Quantum Information Processing

, Volume 8, Issue 6, pp 493–521 | Cite as

Towards measurable bounds on entanglement measures

  • Remigiusz AugusiakEmail author
  • Maciej Lewenstein


While the experimental detection of entanglement provides already quite a difficult task, experimental quantification of entanglement is even more challenging, and has not yet been studied thoroughly. In this paper we discuss several issues concerning bounds on concurrence measurable collectively on copies of a given quantum state. Firstly, we concentrate on the recent bound on concurrence by (Mintert and Buchleitner in Phys Rev Lett 98:140505/1–140505/4, 2007). Relating it to the reduction criterion for separability we provide yet another proof of the bound and point out some possibilities following from the proof which could lead to improvement of the bound. Then, relating concurrence to the generalized robustness of entanglement, we provide a method allowing for construction of lower bounds on concurrence from any positive map (not only the reduction one). All these quantities can be measured as mean values of some two-copy observables. In this sense the method generalizes the Mintert–Buchleitner bound and recovers it when the reduction map is used. As a particular case we investigate the bound obtained from the transposition map. Interestingly, comparison with MB bound performed on the class of \({4\otimes 4}\) rotationally invariant states shows that the new bound is positive in regions in which the MB bound gives zero. Finally, we provide measurable upper bounds on the whole class of concurrences.


Bounds on entanglement measures Concurrence Measurability of entanglement Generalized robustness of entanglement 




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  1. 1.
    Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Zoller P. et al.: Quantum information processing and communication—strategic report on current status, visions and goals for research in Europe. Eur. Phys. J. D 36, 203–228 (2005)ADSCrossRefGoogle Scholar
  3. 3.
    Gühne O., Tóth G.: Entanglement detection. Phys. Rep. 474, 1–75 (2009)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    White A.G., Mitchell J.R., Nairz O., Kwiat P.G.: Interaction-free imaging. Phys. Rev. A 58, 605–613 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    Häffner H., Hänsel W., Roos C.F., Benhelm J., Chek–al–kar D., Chwalla M., Korber T., Rapol U.D., Riebe M., Schmidt P.O., Becher C., Gühne O., Dür W., Blatt R.: Scalable multiparticle entanglement of trapper ions. Nature 438, 643–646 (2005)PubMedADSCrossRefGoogle Scholar
  6. 6.
    Korbicz J.K., Gühne O., Lewenstein M., Häffner H., Roos C.F., Blatt R.: Generalized spin-squeezing inequalities in N-qubit systems: theory and experiment. Phys. Rev. A 74, 052319/1–052319/13 (2006)ADSGoogle Scholar
  7. 7.
    Jaeger G., Horne M.A., Shimony A.: Complementarity of one-particle and two-particle interference. Phys. Rev. A 48, 1023–1027 (1993)PubMedADSCrossRefGoogle Scholar
  8. 8.
    Weinfurter H., Żukowski M.: Four-photon entanglement from down-conversion. Phys. Rev. A 64, 010102/1–010102/4 (2001)ADSCrossRefGoogle Scholar
  9. 9.
    Bell J.S.: Speakable and Unspeakable in Quantum Mechamics. Cambridge University Press, Cambridge (2004)Google Scholar
  10. 10.
    Clauser J.F., Horne M.A., Shimony A., Holt R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969)ADSCrossRefGoogle Scholar
  11. 11.
    Werner R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)PubMedADSCrossRefGoogle Scholar
  12. 12.
    Barrett J.: Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality. Phys. Rev. A 65, 042302/1–042302/4 (2002)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Tóth G., Acín A.: Genuine tripartite entangled states with a local hidden-variable model. Phys. Rev. A 74, 030306(R)/1–030306(R)/4 (2006)ADSCrossRefGoogle Scholar
  14. 14.
    Almeida M.L., Pironio S., Barrett J., Tóth G., Acín A.: Robustness of the nonlocality of entangled quantum states. Phys. Rev. Lett. 99, 040403/1–040403/4 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Werner R.F., Wolf M.: Bell inequalities and entanglement. Quant. Inf. Comp. 1, 1–25 (2001)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Horodecki M., Horodecki P., Horodecki R.: Separability of mixed states: necessary and sufficient condition. Phys. Lett. A 223, 1–8 (1996)zbMATHMathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Terhal B.M.: Bell inequalities and separability criterion. Phys. Lett. A 271, 319–326 (2000)zbMATHMathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Gühne O., Hyllus P., Bruß D., Ekert A., Lewenstein M., Macchiavello C., Sanpera A.: Detection of entanglement with few local measurements. Phys. Rev. A 66, 062305/1–062305/5 (2002)ADSCrossRefGoogle Scholar
  19. 19.
