Many-Body Physics from a Quantum Information Perspective

  • R. AugusiakEmail author
  • F. M. Cucchietti
  • M. LewensteinEmail author
Part of the Lecture Notes in Physics book series (LNP, volume 843)


The quantum information approach to many-body physics has been very successful in giving new insights and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many-body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.


Entangle State Entanglement Entropy Entanglement Measure Bipartite State Entanglement Witness 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We are grateful to Ll. Masanes for helpful discussion. We acknowledge the support of Spanish MEC/MINCIN projects TOQATA (FIS2008-00784) and QOIT (Consolider Ingenio 2010), ESF/MEC project FERMIX (FIS2007-29996-E), EU Integrated Project SCALA, EU STREP project NAMEQUAM, ERC Advanced Grant QUAGATUA, Caixa Manresa, AQUTE, and Alexander von Humboldt Foundation Senior Research Prize.


  1. 1.
    Jaksch, D., Briegel, H.-J., Cirac, J.I., Gardiner, C.W., Zoller, P.: Entanglement of atoms via cold controlled collisions. Phys. Rev. Lett. 82, 1975 (1999)ADSCrossRefGoogle Scholar
  2. 2.
    Osborne, T.J., Nielsen, M.A.: Entanglement, quantum phase transitions, and density matrix renormalization. Quantum Inf. Proc. 1, 45 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A 66, 032110 (2002)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature 416, 608 (2002)ADSCrossRefGoogle Scholar
  5. 5.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)zbMATHADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881 (1992)zbMATHADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1992)ADSCrossRefGoogle Scholar
  8. 8.
    Horodecki, R., Horodecki, M., Horodecki, P., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)zbMATHADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989)ADSCrossRefGoogle Scholar
  10. 10.
    Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333 (1997)zbMATHADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Sanpera, A., Terrach, R., Vidal, G.: Local description of quantum inseparability. Phys. Rev. A 58, 826 (1998)Google Scholar
  12. 12.
    Samsonowicz, J., Kuś, M., Lewenstein, M.: Phys. Rev. A 76, 022314 (2007)Google Scholar
  13. 13.
     Zukowski, M., Zeilinger, A., Horne, M.A., Ekert, A.K.: “Event-ready-detectors” Bell experiment via entanglement swapping. Phys. Rev. Lett. 71, 4287 (1993)ADSCrossRefGoogle Scholar
  14. 14.
    Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474, 1 (2009)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: Distinguishing separable and entangled states. Phys. Rev. Lett. 88, 187904 (2002)ADSCrossRefGoogle Scholar
  16. 16.
    Hulpke, F., Bruß, D.: A two-way algorithm for the entanglement problem. J. Phys. A: Math. Gen. 38, 5573 (2005)zbMATHADSCrossRefGoogle Scholar
  17. 17.
    Gurvits, L.: Classical complexity and quantum entanglement. STOC 69, 448 (2003)MathSciNetGoogle Scholar
  18. 18.
    Choi, M.-D.: Positive linear maps. Proc. Symp. Pure Math. 38, 583 (1982)Google Scholar
  19. 19.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett 77, 1413 (1996)zbMATHADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Horodecki, M., Horodecki, P., Horodecki, R.: Mixed–state entanglement and distillation: is there a “bound” entanglement in nature?. Phys. Rev. Lett. 80, 5239 (1998)zbMATHADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Zyczkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A 58, 883 (1998)Google Scholar
  22. 22.
    DiVincenzo, D.P., Shor, P.W., Smolin, J.A., Terhal, B.M., Thapliyal, A.V.: Evidence for bound entangled states with negative partial transpose. Phys. Rev. A 61, 062312 (2000)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Dür, W., Cirac, J.I., Lewenstein, M., Bruß, D.: Distillability and partial transposition in bipartite systems. Phys. Rev. A 61, 062313 (2000)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1 (1996)zbMATHADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin (1985)zbMATHCrossRefGoogle Scholar
  26. 26.
    Terhal, B.M.: Bell inequalities and the separability criterion. Phys. Lett. A 271, 319 (2000)zbMATHADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N–level systems. J. Math. Phys. 17, 821 (1976)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Alg. Appl. 10, 285 (1975)zbMATHCrossRefGoogle Scholar
  29. 29.
    Kraus, K.: States, Effects and Operations: Fundamental Notions of Quantum Theory. Springer, Berlin (1983)zbMATHCrossRefGoogle Scholar
  30. 30.
    Stinespring, W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211 (1955)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Jamiołkowski, A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275 (1972)zbMATHADSCrossRefGoogle Scholar
  32. 32.
    Horodecki, M., Horodecki, P., Horodecki, R.: Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps. Phys. Lett. A 283, 1 (2001)zbMATHADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Woronowicz, S.L.: Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10, 165 (1976)zbMATHADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Tanahashi, K., Tomiyama, J.: Indecomposable positive maps in matrix algebras. Can. Math. Bull 31, 308 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Horodecki, M., Horodecki, P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206 (2000)ADSCrossRefGoogle Scholar
