An entropic uncertainty principle for positive operator valued measures

TL;DR

This paper establishes a new entropic uncertainty principle for mixed quantum states with respect to pairs of positive operator valued measures, leading to sharp spatial-spectral uncertainty and log-Sobolev inequalities in certain mathematical settings.

Contribution

It extends previous results to mixed states and introduces a general framework for uncertainty principles involving positive operator valued measures.

Findings

Derives a sharp entropic uncertainty principle for mixed states.

Provides spatial-spectral uncertainty principles for invariant operators.

Establishes log-Sobolev inequalities on homogeneous spaces.

Abstract

Extending a recent result by Frank and Lieb, we show an entropic uncertainty principle for mixed states in a Hilbert space relatively to pairs of positive operator valued measures that are independent in some sense. This yields spatial-spectral uncertainty principles and log-Sobolev inequalities for invariant operators on homogeneous spaces, which are sharp in the compact case.