Advertisement

Geometric Measures of Quantum Correlations with Bures and Hellinger Distances

  • D. SpehnerEmail author
  • F. Illuminati
  • M. Orszag
  • W. Roga
Chapter
Part of the Quantum Science and Technology book series (QST)

Abstract

This chapter contains a survey of the geometric approach to quantum correlations. We focus mainly on the geometric measures of quantum correlations based on the Bures and quantum Hellinger distances.

Notes

Acknowledgements

We acknowledge support from the French ANR project No. ANR-13-JS01-0005-01, the EU FP7 Cooperation STREP Projects iQIT No. 270843 and EQuaM No. 323714, the Italian Minister of Scientific Research (MIUR) national PRIN programme, and the Chilean Fondecyt project No. 1140994.

References

  1. 1.
    B. Aaronson, R.L. Franco, G. Adesso, Comparative investigation of the freezing phenomena for quantum correlations under nondissipative decoherence. Phys. Rev. A 88, 012120 (2013)ADSCrossRefGoogle Scholar
  2. 2.
    T. Abad, V. Karimipour, L. Memarzadeh, Power of quantum channels for creating quantum correlations. Phys. Rev. A 86, 062316 (2012)ADSCrossRefGoogle Scholar
  3. 3.
    M. Ali, A.R.P. Rau, G. Alber, Quantum discord for two-qubit \(X\) states. Phys. Rev. A 81, 042105 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    H. Araki, A remark on Bures distance function for normal states. Publ. RIMS Kyoto Univ. 6, 477–482 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    K.M.R. Audenaert, N. Datta, \(\alpha \)-\(z\)-relative Rényi entropies. J. Math. Phys. 56, 022202 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    K.M.R. Audenaert, J. Calsamiglia, R. Muñoz-Tapia, E. Bagan, L.I. Masanes, A. Acin, F. Verstraete, Discriminating States: The Quantum Chernoff Bound. Phys. Rev. Lett. 98, 160501 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    R. Balian, The entropy-based quantum metric. Entropy 2014 16(7), 3878–3888 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    R. Balian, Y. Alhassid, H. Reinhardt, Dissipation in many-body systems: a geometric approach based on information theory. Phys. Rep. 131, 1 (1986)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Ban, K. Kurokawa, R. Momose, O. Hirota, Optimum measurements for discrimination among symmetric quantum states and parameter estimation. Int. J. Theor. Phys. 36, 1269–1288 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    S.M. Barnett, Minimum error discrimination between multiply symmetric states. Phys. Rev. A 64, 030303 (2001)ADSCrossRefGoogle Scholar
  11. 11.
    H. Barnum, E. Knill, Reversing quantum dynamics with near-optimal quantum and classical fidelity. J. Math. Phys. 43, 2097–2106 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)ADSCrossRefGoogle Scholar
  13. 13.
    C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    T. Benoist, V. Jaks̆ić, Y. Pautrat, C.-A. Pillet, On entropy production of repeated quantum measurements I. General theory, arXiv:1607.00162 [math-ph]
  15. 15.
    J.A. Bergou, U. Herzog, M. Hillery, Discrimination of quantum states, in Quantum State Estimation, vol. 649, Lecture Notes in Physics, ed. by M. Paris, J. Rehacek (Springer, Berlin, 2004), pp. 417–465CrossRefGoogle Scholar
  16. 16.
    R. Bhatia, Matrix Analysis (Springer, Berlin, 1991)zbMATHGoogle Scholar
  17. 17.
    S.L. Braunstein, C.M. Caves, Statistical distance and the geometry of quantum states. Phys. Rev. Lett 72, 3439–3443 (1994)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    T.R. Bromley, M. Cianciaruso, R. Lo Franco, G. Adesso, Unifying approach to the quantification of bipartite correlations by Bures distance. J. Phys. A: Math. Theor. 47, 405302 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    D. Bures, An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite \( w^{\ast } \)-algebras. Trans. Am. Math. Soc. 135, 199–212 (1969)MathSciNetzbMATHGoogle Scholar
  20. 20.
    D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, A. Winter, Operational interpretations of quantum discord. Phys. Rev. A 83, 032324 (2011)ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    N.N. Cencov, Statistical Decision Rules and Optimal Interferences, vol. 53, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1982)Google Scholar
  22. 22.
