Majorization approach to entropic uncertainty relations for coarse-grained observables

TL;DR

This paper develops improved entropic uncertainty relations for coarse-grained position and momentum measurements using majorization, providing tighter bounds that enhance entanglement detection in continuous variables.

Contribution

It introduces a majorization-based method to derive continuous, coarse-grained entropic uncertainty relations involving Rènyi entropies, surpassing existing bounds especially for larger coarse graining.

Findings

Majorization-based bounds outperform existing results for large coarse graining.

Derived entropic inequalities involve two Rènyi entropies of the same order.

Results are applicable to entanglement detection in continuous variable systems.

Abstract

We improve the entropic uncertainty relations for position and momentum coarse-grained measurements. We derive the continuous, coarse-grained counterparts of the discrete uncertainty relations based on the concept of majorization. The obtained entropic inequalities involve two R\'enyi entropies of the same order, and thus go beyond the standard scenario with conjugated parameters. In a special case describing the sum of two Shannon entropies the majorization-based bounds significantly outperform the currently known results in the regime of larger coarse graining, and might thus be useful for entanglement detection in continuous variables.