    Barbieri M., De Martini F., Di Nepi G., Mataloni P., D’Ariano G.M., Macchiavello C.: Detection of entanglement with polarized photons: experimental realization of an entanglement witness. Phys. Rev. Lett. 91, 227901/1–227901/4 (2003)ADSCrossRefGoogle Scholar
  20. 20.
    Bourennane M., Eibl M., Kurtsiefer C., Gaertner S., Weinfurter H., Gühne O., Hyllus P., Bruß D., Lewenstein M., Sanpera A.: Experimental detection of multipartite entanglement using witness operators. Phys. Rev. Lett. 92, 087902/1–087902/4 (2004)ADSGoogle Scholar
  21. 21.
    Sancho J.M., Huelga S.F.: Measuring the entanglement of bipartite pure states. Phys. Rev. A 61, 042303/1–042303/7 (2000)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Acín A., Tarrach R., Vidal G.: Optimal estimation of two-qubit pure-state entanglement. Phys. Rev. A 61, 062307/1–062307/8 (2000)ADSGoogle Scholar
  23. 23.
    Walborn S.P., Souto Ribeiro P.H., Davidovich L., Mintert F., Buchleitner A.: Experimental determination of entanglement with a single measurement. Nature 440, 1022–1024 (2006)PubMedADSCrossRefGoogle Scholar
  24. 24.
    Horodecki P.: Measuring quantum entanglement without prior state reconstruction. Phys. Rev. Lett. 90, 167901/1–167901/4 (2003)ADSGoogle Scholar
  25. 25.
    Mintert F., Buchleitner A.: Observable entanglement measure for mixed quantum states. Phys. Rev. Lett. 98, 140505/1–140505/4 (2007)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Augusiak R., Demianowicz M., Horodecki P.: Universal observable detecting all two-qubit entanglement and determinant-based separability tests. Phys. Rev. A 77, 030301(R)/1–030301(R)/4 (2008)ADSGoogle Scholar
  27. 27.
    Horodecki P.: From limits of quantum operations to multicopy entanglement witnesses and state-spectrum estimation. Phys. Rev. A 68, 052101/1–052101/6 (2003)ADSCrossRefGoogle Scholar
  28. 28.
    Horodecki P., Ekert A.: Method for direct detection of quantum entanglement. Phys. Rev. Lett. 89, 127902/1–127902/4 (2002)MathSciNetADSGoogle Scholar
  29. 29.
    Korbicz J.K. et al.: Structural approximations to positive maps and entanglement-breaking channels. Phys. Rev. A 78, 062105/1–062105/17 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    Gühne O., Lütkenhaus N.: Nonlinear entanglement witnesses. Phys. Rev. Lett. 96, 170502/1–170502/4 (2006)CrossRefGoogle Scholar
  31. 31.
    Gühne O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903/1–117903/4 (2004)ADSCrossRefGoogle Scholar
  32. 32.
    Tóth G., Knapp C., Gühne O., Briegel H.J.: Optimal spin squeezing inequalities detect bound entanglement in spin models. Phys. Rev. Lett. 99, 250405/1–250405/4 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    Korbicz J.K., Cirac J.I., Lewenstein M.: Spin squeezing inequalities and N Qubit states. Phys. Rev. Lett. 95, 120502/1–120502/4 (2005)ADSGoogle Scholar
  34. 34.
    Korbicz J.K., Cirac J.I., Lewenstein M.: Spin squeezing inequalities and N Qubit states. Phys. Rev. Lett. 95, 259901/1 (2005)ADSGoogle Scholar
  35. 35.
    Hofmann H.F., Takeuchi S.: Violation of local uncertainty relations as a signature of entanglement. Phys. Rev. A 68, 032103/1–032103/6 (2003)MathSciNetADSGoogle Scholar
  36. 36.
    Gühne O., Lewenstein M.: Entropic uncertainty relations and entanglement. Phys. Rev. A 70, 022316/1–022316/8 (2004)ADSCrossRefGoogle Scholar
  37. 37.
    de Vicente J.I., Sánchez–Ruiz J.: Separability conditions from the Landau-Pollack uncertainty relations. Phys. Rev. A 71, 052325/1–052325/8 (2005)ADSGoogle Scholar
  38. 38.
    Acín A., Bruß D., Lewenstein M., Sanpera A.: Classification of mixed three-qubit states. Phys. Rev. Lett. 87, 040401/1–040401/4 (2001)ADSGoogle Scholar
  39. 39.
