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State-independent uncertainty relations and entanglement detection

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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 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.

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Acknowledgements

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. Department of Physics, University of Chinese Academy of Sciences, YuQuan Road 19A, Beijing, 100049, China

    Chen Qian, Jun-Li Li & Cong-Feng Qiao

  2. Key Laboratory of Vacuum Physics, University of Chinese Academy of Sciences, Beijing, China

    Chen Qian, Jun-Li Li & Cong-Feng Qiao

Authors
  1. Chen Qian
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  2. Jun-Li Li
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  3. Cong-Feng Qiao
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Correspondence to Cong-Feng Qiao.

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Qian, C., Li, JL. & Qiao, CF. State-independent uncertainty relations and entanglement detection. Quantum Inf Process 17, 84 (2018). https://doi.org/10.1007/s11128-018-1855-4

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  • DOI: https://doi.org/10.1007/s11128-018-1855-4

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