Quantum Information Processing

, Volume 15, Issue 4, pp 1585–1599 | Cite as

Quantifying nonclassicality of correlations based on the concept of nondisruptive local state identification

  • Azam Kheirollahi
  • Seyed Javad Akhtarshenas
  • Hamidreza Mohammadi


A bipartite state is classical with respect to party A if and only if party A can perform nondisruptive local state identification (NDLID) by a projective measurement. Motivated by this we introduce a class of quantum correlation measures for an arbitrary bipartite state. The measures utilize the general Schatten p-norm to quantify the amount of departure from the necessary and sufficient condition of classicality of correlations provided by the concept of NDLID. We show that for the case of Hilbert–Schmidt norm, i.e., \(p=2\), a closed formula is available for an arbitrary bipartite state. The reliability of the proposed measures is checked from the information-theoretic perspective. Also, the monotonicity behavior of these measures under LOCC is exemplified. The results reveal that for the general pure bipartite states these measures have an upper bound which is an entanglement monotone in its own right. This enables us to introduce a new measure of entanglement, for a general bipartite state, by convex roof construction. Some examples and comparison with other quantum correlation measures are also provided.


Quantum correlation Nondisruptive local state identification Schatten p-norm Entanglement monotone 



The authors wish to thank The Office of Graduate Studies of The University of Isfahan for their support.


