State-independent uncertainty relations and entanglement detection

  • Chen Qian
  • Jun-Li Li
  • Cong-Feng Qiao


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 the 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.


Uncertainty relations Entanglement detection Bloch vector Horn’s inequalities 



This work was supported in part by the Ministry of Science and Technology of the People’s Republic of China(2015CB856703); by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No.XDB23030100; and by the National Natural Science Foundation of China(NSFC) under the Grants 11375200 and 11635009.


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Authors and Affiliations

  1. 1.Department of PhysicsUniversity of Chinese Academy of SciencesBeijingChina
  2. 2.Key Laboratory of Vacuum PhysicsUniversity of Chinese Academy of SciencesBeijingChina

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