    Gühne O., Hyllus P., Bruss D., Ekert A., Lewenstein M., Macchiavello C., Sanpera A.: Experimental detection of entanglement via witness operators and local measurement. J. Mod. Opt. 50, 1079–1102 (2003)zbMATHADSCrossRefGoogle Scholar
  40. 40.
    Horodecki M.: Entanglement measures. Quant. Inf. Comp. 1, 3–26 (2001)zbMATHMathSciNetGoogle Scholar
  41. 41.
    Virmani S., Plenio M.B.: An introduction to entanglement measures. Quant. Inf. Comp. 7, 1–51 (2007)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Mintert F.: Entanglement measures as physical observables. Appl. Phys. B 89, 493–497 (2007)ADSCrossRefGoogle Scholar
  43. 43.
    Breuer H.-P.: Separability criteria and bounds for entanglement measure. J. Phys. A 39, 11847–11860 (2006)zbMATHMathSciNetADSCrossRefGoogle Scholar
  44. 44.
    Mintert F.: Concurrence via entanglement witness. Phys. Rev. A 75, 052302/1–052302/4 (2007)ADSCrossRefGoogle Scholar
  45. 45.
    Aolita L., Buchleitner A., Mintert F.: Scalable method to estimate experimentally the entanglement of multipartite systems. Phys. Rev. A 78, 022308/1–022308/4 (2008)ADSCrossRefGoogle Scholar
  46. 46.
    Zhang C.-J., Gong Y.-X., Zhang Y.-S., Guo G.-C.: Observable estimation of entanglement for arbitrary finite-dimensional mixed states. Phys. Rev. A 78, 042308/1–042308/5 (2008)ADSGoogle Scholar
  47. 47.
    Bovino F., Castagnoli G., Ekert A., Horodecki P., Moura Alves C., Sergienko A.V.: Direct measurement of nonlinear properties of bipartite quantum states. Phys. Rev. Lett. 95, 240407/1–240407/4 (2005)MathSciNetADSCrossRefGoogle Scholar
  48. 48.
    Schmid C., Kiesel N., Wieczorek W., Weinfurter H., Mintert F., Buchleitner A.: Experimental direct observation of mixed state entanglement. Phys. Rev. Lett. 101, 260505 (2008)PubMedADSCrossRefGoogle Scholar
  49. 49.
    Moura Alves C., Jaksch D.: Multipartite entanglement detection in bosons. Phys. Rev. Lett. 93, 110501/1–110501/4 (2004)ADSCrossRefGoogle Scholar
  50. 50.
    van Enk S.J., Lütkenhaus N., Kimble H.J.: Experimental procedures for entanglement verification. Phys. Rev. A 75, 052318/1–052318/14 (2007)ADSGoogle Scholar
  51. 51.
    van Enk S.J.: Direct measurements of entanglement and permutation symmetry. Phys. Rev. Lett. 102, 190503/1–190503/4 (2009)Google Scholar
  52. 52.
    Brandão F.S.G.L.: Quantifying entanglement with witness operator. Phys. Rev. A 72, 022310/1–022310/15 (2005)ADSCrossRefGoogle Scholar
  53. 53.
    Brandão F.S.G.L., Vianna R.O.: Witnessed entanglement. Int. J. Quant. Inf. 4, 331–340 (2006)zbMATHCrossRefGoogle Scholar
  54. 54.
    Audenaert K.M.R., Plenio M.B.: When are correlations quantum?—verification and quantification of entanglement by simple measurements. New J. Phys. 8, 266/1–266/20 (2006)ADSCrossRefGoogle Scholar
  55. 55.
    Gühne O., Reimpell M., Werner R.F.: Estimating entanglement measures in experiments. Phys. Rev. Lett. 98, 110502/1–110502/4 (2007)CrossRefGoogle Scholar
  56. 56.
    Gühne O., Reimpell M., Werner R.F.: Lower bounds on entanglement measures from incomplete information. Phys. Rev. A 77, 052317/1–052317/8 (2008)ADSGoogle Scholar
  57. 57.
    Eisert J., Brandão F.G.S.L., Audenaert K.M.R.: Quantitative entanglement witnesses. New. J. Phys. 9, 46/1–46/19 (2007)ADSCrossRefGoogle Scholar
  58. 58.
    Wunderlich, H., Plenio, M.B.: Quantitative verification of entanglement and fidelities from incomplete measurement data. arXiv:0902.1848 (2009)Google Scholar
  59. 59.