    L. Chang, S. Luo, Remedying the local ancilla problem with geometric discord. Phys. Rev. A 87, 062303 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    C.-L. Chou, L.Y. Hsu, Minimal-error discrimination between symmetric mixed quantum states. Phys. Rev. A 68, 042305 (2003)ADSCrossRefGoogle Scholar
  24. 24.
    F. Ciccarello, T. Tufarelli, V. Giovannetti, Towards computability of trace distance discord. New J. Phys. 16, 013038 (2014)ADSCrossRefGoogle Scholar
  25. 25.
    B. Dakić, V. Vedral, C. Brukner, Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)ADSzbMATHCrossRefGoogle Scholar
  26. 26.
    A. Datta, S.T. Flammia, C.M. Caves, Entanglement and the power of one qubit. Phys. Rev. A 72, 042316 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    A. Datta, A. Shaji, C.M. Caves, Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    Y.C. Eldar, von Neumann measurement is optimal for detecting linearly independent mixed quantum states. Phys. Rev. A 68, 052303 (2003)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Y.C. Eldar, G.D. Forney Jr., On quantum detection and the square-root measurement. IEEE Trans. Inf. Theory 47, 858–872 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    B.M. Escher, R.L. de Matos Filho, L. Davidovish, General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys. 7, 406–411 (2011)Google Scholar
  31. 31.
    R.L. Frank, E.H. Lieb, Monotonicity of a relative Rényi entropy. J. Math. Phys. 54, 122201 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    S. Gharibian, Quantifying nonclassicality with local unitary operations. Phys. Rev. A 86, 042106 (2012)ADSCrossRefGoogle Scholar
  33. 33.
    S.M. Giampaolo, F. Illuminati, Characterization of separability and entanglement in (\(2 \times D\))- and (\(3\times D\))-dimensional systems by single-qubit and single-qutrit unitary transformations. Phys. Rev. A 76, 042301 (2007)ADSCrossRefGoogle Scholar
  34. 34.
    S.M. Giampaolo, A. Streltsov, W. Roga, D. Bruß, F. Illuminati, Quantifying nonclassicality: global impact of local unitary evolutions. Phys. Rev. A 87, 012313 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    D. Girolami, G. Adesso, Quantum discord for general two-qubit states: analytical progress. Phys. Rev. A 83, 052108 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    D. Girolami, T. Tufarelli, G. Adesso, Characterizing nonclassical correlations via local quantum uncertainty. Phys. Rev. Lett. 110, 240402 (2013)ADSCrossRefGoogle Scholar
  37. 37.
    D. Girolami, A.M. Souza, V. Giovannetti, T. Tufarelli, J.G. Filgueiras, R.S. Sarthour, D.O. Soares-Pinto, I.S. Oliveira, G. Adesso, Quantum discord determines the interferometric power of quantum states. Phys. Rev. Lett. 112, 210401 (2014)ADSCrossRefGoogle Scholar
  38. 38.
    P. Hausladen, W.K. Wootters, A “pretty good” measurement for distinguishing quantum states. J. Mod. Opt. 41, 2385–2390 (1994)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys. 246, 359–374 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)zbMATHGoogle Scholar
  41. 41.
    L. Henderson, V. Vedral, Classical, quantum and total correlations. J. Phys. A: Math. Gen. 34, 6899–6905 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    A.S. Holevo, On quasiequivalence of locally normal states. Theor. Math. Phys. 13(2), 1071–1082 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    A.S. Holevo, On asymptotically optimal hypothesis testing in quantum statistics. Theory Probab. Appl. 23, 411–415 (1979)zbMATHCrossRefGoogle Scholar
  44. 44.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Y. Huang, Quantum discord for two-qubit \(X\) states: analytical formula with very small worst-case error. Phys. Rev. A 88, 014302 (2013)ADSCrossRefGoogle Scholar
  46. 46.
    Y. Huang, Computing quantum discord is NP-complete. New J. Phys. 16, 033027 (2014)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    M. Hübner, Explicit computation of the Bures distance for density matrices. Phys. Lett. A 163, 239–242 (1992)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, A. Smerzi, Fisher information and multiparticle entanglement. Phys. Rev. A 85, 022321 (2012)ADSCrossRefGoogle Scholar
  49. 49.
