International Journal of Theoretical Physics

, Volume 57, Issue 5, pp 1498–1515 | Cite as

Λ-Nonlocality of Multipartite States and the Related Nonlocality Inequalities

  • Ying Yang
  • Huai-xin Cao
  • Liang Chen
  • Yongfeng Huang


Correlations between subsystems of a composite quantum system include Bell nonlocality, steerability, entanglement and quantum discord. Bell nonlocality of a bipartite state is one of important quantum correlations demonstrated by some local quantum measurements. In this paper, we discuss nonlocality of a multipartite quantum system. The Λ-locality and Λ-nonlocality of multipartite states are firstly introduced, some related properties are discussed. Some related nonlocality inequalities are established for {1,2;3}-local, {1;2,3}-local, and Λ-local states, respectively. The violation of one of these inequalities gives a sufficient condition for Λ-nonlocal states. As application, genuinely nonlocality of a tripartite state is checked. Finally, a class of 2-separable nonlocal states are given, which shows that a 2-separable tripartite state is not necessarily local.


Λ-Locality Λ-nonlocality Nonlocality inequality Multipartite state 



This subject was supported by the National Natural Science Foundation of China (Nos. 11371012, 11401359,11571213,11601300)


  1. 1.
    Bell, J.S.: Speakable and unspeakable in quantum mechanics. Physics 1, 195 (1964)CrossRefGoogle Scholar
  2. 2.
    Ekert, A.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Acin, A., Brunner, N., Gisin, N., et al.: Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Buhrman, H., Cleve, R., Massar, S., Wolf, R.: Nonlocality and communication complexity. Rev. Mod. Phys. 82, 665 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    Bardyn, C.E., Liew, T.C.H., Massar, S., et al.: Device-independent state estimation based on Bell’s inequalities. Phys. Rev. A 80, 062327 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    Pironio, S., Acín, A., Massar, S., et al.: Random numbers certified by Bell’s theorem. Nature 464, 1021–1024 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Bell, J.S.: Speakable and unspeakable in quantum mechanics. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  8. 8.
    Genovese, M.: Research on hidden variable theories: A review of recent progresses. Phys. Rep. 413, 319 (2005)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Aspect, A.: Bell’s inequality test: more ideal than ever. Nature 398, 189 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    Barrett, J., Linden, N., Massar, S., et al.: Nonlocal correlations as an information-theoretic resource. Phys. Rev. A 71, 022101 (2005)ADSCrossRefGoogle Scholar
  11. 11.
    Brukner C̆, żukowski, M., Pan, J.W., et al.: Bell’s inequalities and quantum communication complexity. Phys. Rev. Lett. 92, 127901 (2004)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Barrett, J., Hardy, L., Kent, A.: No-signaling and quantum key distribution. Phys. Rev. Lett. 95, 010503 (2005)ADSCrossRefGoogle Scholar
  13. 13.
    Masanes, L.: Universally composable privacy amplification from causality constraints. Phys. Rev. Lett. 102, 140501 (2009)ADSCrossRefGoogle Scholar
  14. 14.
    Du̇r, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature 416, 608 (2002)ADSCrossRefGoogle Scholar
  17. 17.
    Vidal, G.: Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    Popescu, S., Rohrlich, D.: Generic quantum nonlocality. Phys. Lett. A 166, 293 (1992)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Gisin, N., Bechmann-Pasquinucci, H.: Bell inequality, Bell states and maximally entangled states for n qubits. Phys. Lett. A 246, 1 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mermin, N.D.: Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65, 1838 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Saha, D., Cabello, A., Choudhary, S.K., Pawlowski, M.: Quantum nonlocality via local contextuality with qubit-qubit entanglement. Phys. Rev. A. 93, 042123 (2016)ADSCrossRefGoogle Scholar
  22. 22.
    Svetlichny, G.: Distinguishing three-body from two-body nonseparability by a Bell-type inequality. Phys. Rev. D 35, 3066 (1987)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Gao, T., Hong, Y., Lu, Y., Yan, F.L.: Efficient k-separability criteria for mixed multipartite quantum states. Euro. Phys. Lett. 104, 20007 (2013)ADSCrossRefGoogle Scholar
  24. 24.
    Wang, Y.Z., Hou, J.C.: Some necessary and sufficient conditions for k-separability of multipartite pure states. Quan. Inf. Proc. 14, 3711–3722 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Ying Yang
    • 1
    • 2
  • Huai-xin Cao
    • 1
  • Liang Chen
    • 1
    • 3
  • Yongfeng Huang
    • 1
    • 3
  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina
  2. 2.Department of Applied MathematicsYuncheng CollegeYunchengChina
  3. 3.Department of MathematicsChangji CollegeChangjiChina

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