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Generalized approach to quantify correlations in bipartite quantum systems

  • D. G. Bussandri
  • A. P. MajteyEmail author
  • P. W. Lamberti
  • T. M. Osán
Article
  • 84 Downloads

Abstract

In this work, we developed a general approach to the problem of detecting and quantifying different types of correlations in bipartite quantum systems. Our method is based on the use of distances between quantum states and processes. We rely upon the premise that total correlations can be separated into classical and quantum contributions due to their different nature. In addition, according to recently discussed criteria, we determined the requirements to be satisfied by distances in order to generate correlation measures physically well behaved. The proposed measures allow us to quantify quantum, classical and total correlations. Besides the well-known case of relative entropy, we introduce some additional examples of distances which can be used to build bona fide quantifiers of correlations.

Keywords

Quantum correlations Quantum discord Correlations measures Distance measures Bipartite quantum systems 

Notes

Acknowledgements

D.B., A.P.M., P.W.L. and T.M.O acknowledge the Argentinian agency SeCyT-UNC and CONICET for financial support. D. B. has a fellowship from CONICET.

References

  1. 1.
    Vedral, V.: Quantum entanglement. Nat. Phys. 10, 256 (2014)CrossRefGoogle Scholar
  2. 2.
    Yu, T., Eberly, J.H.: The end of an entanglement. Science 316, 555 (2007)CrossRefGoogle Scholar
  3. 3.
    Yu, T., Eberly, J.H.: Sudden death of entanglement. Science 323, 598 (2009)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Plenio, M., Virmani, S.: An introduction to entanglement measures. Quantum Inf. Comput. 7, 1 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brunner, N., Calvacanti, D., Pironio, F., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419 (2014)ADSCrossRefGoogle Scholar
  6. 6.
    Orieaux, A., D’Arrigo, A., Ferranti, G., Lo Franco, R., Benenti, G., Paladino, E., Falci, G., Sciarrino, F., Mataloni, P.: Experimental on-demand recovery of entanglement by local operations within non-Markovian dynamics. Sci. Rep. 5, 8575 (2015)ADSCrossRefGoogle Scholar
  7. 7.
    Man, Z.-X., Xia, Y.-J., Lo Franco, R.: Cavity-based architecture to preserve quantum coherence and entanglement. Sci. Rep. 5, 13843 (2015)ADSCrossRefGoogle Scholar
  8. 8.
    Dijkstra, A.G., Tanimura, Y.: Non-Markovian entanglement dynamics in the presence of system-bath coherence. Phys. Rev. Lett. 104, 250401 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    de Aolina, L., Melo, F., Davidovich, L.: Open-system dynamics of entanglement: a key issues review. Rep. Prog. Phys. 78, 042001 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    D’Arrigo, A., Lo Franco, R., Benenti, G., Paladino, E., Falci, G.: Hidden entanglement in the presence of random telegraph dephasing noise. Phys. Scr. T153, 014014 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    Xu, J.-S., Sun, K., Li, C.-F., Xu, X.-Y., Guo, G.-C., Andersson, E., Lo Franco, R., Compagno, G.: Experimental recovery of quantum correlations in absence of system-environment back-action. Nat. Commun. 4, 2851 (2013)CrossRefGoogle Scholar
  12. 12.
    D’Arrigo, A., Benenti, G., Lo Franco, R., Falci, G.: Hidden entanglement, system-environment information flow and non-Markovianity. Int. J. Quantum Inf. 12, 1461005 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lo Franco, R.: Nonlocality threshold for entanglement under general dephasing evolutions: a case study. Quantum Inf. Process. 15, 2393 (2016)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Rab, A.S., Polino, E., Man, Z.-X., An, N.B., Xia, Y.-J., Spagnolo, N., Lo Franco, R., Sciarrino, F.: Entanglement of photons in their dual wave-particle nature. Nat. Commun. 8, 915 (2017)ADSCrossRefGoogle Scholar
  15. 15.
    Knill, E., Laflamme, R.: Power of one bit of quantum information. Phys. Rev. Lett. 81, 5672 (1998)ADSCrossRefGoogle Scholar
  16. 16.
    Laflamme, R., Cory, D.G., Negrevergne, C., Viola, L.: NMR quantum information processing and entanglement. Quantum Inf. Comput. 2, 166 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Braunstein, S.L., Caves, C.M., Jozsa, R., Linden, N., Popescu, S., Schack, R.: Separability of very noisy mixed states and implications for NMR quantum computing. Phys. Rev. Lett. 83, 1054 (1999)ADSCrossRefGoogle Scholar