    V. Jaks̆ić, C.-A. Pillet, Entropic Functionals in Quantum Statistical Mechanics, in Proceedings of XVIIth International Congress of Mathematical Physics (Aalborg 2012) (World Scientific, Singapore, 2013), 336–343Google Scholar
  50. 50.
    R. Jozsa, Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–2323 (1994)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    E.H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math. 11, 267–288 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    E.H. Lieb, M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    S. Luo, Wigner-Yanase skew information and uncertainty relations. Phys. Rev. Lett. 91, 180403 (2003)ADSCrossRefGoogle Scholar
  54. 54.
    S. Luo, Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)ADSCrossRefGoogle Scholar
  55. 55.
    S. Luo, S. Fu, Geometric measure of quantum discord. Phys. Rev. A 82, 034302 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    V. Madhok, A. Datta, Interpreting quantum discord through quantum state merging. Phys. Rev. A 83, 032323 (2011)ADSCrossRefGoogle Scholar
  57. 57.
    P. Marian, T.A. Marian, Hellinger distance as a measure of gaussian discord. J. Phys. A: Math. Theor. 48, 115301 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    J.A. Miszczak, Z. Puchala, P. Horodecki, A. Uhlmann, K. \(\dot{\rm Z}\)yczkowski, Sub- and super-fidelity as bounds for quantum fidelity. Quantum Inf. Comput. 9(1–2), 0103–0130 (2009)Google Scholar
  59. 59.
    K. Modi, T. Parerek, W. Son, V. Vedral, M. Williamson, Unified view of quantum and classical correlations. Phys. Rev. Lett. 104, 080501 (2010)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    A. Monras, G. Adesso, S.M. Giampaolo, G. Gualdi, G.B. Davies, F. Illuminati, Entanglement quantification by local unitary operations. Phys. Rev. A 84, 012301 (2011)ADSCrossRefGoogle Scholar
  62. 62.
    M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, M. Tomamichel, On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54, 122203 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    T. Nakano, M. Piani, G. Adesso, Negativity of quantumness and its interpretations. Phys. Rev. A 88, 012117 (2013)ADSCrossRefGoogle Scholar
  64. 64.
    N.A. Nielsen, I.L. Chuang, Quantum Computation and Information (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
  65. 65.
    M. Nussbaum, A. Szkola, The Chernoff lower bound for symmetric quantum hypothesis testing, vol. 37, The Annals of Statistics (Institute of Mathematical Statistics, 2009), pp. 1040–1057Google Scholar
  66. 66.
    H. Ollivier, W.H. Zurek, Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSzbMATHCrossRefGoogle Scholar
  67. 67.
    M. Ozawa, Entanglement measures and the Hilbert-Schmidt distance. Phys. Lett. A 268, 158–160 (2000)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    F.M. Paula, T.R. de Oliveira, M.S. Sarandy, Geometric quantum discord through the Schatten 1-norm. Phys. Rev. A 87, 064101 (2013)ADSCrossRefGoogle Scholar
  69. 69.
    D. Pérez-Garcia, M.M. Wolf, D. Petz, M.B. Ruskai, Contractivity of positive and trace-preserving maps under \(L^p\)-norms. J. Math. Phys. 47, 083506 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    D. Petz, Monotone metrics on matrix spaces. Lin. Alg. Appl. 244, 81–96 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    D. Petz, Monotonicity of quantum relative entropy revisited. Rev. Math. Phys. 15, 79–91 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    L. Pezzé, A. Smerzi, Entanglement, nonlinear dynamics, and the Heisenberg limit. Phys. Rev. Lett. 102, 100401 (2009)Google Scholar
  73. 73.
    M. Piani, Problem with geometric discord. Phys. Rev. A 86, 034101 (2012)ADSCrossRefGoogle Scholar
  74. 74.
    M. Piani, S. Gharibian, G. Adesso, J. Calsamiglia, P. Horodecki, A. Winter, All nonclassical correlations can be activated into distillable entanglement. Phys. Rev. Lett. 106, 220403 (2011)ADSCrossRefGoogle Scholar
  75. 75.
    M. Piani, V. Narasimhachar, J. Calsamiglia, Quantumness of correlations, quantumness of ensembles and quantum data hiding. New J. Phys. 16, 113001 (2014)ADSCrossRefGoogle Scholar
  76. 76.
    W. Roga, S.M. Giampaolo, F. Illuminati, Discord of response. J. Phys A: Math. Theor. 47, 365301 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    W. Roga, D. Buono, F. Illuminati, Device-independent quantum reading and noise-assisted quantum transmitters. New J. Phys. 17, 013031 (2015)ADSCrossRefGoogle Scholar
