# Entropy-power uncertainty relations : towards a tight inequality for all Gaussian pure states

## TL;DR

This paper introduces a new entropy-power-based uncertainty relation for continuous variables, providing a tighter, rotation-invariant inequality that is saturated by all Gaussian pure states, improving upon previous entropic formulations.

## Contribution

It presents a novel entropy-power uncertainty relation that extends to rotated variables and offers a tighter bound, validated through numerical evidence and applicable to Gaussian states.

## Key findings

- The relation is equivalent to the Bialynicki-Birula and Mycielski entropic uncertainty.
- It implies the Schrödinger-Robertson uncertainty relation.
- It is saturated for all Gaussian pure states.

## Abstract

We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity. Interestingly, it implies the generalized (rotation-invariant) Schr\"odinger-Robertson uncertainty relation exactly as the original entropy-power uncertainty relation implies Heisenberg relation. It is saturated for all Gaussian pure states, in contrast with hitherto known entropic formulations of the uncertainty principle.