Multi-observable Uncertainty Relations in Product Form of Variances

TL;DR

This paper derives new, tighter uncertainty relations in product form for multiple quantum observables, improving upon existing relations and applicable to systems like spin-half particles and Gell-Mann matrices.

Contribution

It introduces novel tight uncertainty relations for three and more observables, surpassing previous bounds and including explicit examples like Gell-Mann matrices.

Findings

Derived tighter uncertainty relations for three observables.

Extended uncertainty relations to an arbitrary number of observables.

Provided explicit example with Gell-Mann matrices.

Abstract

We investigate the product form uncertainty relations of variances for $n\,(n\geq 3)$ quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schr\"{o}dinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.