Variance-based uncertainty relations for incompatible observables

Abstract

We formulate uncertainty relations for arbitrary finite number of incompatible observables. Based on the sum of variances of the observables, both Heisenberg-type and Schrödinger-type uncertainty relations are provided. These new lower bounds are stronger in most of the cases than the ones derived from some existing inequalities. Detailed examples are presented.

A Stronger Multi-observable Uncertainty Relation

“…Recently, some three observables uncertainty relations were studied, such as Heisenberg uncertainty relation for three canonical observables 25 , uncertainty relations for three angular momentum components 26 , uncertainty relation for three arbitrary observables 14 . Furthermore, some multiple observables uncertainty relations were proposed, which include multi-observable uncertainty relation in product 27 28 and sum 29 30 form of variances. It is worth noting that Chen and Fei derived an variance-based uncertainty relation 30 …”

Tighter uncertainty relations based on Wigner-Yanase skew information for observables and channels

“…Uncertainty relations for general multiple observables have been further studied either in product form [30,31] or sum form of variances [32][33][34][35][36].…”

Experimental investigation of multi-observable uncertainty relations

“…The generalization and improvement mainly focused on the uncertainty relation capable of dealing with more than two incompatible observables, i.e., the Heisenberg-type uncertainty relation for three canonical observables [27], uncertainty relations for angular momentum [28], and arbitrary incompatible observables [17]. There are also uncertainty relations with even more incompatible observables formulated in products [29,30] or sums [31][32][33] of variances.…”