State-independent Uncertainty Relations and Entanglement Detection

Abstract

The uncertainty relation is one of the key ingredients of quantum theory. Despite the great efforts devoted to this subject, most of the variance-based uncertainty relations are state-dependent and suffering from the triviality problem of zero lower bounds. Here we develop a method to get uncertainty relations with state-independent lower bounds. The method works by exploring the eigenvalues of a Hermitian matrix composed by Bloch vectors of incompatible observables and is applicable for both pure and mixed states and for arbitrary number of N- dimensional observables. The uncertainty relation for incompatible observables can be explained by geometric relations related to the parallel postulate and the inequalities in Horn's conjecture on Hermitian matrix sum. Practical entanglement criteria are also presented based on the derived uncertainty relations.

15 pages, no figures

References in corpus (7)

• Heisenberg's Uncertainty Principle
• Stronger uncertainty relations for the sum of variances
• Covariance matrices and the separability problem
• State-independent Uncertainty Relations and Entanglement Detection in Noisy Systems
• Variance-based uncertainty relations for incompatible observables
• Multi-observable Uncertainty Relations in Product Form of Variances
• Uncertainty equalities and uncertainty relation in weak measurement

Cited by in corpus (8)

• The Optimal Uncertainty Relation
• Tight $N$-observable uncertainty relations and their experimental demonstrations
• Strong unitary uncertainty relations
• Uncertainty regions of observables and state-independent uncertainty relations
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Probability density functions of quantum mechanical observable uncertainties
• Uncertainty relation and the constrained quadratic programming
• Multi-observable Uncertainty Equality based on the sum of standard deviations in the qubit system