Uncertainty relations for any multi observables

TL;DR

This paper introduces a universal uncertainty relation for any number of quantum observables, linking standard deviations to the numerical radius of operators, and provides tight bounds for specific cases.

Contribution

It establishes a new universal uncertainty relation applicable to any number of observables, extending and refining existing principles like Schrödinger's.

Findings

Universal uncertainty relation formulated for any $k$ observables.

Relation depends on whether $k$ is even or odd.

Tight bounds achieved for cases $k=2n$ and $k=3$.

Abstract

Uncertainty relations describe the lower bound of product of standard deviations of observables. By revealing a connection between standard deviations of quantum observables and numerical radius of operators, we establish a universal uncertainty relation for any $k$ observables, of which the formulation depends on the even or odd quality of $k$. This universal uncertainty relation is tight at least for the cases $k=2n$ and $k=3$. For two observables, the uncertainty relation is exactly a simpler reformulation of Schr\"odinger's uncertainty principle.