# The Optimal Uncertainty Relation

## TL;DR

This paper introduces a universal quantum uncertainty relation framework based on lattice theory and majorization, revealing intrinsic structures and connections to entanglement and economic concepts like Lorenz curves.

## Contribution

It develops a lattice-theoretic approach to define optimal uncertainty bounds and links quantum uncertainty with entanglement transformation and economic measures.

## Key findings

- Least bounds of uncertainty relations are properly defined in lattice theory.
- Majorization lattice reveals intrinsic quantum uncertainty structure.
- Optimal uncertainty can be mimicked by Lorenz curves.

## Abstract

Employing the lattice theory on majorization, we investigate the universal quantum uncertainty relation for any number observables and general measurement. We find: 1. The least bounds of the universal uncertainty relations can only be properly defined in the lattice theory; 2. Contrary to variance and entropy, the metric induced by the majorization lattice implies an intrinsic structure of the quantum uncertainty; 3. The lattice theory correlates the optimization of uncertainty relation with the entanglement transformation under local quantum operation and classical communication. Interestingly, the optimality of the universal uncertainty relation is found can be mimicked by the Lorenz curve, initially introduced in economics to measure the wealth concentration degree of a society.