    Uhlmann A.: The “transition probability” in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976)zbMATHMathSciNetADSCrossRefGoogle Scholar
  60. 60.
    Jozsa R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–2323 (1994)zbMATHMathSciNetADSCrossRefGoogle Scholar
  61. 61.
    Miszczak J.A., Puchała Z., Horodecki P., Uhlmann A., Życzkowski K.: Sub-and super-fidelity as bounds for quantum fidelity. Quant. Inf. Comp. 9, 103–130 (2009)zbMATHGoogle Scholar
  62. 62.
    Ma Z., Deng D.-L., Zhang F.-L., Chen J.-L.: Bounds of concurrence and their relation with fidelity and frontier states. Phys. Lett. A 373, 1616–1620 (2009)ADSCrossRefGoogle Scholar
  63. 63.
    Sinołȩcka M., Kuś M., Życzkowski K.: Manifolds of equal entanglement for composite quantum systems. Acta Phys. Pol. B 33, 2081–2095 (2002)ADSGoogle Scholar
  64. 64.
    Fan H., Matsumoto K., Imai H.: Quantify entanglement by concurrence hierarchy. J. Phys. A 36, 4151–4158 (2003)zbMATHMathSciNetADSCrossRefGoogle Scholar
  65. 65.
    Gour G.: Family of concurrence monotones and its applications. Phys. Rev. A 71, 012318/1–012318/8 (2005)MathSciNetADSCrossRefGoogle Scholar
  66. 66.
    Rungta P., Bužek V., Caves C.M., Hillery M., Milburn G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315/1–042315/13 (2001)ADSCrossRefGoogle Scholar
  67. 67.
    Hill S., Wootters W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022–5025 (1997)ADSCrossRefGoogle Scholar
  68. 68.
    Carvalho A.R.R., Mintert F., Buchleitner A.: Decoherence and multipartite entanglement. Phys. Rev. Lett. 93, 230501/1–230501/4 (2004)ADSGoogle Scholar
  69. 69.
    Borras A., Majtey A.P., Plastino A.R., Casas M., Plastino A.: Typical features of the Mintert-Buchleitner lower bound for concurrence. Phys. Rev. A 79, 022112/1–0222112/6 (2009)ADSGoogle Scholar
  70. 70.
    Horodecki R., Horodecki P., Horodecki M.: Quantum α-entropy inequalities: independent condition for local realism?. Phys. Lett. A 210, 377–381 (1996)zbMATHMathSciNetADSCrossRefGoogle Scholar
  71. 71.
    Horodecki R., Horodecki P.: Quantum redundancies and local realism. Phys. Lett. A 194, 147–152 (1994)zbMATHMathSciNetADSCrossRefGoogle Scholar
  72. 72.
    Horodecki M., Horodecki P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206–4216 (1999)MathSciNetADSCrossRefGoogle Scholar
  73. 73.
    Terhal B.M.: Detecting quantum entanglement. Theor. Comput. Sci. 287, 313–326 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Vollbrecht K.G.H., Wolf M.M.: Conditional entropies and their relation to entanglement criteria. J. Math. Phys. 43, 4299–4306 (2002)zbMATHMathSciNetADSCrossRefGoogle Scholar
  75. 75.
    Abe S., Rajagopal A.K.: Nonadditive conditional entropy and its significance for local realism. Phys. A 289, 157–164 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    Tsallis C., Lloyd S., Baranger M.: Peres criterion for separability through nonextensive entropy. Phys. Rev. A 63, 042104/1–042104/6 (2001)ADSCrossRefGoogle Scholar
  77. 77.
    Batle J., Casas M., Plastino A., Plastino A.R.: Maximally entangled mixed states and conditional entropies. Phys. Rev. A 71, 024301/1–024301/4 (2005)MathSciNetADSCrossRefGoogle Scholar
  78. 78.
    Cerf N.J., Adami C., Gingrich R.M.: Reduction criterion for separability. Phys. Rev. A 60, 898–909 (1999)MathSciNetADSCrossRefGoogle Scholar
  79. 79.
    Augusiak R., Stasińska J., Horodecki P.: Beyond the standard entropic inequalities: stronger scalar separability criteria and their applications. Phys. Rev. A 77, 012333/1–012333/14 (2008)ADSGoogle Scholar
  80. 80.
    Augusiak R., Stasińska J.: General scheme for construction of scalar separability criteria from positive maps. Phys. Rev. A 77, 010303(R)/1–010303(R)/4 (2008)ADSGoogle Scholar
  81. 81.
