Variance uncertainty relations without covariances for three and four observables

TL;DR

This paper introduces new variance-based uncertainty relations for three and four observables that do not explicitly involve covariances, providing potentially tighter bounds than traditional relations in certain cases.

Contribution

The authors derive novel sum and product uncertainty relations involving multiple observables without explicit covariances, including a new inequality with a nonzero lower bound when the mean commutator is zero.

Findings

New uncertainty relations for three and four observables without covariances

A novel inequality with a nonzero lower bound for variance products

Examples showing improved bounds over Robertson--Schrödinger relations

Abstract

New sum and product uncertainty relations, containing variances of three or four observables, but not containing explicitly their covariances, are derived. One of consequences is the new inequality, giving a nonzero lower bound for the product of two variances in the case of zero mean value of the commutator between the related operators. Moreover, explicit examples show that in some cases this new bound can be better than the known Robertson--Schr\"odinger one.