy power to derive a new one-parameter class of information-theoretic uncertainty relations for pairs of conjugate observables in an infinite-dimensional Hilbert space. This class constitutes an infinite tower of higher-order statistics uncertainty relations, which allows one in principle to determine the shape of the underlying information-distribution function by measuring the relevant entropy powers. We illustrate the capability of the new class by discussing two examples: superpositions of vacuum and squeezed states and the Cauchy-type heavy-tailed wave function.