# Universal quantum uncertainty relations: minimum-uncertainty wave packet depends on measure of spread

## TL;DR

This paper introduces a universal quantum uncertainty relation based on the median, which remains effective where traditional mean-based measures fail, revealing that minimum-uncertainty states depend on the chosen measure of spread.

## Contribution

It proposes a median-based quantum uncertainty relation that is universal and applicable to all quantum states, contrasting with traditional mean-based relations.

## Key findings

- Cauchy-Lorentz distributions minimize median-based uncertainty
- Minimum-uncertainty state depends on the measure of spread used
- Median-based relation is not saturated by Gaussian states

## Abstract

Based on the statistical concept of the median, we propose a quantum uncertainty relation between semi-interquartile ranges of the position and momentum distributions of arbitrary quantum states. The relation is universal, unlike that based on the mean and standard deviation, as the latter may become non-existent or ineffective in certain cases. We show that the median-based one is not saturated for Gaussian distributions in position. Instead, the Cauchy-Lorentz distributions in position turn out to be the one with the minimal uncertainty, among the states inspected, implying that the minimum-uncertainty state is not unique but depends on the measure of spread used. Even the ordering of the states with respect to the distance from the minimum uncertainty state is altered by a change in the measure. We invoke the completeness of Hermite polynomials in the space of all quantum states to probe the median-based relation. The results have potential applications in a variety of studies including those on the quantum-to-classical boundary and on quantum cryptography.