  78. 78.
    W. Roga, D. Spehner, F. Illuminati, Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations. J. Phys. A: Math. Theor. 49, 235301 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    M.B. Ruskai, Beyond strong subadditivity: improved bounds on the contraction of the generalized relative entropy. Rev. Math. Phys. 6(5a), 1147–1161 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    H.-J. Sommers, K. \(\dot{\rm Z}\)yczkowski, Bures volume of the set of mixed quantum states. J. Phys. A: Math. Gen. 36, 10083–10100 (2003)Google Scholar
  81. 81.
    D. Spehner, Quantum correlations and distinguishability of quantum states. J. Math. Phys. 55, 075211 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    D. Spehner, M. Orszag, Geometric quantum discord with Bures distance. New J. Phys. 15, 103001 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    D. Spehner, M. Orszag, Geometric quantum discord with Bures distance: the qubit case. J. Phys. A: Math. Theor. 47, 035302 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    W.F. Stinespring, Positive functions on \( C^*\)-algebras. Proc. Am. Soc. 6, 211–216 (1955)MathSciNetzbMATHGoogle Scholar
  85. 85.
    A. Streltsov, H. Kampermann, D. Bruß, Linking a distance measure of entanglement to its convex roof. New J. Phys. 12, 123004 (2010)ADSCrossRefGoogle Scholar
  86. 86.
    A. Streltsov, H. Kampermann, D. Bruß, Linking quantum discord to entanglement in a measurement. Phys. Rev. Lett. 106, 160401 (2011)ADSCrossRefGoogle Scholar
  87. 87.
    A. Streltsov, H. Kampermann, D. Bruß, Behavior of quantum correlations under local noise. Phys. Rev. Lett. 107, 170502 (2011)ADSCrossRefGoogle Scholar
  88. 88.
    A. Streltsov, G. Adesso, M. Piani, D. Bruß, Are general quantum correlations monogamous? Phys. Rev. Lett. 109, 050503 (2012)ADSCrossRefGoogle Scholar
  89. 89.
    G. Toth, Multipartite entanglement and high-precision metrology. Phys. Rev. A 85, 022322 (2012)ADSCrossRefGoogle Scholar
  90. 90.
    G. Tóth, D. Petz, Extremal properties of the variance and the quantum Fisher information. Phys. Rev. A 87, 032324 (2013)ADSCrossRefGoogle Scholar
  91. 91.
    A. Uhlmann, Endlich-dimensionale Dichtematrizen II. Wiss. Z. Karl-Marx-Univ. Leipzig, Math.-Nat R. 22, 139–177 (1973)Google Scholar
  92. 92.
    A. Uhlmann, The “transition probability” in the state space of a \(\ast \)-algebra. Rep. Math. Phys. 9, 273–279 (1976)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    A. Uhlmann, Parallel transport and “quantum holonomy” along density operators. Rep. Math. Phys. 24, 229–240 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    V. Vedral, M.B. Plenio, Entanglement measures and purifications procedures. Phys. Rev. A 57, 1619–1633 (1998)ADSCrossRefGoogle Scholar
  95. 95.
    V. Vedral, M.B. Plenio, M.A. Rippin, P.L. Knight, Quantifying entanglement. Phys. Rev. Lett. 78, 2275–2279 (1997)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    T.C. Wei, P.M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)ADSCrossRefGoogle Scholar
  97. 97.
    E.P. Wigner, M.M. Yanase, Information contents of distributions. Proc. Natl. Acad. Sci. U.S.A. 49, 910–918 (1963)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    M.M. Wilde, A. Winter, D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy. Commun. Math. Phys. 331, 593–622 (2014)ADSzbMATHCrossRefGoogle Scholar
  99. 99.
    M.M. Wolf, Quantum Channels and Operations Guided Tour (2002). http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • D. Spehner
    • 1
    • 2
    Email author
  • F. Illuminati
    • 3
    • 4
  • M. Orszag
    • 5
  • W. Roga
    • 6
  1. 1.Université Grenoble Alpes, Institut FourierGrenobleFrance
  2. 2.CNRS, Laboratoire de Physique et Modélisation des Milieux CondensésGrenobleFrance
  3. 3.Dipartimento di Ingegneria IndustrialeUniversità degli Studi di SalernoFisciano (SA)Italy
  4. 4.INFN, Sezione di Napoli, Gruppo collegato di SalernoFisciano (SA)Italy
  5. 5.Pontificia Universidad Católica, Instituto de FísicaSantiago 22Chile
  6. 6.Department of PhysicsUniversity of StrathclydebGlasgowUK

Personalised recommendations