Uncertainty regions of observables and state-independent uncertainty relations

TL;DR

This paper introduces a framework for calculating tight, state-independent uncertainty bounds for multiple quantum observables using uncertainty regions, aiding experimental detection of entanglement.

Contribution

It provides a novel analytical approach to determine uncertainty regions and derive optimal state-independent uncertainty relations for multiple observables.

Findings

Derived analytical formulas for uncertainty regions.

Established state-independent uncertainty inequalities for multiple observables.

Proposed entanglement detection criteria based on these bounds.

Abstract

The optimal state-independent lower bounds for the sum of variances or deviations of observables are of significance for the growing number of experiments that reach the uncertainty limited regime. We present a framework for computing the tight uncertainty relations of variance or deviation via determining the uncertainty regions, which are formed by the tuples of two or more of quantum observables in random quantum states induced from the uniform Haar measure on the purified states. From the analytical formulae of these uncertainty regions, we present state-independent uncertainty inequalities satisfied by the sum of variances or deviations of two, three and arbitrary many observables, from which experimentally friend entanglement detection criteria are derived for bipartite and tripartite systems.