Sum uncertainty relations for arbitrary $N$ incompatible observables

TL;DR

This paper develops new sum uncertainty relations for any number of incompatible observables, providing tighter bounds than previous two-observable inequalities, with explicit formulas and illustrative examples.

Contribution

It introduces a general framework for sum uncertainty relations for arbitrary N observables, with explicit lower bounds and improved tightness over existing inequalities.

Findings

Derived explicit lower bounds for sum uncertainty relations.

Showed bounds are tighter than previous two-observable inequalities.

Provided examples demonstrating the effectiveness of the new bounds.

Abstract

We formulate uncertainty relations for arbitrary $N$ observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The lower bounds of the corresponding sum uncertainty relations are explicitly derived. These bounds are shown to be tighter than the ones such as derived from the uncertainty inequality for two observables [Phys. Rev. Lett. 113, 260401 (2014)]. Detailed examples are presented to compare among our results with some existing ones.