    Augusiak R., Stasińska J.: Positive maps, majorization, entropic inequalities and detection of entanglement. New J. Phys. 11, 053018/1–053018/25 (2009)CrossRefGoogle Scholar
  82. 82.
    Datta A., Flammia S.T., Shaji A., Caves C.M.: Constrained bounds on measures of entanglement. Phys. Rev. A 75, 062117/1–062117/15 (2007)MathSciNetADSGoogle Scholar
  83. 83.
    Chen K., Albeverio S., Fei S.-M.: Concurrence of arbitrary dimensional bipartite quantum states. Phys. Rev. Lett. 95, 040504/1–040504/4 (2005)ADSGoogle Scholar
  84. 84.
    de Vicente J.I.: Lower bounds on concurrence and separability conditions. Phys. Rev. A 75, 052320/1–052320/5 (2007)ADSGoogle Scholar
  85. 85.
    de Vicente J.I.: Erratum: Lower bounds on concurrence and separability conditions [Phys. Rev. A 75, 052320 (2007)]. Phys. Rev. A 77, 039903(E)/1 (2008)ADSGoogle Scholar
  86. 86.
    Breuer H.-P.: Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett. 97, 080501/1–080501/4 (2006)ADSCrossRefGoogle Scholar
  87. 87.
    Hall W.: A new criterion for indecomposability of positive maps. J. Phys. A 39, 14119–14131 (2006)zbMATHMathSciNetADSCrossRefGoogle Scholar
  88. 88.
    Boyd S., Vandenberghe L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  89. 89.
    Audenaert K.M.R., Braunstein S.L.: On strong superadditivity of the entanglement of formation. Commun. Math. Phys. 246, 443–452 (2004)zbMATHMathSciNetADSCrossRefGoogle Scholar
  90. 90.
    Rungta P., Caves C.M.: Concurrence-based entanglement measures for isotropic states. Phys. Rev. A 67, 012307/1–012307/9 (2003)ADSCrossRefGoogle Scholar
  91. 91.
    Uhlmann A.: On 1-qubit channels. J. Phys. A 34, 7047–7055 (2001)zbMATHMathSciNetADSCrossRefGoogle Scholar
  92. 92.
    Hellmund M., Uhlmann A.: Concurrence and entanglement entropy of stochastic one-qubit maps. Phys. Rev. A 79, 052319/1–052319/10 (2009)ADSCrossRefMathSciNetGoogle Scholar
  93. 93.
    Hildebrand R.: Concurrence revisited. J. Math. Phys. 48, 102108/1–102108/23 (2007)MathSciNetADSCrossRefGoogle Scholar
  94. 94.
    Dodd J., Nielsen M.A.: Simple operational interpretation of the fidelity of mixed states. Phys. Rev. A 66, 044301/1–044301/2 (2002)MathSciNetADSCrossRefGoogle Scholar
  95. 95.
    Steiner M.: Generalized robustness of entanglement. Phys. Rev. A 67, 054305/1–054305/4 (2003)ADSCrossRefGoogle Scholar
  96. 96.
    Vidal G., Tarrach R.: Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999)MathSciNetADSCrossRefGoogle Scholar
  97. 97.
    Choi M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)zbMATHCrossRefGoogle Scholar
  98. 98.
    Jamiołkowski A.: Linear transformations which preserve trace and positive semi definiteness of operators. Rep. Math. Phys. 3, 275–278 (1972)zbMATHADSCrossRefGoogle Scholar
  99. 99.
    Bengtsson I., Życzkowski K.: Geometry of Quantum States. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  100. 100.
    Werner R.F., Holevo A.S.: Counterexample of an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43, 4353–4357 (2002)zbMATHMathSciNetADSCrossRefGoogle Scholar
  101. 101.
    Isham C.J., Linden N., Scheckenberg S.: The classification of decoherence functionals: an analog of Gleason’s theorem. J. Math. Phys. 35, 6360–6370 (1994)zbMATHMathSciNetADSCrossRefGoogle Scholar
  102. 102.
    Terhal B.M., Horodecki P.: Schmidt number for density matrices. Phys. Rev. A 61, 040301(R)/1–040301(R)/4 (2000)MathSciNetADSGoogle Scholar
  103. 103.
    Nielsen M.A.: Condition for a class of entanglement transformations. Phys. Rev. Lett. 83, 436–439 (1999)ADSCrossRefGoogle Scholar
  104. 104.
    Brun T.A.: Measuring polynomial functions of states. Quant. Inf. Comp. 4, 401–408 (2004)zbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.ICFO, Institute Ciéncies FotóniquesMediterranean Technology ParkCastelldefels, BarcelonaSpain

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