  1. 1.
    Barnnett, S.: Quantum Information. Oxford University Press, New York (2009)Google Scholar
  2. 2.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A Math. Gen. 34, 6899–6905 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    Céleri, L.C., Maziero, J., Serra, R.M.: Theoretical and experimental aspects of quantum discord and related measures. Int. J. Quantum Inf. 9, 1837–1873 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Luo, S., Fu, S.: Measurement-induced nonlocality. Phys. Rev. Lett. 106, 120401 (2011)ADSCrossRefMATHGoogle Scholar
  7. 7.
    Dakic, B., Vedral, V., Brukner, C.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)ADSCrossRefMATHGoogle Scholar
  8. 8.
    Akhtarshenas, S.J., Mohammadi, H., Karimi, S., Azmi, Z.: Computable measure of quantum correlation. Quantum Inf. Process. 14, 247–267 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Li, B., Fei, S.-M., Wang, Z.-X., Fan, H.: Assisted state discrimination without entanglement. Phys. Rev. A 85, 022328 (2012)ADSCrossRefGoogle Scholar
  10. 10.
    Knill, E., Laflamme, R.: Power of one bit of quantum information. Phys. Rev. Lett. 81, 5672–5675 (1998)ADSCrossRefGoogle Scholar
  11. 11.
    Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    Fanchini, F.F., Cornelio, M.F., de Oliveira, M.C., Caldeira, A.O.: Conservation law for distributed entanglement of formation and quantum discord. Phys. Rev. A 84, 012313 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    Fazio, R., Modi, K., Pascazio, S., Vedral, V., Yuasu, K.: Witnessing the quantumness of a single system: from anticommutators to interference and discord. Phys. Rev. A 87, 052132 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Piani, M., Horodecki, P., Horodecki, R.: Nonlocal-broadcasting theorem for multipartite quantum correlations. Phys. Rev. Lett. 100, 090502 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    Piani, M., Christandl, M., Mora, C.E., Horodecki, P.: Broadcast copies reveal the quantumness of correlations. Phys. Rev. Lett. 102, 250503 (2009)ADSCrossRefMATHGoogle Scholar
  16. 16.
    Luo, S.: On quantum no-broadcasting. Lett. Math. Phys. 92, 143–153 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Modi, K., Cable, H., Williamson, M., Vedral, V.: Quantum correlations in mixed-state metrology. Phys. Rev. X 1, 021022 (2011)Google Scholar
  18. 18.
    Horodecki, M., Oppenheim, J., Winter, A.: Partial quantum information. Nature 436, 673–676 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Cavalcanti, D., Aolita, L., Boixo, S., Modi, K., Piani, M., Winter, A.: Operational interpretations of quantum discord. Phys. Rev. A 83, 032324 (2011)ADSCrossRefMATHGoogle Scholar
  20. 20.
    Madhok, V., Datta, A.: Interpreting quantum discord through quantum state merging. Phys. Rev. A 83, 032323 (2011)ADSCrossRefGoogle Scholar
  21. 21.
    Zurek, W.H.: Quantum discord and Maxwell’s demons. Phys. Rev. A 67, 012320 (2003)ADSCrossRefGoogle Scholar
  22. 22.
    Brodutch, A., Terno, D.R.: Quantum discord, local operations, and Maxwell’s demons. Phys. Rev. A 81, 062103 (2010)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lanyon, B.P., Barbieri, M., Almeida, M.P., White, A.G.: Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501 (2008)ADSCrossRefGoogle Scholar
  24. 24.
    Xu, J.-S., Xu, X.-Y., Li, C.-F., Zhang, C.-J., Zou, X.-B., Guo, G.-C.: Experimental investigation of classical and quantum correlations under decoherence. Nat. Commun. 1, 1–6 (2010)Google Scholar
  25. 25.
    Rahimi, R., SaiToh, A.: Single-experiment-detectable nonclassical correlation witness. Phys. Rev. A 82, 022314 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    Yu, S., Zhang, C., Chen, Q., Oh, C.H.: Witnessing the quantum discord of all the unknown states (2011). arXiv:1102.4710
  27. 27.
    Auccaise, R., Maziero, J., Celeri, L.C., Soares-Pinto, D.O., deAzevedo, E.R., Bonagamba, T.J., Sarthour, R.S., Oliveira, I.S., Serra, R.M.: Experimental witnessing the quantumness of correlations. Phys. Rev. Lett. 107, 070501 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    Auccaise, R., Celeri, L.C., Soares-Pinto, D.O., deAzevedo, E.R., Maziero, J., Souza, A.M., Bonagamba, T.J., Sarthour, R.S., Oliveira, I.S., Serra, R.M.: Environment-induced sudden transition in quantum discord dynamics. Phys. Rev. Lett. 107, 140403 (2011)ADSCrossRefGoogle Scholar
  29. 29.
    Dakic, B., Ole Lipp, Y., Ma, X., Ringbauer, M., Kropatschek, S., Barz, S., Paterek, T., Vedral, V., Zeilinger, A., Brukner, C., Walther, P.: Quantum discord as resource for remote state preparation. Nat. Phys. 8, 666–670 (2012)CrossRefGoogle Scholar
  30. 30.
    Ferraro, A., Aolita, L., Cavalcanti, D., Cucchietti, F.M., Acin, A.: Almost all quantum states have nonclassical correlations. Phys. Rev. A 81, 052318 (2010)ADSCrossRefGoogle Scholar
  31. 31.
    Chen, L., Chitambar, E., Modi, K., Vacanti, G.: Detecting multipartite classical states and their resemblances. Phys. Rev. A 83, 020101 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    Wu, Y.C., Guo, G.C.: Norm-based measurement of quantum correlation. Phys. Rev. A 83, 062301 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    Guo, Z., Cao, H., Chen, Z.: Distinguishing classical correlations from quantum correlations. J. Phys. A Math. Theor. 45, 145301 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Brodutch, A., Modi, K.: Criteria for measures of quantum correlations. Quantum Inf. Comput. 12, 721–742 (2012)MathSciNetMATHGoogle Scholar
  35. 35.
    Schatten, R.: Norm Ideals of Completely Continuous Operators. Springer, Berlin (1970)CrossRefMATHGoogle Scholar
  36. 36.
    Georgi, H.: Lie Algebras in Particle Physics. Advanced Book Program, Reading (1999)Google Scholar
  37. 37.
    Bengtsson, I., Życzkowski, K.: Geometry of Quantum States. Cambridge University Press, New York (2006)CrossRefMATHGoogle Scholar
  38. 38.
    Buchleitner, A., Viviescas, C., Tiersch, M.: Entanglement and Decoherence. Springer, Berlin (2009)CrossRefMATHGoogle Scholar
  39. 39.
    Uhlmann, A.: Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open Syst. Inf. Dyn. 5, 209–227 (1998)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)ADSCrossRefGoogle Scholar
  41. 41.
    Piani, M.: Problem with geometric discord. Phys. Rev. A 86, 034101 (2012)ADSCrossRefGoogle Scholar
  42. 42.
    Paula, F.M., de Oliveira, T.R., Sarandy, M.S.: Geometric quantum discord through the Schatten 1-norm. Phys. Rev. A 87, 064101 (2013)ADSCrossRefGoogle Scholar
  43. 43.
    Ciccarello, F., Tufarelli, T., Giovannetti, V.: Toward computability of trace distance discord. New J. Phys. 16, 013038 (2014)ADSCrossRefGoogle Scholar
  44. 44.
    Abad, T., Karimipour, V., Memarzadeh, L.: Power of quantum channels for creating quantum correlations. Phys. Rev. A 86, 062316 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Azam Kheirollahi
    • 1
  • Seyed Javad Akhtarshenas
    • 2
  • Hamidreza Mohammadi
    • 1
    • 3
  1. 1.Department of PhysicsUniversity of IsfahanIsfahanIran
  2. 2.Department of PhysicsFerdowsi University of MashhadMashhadIran
  3. 3.Quantum Optics GroupUniversity of IsfahanIsfahanIran

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