  18. 18.
    Meyer, D.A.: Sophisticated quantum search without entanglement. Phys. Rev. Lett. 85, 2014 (2000)ADSCrossRefGoogle Scholar
  19. 19.
    Datta, A., Flammia, S.T., Caves, C.M.: Entanglement and the power of one qubit. Phys. Rev. A 72, 042316 (2005)ADSCrossRefGoogle Scholar
  20. 20.
    Datta, A., Vidal, G.: Role of entanglement and correlations in mixed-state quantum computation. Phys. Rev. A 75, 042310 (2007)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)ADSCrossRefGoogle Scholar
  22. 22.
    Lanyon, B.P., Barbieri, M., Almeida, M.P., White, A.G.: Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501 (2008)ADSCrossRefGoogle Scholar
  23. 23.
    Silva, I.A., Souza, A.M., Bromley, T.R., Cianciaruso, M., Marx, R., Sarthour, R.S., Oliveira, I.S., Lo Franco, R., Glaser, S.J., deAzevedo, E.R., Soares-Pinto, D.O., Adesso, G.: Observation of time-invariant coherence in a nuclear magnetic resonance quantum simulator. Phys. Rev. Lett. 117, 160402 (2016)ADSCrossRefGoogle Scholar
  24. 24.
    Aaronson, B., Lo Franco, R., Adesso, G.: Comparative investigation of the freezing phenomena for quantum correlations under nondissipative decoherence. Phys. Rev. A 88, 012120 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    Aaronson, B., Lo Franco, R., Compagno, G., Adesso, G.: Hierarchy and dynamics of trace distance correlations. New J. Phys. 15, 093022 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    Haikka, P., Johnson, T.H., Maniscalco, S.: Non-Markovianity of local dephasing channels and time-invariant discord. Phys. Rev. A 87, 010103(R) (2013)ADSCrossRefGoogle Scholar
  27. 27.
    Cianciaruso, M., Bromley, T.R., Roga, W., Lo Franco, R., Adesso, G.: Universal freezing of quantum correlations within the geometric approach. Sci. Rep. 5, 10177 (2015)ADSCrossRefGoogle Scholar
  28. 28.
    Bennett, C.H., DiVincenzo, D.P., Fuchs, C.A., Mor, T., Rains, E., Shor, P.W., Smolin, J.A., Wootters, W.K.: Mixed state entanglement and quantum error correction. Phys. Rev. A 59, 1070 (1999)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Groisman, B., Popescu, S., Winter, A.: Quantum, classical, and total amount of correlations in a quantum state. Phys. Rev. A 72, 032317 (2005)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A 34, 6899 (2001)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Oppenheim, J., Horodecki, M., Horodecki, P., Horodecki, R.: Thermodynamical approach to quantifying quantum correlations. Phys. Rev. Lett. 89, 180402 (2002)ADSCrossRefGoogle Scholar
  32. 32.
    Yang, D., Horodecki, M., Wang, Z.D.: An additive and operational entanglement measure: conditional entanglement of mutual information. Phys. Rev. Lett. 101, 140501 (2008)ADSCrossRefGoogle Scholar
  33. 33.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefGoogle Scholar
  34. 34.
    Luo, S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)ADSCrossRefGoogle Scholar
  35. 35.
    Modi, K., Paterek, T., Son, W., Vedral, V., Williamson, M.: Unified view of quantum and classical correlations. Phys. Rev. Lett. 104, 080501 (2010)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Piani, M.: Problem with geometric discord. Phys. Rev. A 86, 034101 (2012)ADSCrossRefGoogle Scholar
  37. 37.
    Tufarelli, T., MacLean, T.M., Girolami, D., Vasile, R., Adesso, G.: The geometric approach to quantum correlations: computability versus reliability. J. Phys. A Math. Theor. 46, 275308 (2013)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Paula, F.M., Saguia, A., De Oliveira, T.R., Sarandy, M.S.: Overcoming ambiguities in classical and quantum correlation measures. EPL 108, 10003 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    Lang, M.D., Caves, C.M.: Quantum discord and the geometry of bell-diagonal states. Phys. Rev. Lett. 105, 150501 (2010)ADSCrossRefGoogle Scholar
  40. 40.
    Cen, L.-X., Li, X.Q., Shao, J., Yan, Y.J.: Quantifying quantum discord and entanglement of formation via unified purifications. Phys. Rev. A 83, 054101 (2011)ADSCrossRefGoogle Scholar
  41. 41.
    Adesso, G., Datta, A.: Quantum versus classical correlations in Gaussian states. Phys. Rev. Lett. 105, 030501 (2010); Giorda, P., Paris, M.G.A.: Gaussian Quantum Discord. ibid 105, 020503 (2010)Google Scholar
  42. 42.
    Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010); Ali, M., Rau, A.R.P., Alber G.: ibid. 82, 069902(E) (2010)Google Scholar
  43. 43.
    Shi, M., Yang, W., Jiang, F., Du, J.: Quantum discord of two-qubit rank-2 states. J. Phys. A Math. Theor. 44, 415304 (2011)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Chen, Q., Zhang, C., Yu, S., Yi, X.X., Oh, C.H.: Quantum discord of two-qubit X states. Phys. Rev. A 84, 042313 (2011)ADSCrossRefGoogle Scholar
  45. 45.
    Lu, X.M., Ma, J., Xi, Z., Wang, X.: Optimal measurements to access classical correlations of two-qubit states. Phys. Rev. A 83, 012327 (2011)ADSCrossRefGoogle Scholar
  46. 46.
    Girolami, D., Adesso, G.: Quantum discord for general two-qubit states: analytical progress. Phys. Rev. A 83, 052108 (2011)ADSCrossRefGoogle Scholar
  47. 47.
    Li, B., Wang, Z.X., Fei, S.M.: Quantum discord and geometry for a class of two-qubit states. Phys. Rev. A 83, 022321 (2011)ADSCrossRefGoogle Scholar
  48. 48.
    Huang, Y.: Computing quantum discord is NP-complete. New J. Phys. 16, 033027 (2014)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Dakić, B., Vedral, V., Brukner, Č.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)ADSCrossRefGoogle Scholar
  50. 50.
    Gessner, M., Laine, E.-M., Breuer, H.-P., Piilo, J.: Correlations in quantum states and the local creation of quantum discord. Phys. Rev. A 85, 052122 (2012)ADSCrossRefGoogle Scholar
  51. 51.
    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 (2012)ADSCrossRefGoogle Scholar
  52. 52.
    Luo, S.: Using measurement-induced disturbance to characterize correlations as classical or quantum. Phys. Rev. A 77, 022301 (2008)ADSCrossRefGoogle Scholar
  53. 53.
    Modi, K., Vedral, V.: Unification of quantum and classical correlations and quantumness measures. AIP Conf. Proc. 1384, 69 (2011)ADSCrossRefGoogle Scholar
  54. 54.
    Brodutch, A., Modi, K.: Criteria for measures of quantum correlations. Quantum Inf. Comput. 12, 721–742 (2012)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Adesso, G., Bromley, T.R., Cianciaruso, M.: Measures and applications of quantum correlations. J. Phys. A Math. Theor. 49, 473001 (2016)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  57. 57.
    Hayashi, M.: Quantum Information: An Introduction. Springer, Berlin (2006)zbMATHGoogle Scholar
  58. 58.
    Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  59. 59.
    Uhlmann, A.: The transition probability in the state space of a \(\star \)-algebra. Rep. Math. Phys. 9, 273 (1976)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt 41, 2315 (1994)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    Bures, D.J.C.: An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite \(\text{ W }^*\)-algebras. Trans. Am. Math. Soc. 135, 199 (1969)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Dodonov, V.V., Man’ko, O.V., Man’ko, V.I., Wunsche, A.: Hilbert–Schmidt distance and non-classicality of states in quantum optics. J. Mod. Opt. 47, 633 (2000)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    Luo, S., Zhang, Q.: Informational distance on quantum-state space. Phys. Rev. A 69, 032106 (2004)ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    Majtey, A.P., Lamberti, P.W., Prato, D.P.: Jensen–Shannon divergence as a measure of distinguishability between mixed quantum states. Phys. Rev. A 72, 052310 (2005)ADSCrossRefGoogle Scholar
  65. 65.
    Spehner, D., Illuminati, F., Orszag, M., Roga, W.: Geometric measures of quantum correlations with Bures and Hellinger distances. In: Fanchini, F., Soares Pinto, D., Adesso, G. (eds.) Lectures on General Quantum Correlations and Their Applications, Quantum Science and Technology. Springer, Cham (2017)zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Facultad de Matemática, Astronomía, Física y ComputaciónUniversidad Nacional de CórdobaCórdobaArgentina
  2. 2.Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET)CABAArgentina
  3. 3.Instituto de Física Enrique Gaviola (IFEG)Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET)CórdobaArgentina

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