  36. 36.
    Cerf, N.J., Adami, C., Gingrich, R.M.: Reduction criterion for separability. Phys. Rev. A 60, 898 (1999)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Breuer, H.-P.: Optimal entanglement criterion for mixed states. Phys. Rev. Lett. 97, 080501 (2006)ADSCrossRefGoogle Scholar
  38. 38.
    Hall, W.: A new criterion for indecomposability of positive maps. J. Phys. A 39, 14119 (2006)zbMATHADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Woronowicz, S.: Nonextendible positive maps. Comm. Math. Phys. 51, 243 (1976)zbMATHADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Lewenstein, M., Kraus, B., Cirac, J.I., Horodecki, P.: Optimization of entanglement witnesses. Phys. Rev. A 62, 052310 (2000)ADSCrossRefGoogle Scholar
  41. 41.
    Terhal, B.M.: A family of indecomposable positive linear maps based on entangled quantum states. Lin. Alg. Appl. 323, 61 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Vedral, V., Plenio, M.B., Rippin, M.A., Knight, P.L.: Quantifying entanglement. Phys. Rev. Lett 78, 2275 (1997)zbMATHADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    DiVincenzo, D.P., Fuchs, C.A., Mabuchi, H., Smolin, J.A., Thapliyal, A., Uhlmann, A.: In Proceedings of the first NASA International Conference on Quantum Computing and Quantum Communication. Springer (1998)Google Scholar
  45. 45.
    Laustsen, T., Verstraete, F., van Enk, S.J.: Local vs. joint measurements for the entanglement of assistance. Quantum Inf. Comput. 3, 64 (2003)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Nielsen, M.A.: Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83, 436 (1999)ADSCrossRefGoogle Scholar
  47. 47.
    Vidal, G.: Entanglement monotones. J. Mod. Opt. 47, 355 (2000)ADSMathSciNetGoogle Scholar
  48. 48.
    Jonathan, D., Plenio, M.B.: Minimal conditions for local pure-state entanglement manipulation. Phys. Rev. Lett. 83, 1455 (1999)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Horodecki, M., Sen(De), A., Sen, U.: Dual entanglement measures based on no local cloning and no local deleting. Phys. Rev. A 70, 052326 (2004)ADSCrossRefGoogle Scholar
  50. 50.
    Horodecki, M.: Distillation and bound entanglement. Quantum Inf. Comput. 1, 3 (2001)zbMATHMathSciNetGoogle Scholar
  51. 51.
    Plenio, M.B., Virmani, S.: An introduction to entanglement measures. Quant. Inf. Comp. 7, 1 (2007)zbMATHMathSciNetGoogle Scholar
  52. 52.
    Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)ADSCrossRefGoogle Scholar
  53. 53.
    Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022 (1997)ADSCrossRefGoogle Scholar
  54. 54.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)ADSCrossRefGoogle Scholar
  55. 55.
    Terhal, B.M., Vollbrecht, K.G.H.: Entanglement of formation for isotropic states. Phys. Rev. Lett. 85, 2625 (2000)ADSCrossRefGoogle Scholar
  56. 56.
    Vollbrecht, K.G.H., Werner, R.F.: Entanglement measures under symmetry. Phys. Rev. A 64, 062307 (2001)ADSCrossRefGoogle Scholar
  57. 57.
    Rungta, P., Bužek, V.V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)Google Scholar
  58. 58.
    Rungta, P., Caves, C.M.: Concurrence-based entanglement measures for isotropic states. Phys. Rev. A 67, 012307 (2003)ADSCrossRefGoogle Scholar
  59. 59.
    Aolita, L., Mintert, F.: Measuring multipartite concurrence with a single factorizable observable. Phys. Rev. Lett. 97, 050501 (2006)ADSCrossRefGoogle Scholar
  60. 60.
    Walborn, S.P., Ribero, P.H.S., Davidovich, L., Mintert, F., Buchleitner, A.: Experimental determination of entanglement with a single measurement. Nature 440, 1022 (2006)ADSCrossRefGoogle Scholar
  61. 61.
    Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)ADSCrossRefGoogle Scholar
  62. 62.
    Plenio, M.B.: Logarithmic negativity: a full entanglement monotone that is not convex. Phys. Rev. Lett. 95, 090503 (2005)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    Bombelli, L., Koul, R.K., Lee, J., Sorkin, R.D.: Quantum source of entropy for black holes. Phys. Rev. D 34, 373 (1986)zbMATHADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    Srednicki, M.: Entropy and area. Phys. Rev. Lett. 71, 666 (1993)zbMATHADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D 7, 2333 (1973)ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    Bekenstein, J.D.: Black holes and information theory. Contemp. Phys. 45, 31 (2004)ADSCrossRefGoogle Scholar
  67. 67.
    Hawking, S.W.: Black hole explosions?. Nature 248, 30 (1974)ADSCrossRefGoogle Scholar
  68. 68.
    Bousso, R.: The holographic principle. Rev. Mod. Phys. 74, 825 (2002)zbMATHADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    Eisert, J., Cramer, M., Plenio, M.B.: Area laws for the entanglement entropy – a review. Rev. Mod. Phys. 82, 277 (2010)zbMATHADSMathSciNetCrossRefGoogle Scholar
  70. 70.
    Calabrese, P., Cardy, J., Doyon, B.: Special issue: entanglement entropy in extended quantum systems. J. Phys. A 42, 500301 (2009)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Lubkin, E.: Entropy of an n–system from its correlation with a k–reservoir. J. Math. Phys. 19, 1028 (1978)zbMATHADSCrossRefGoogle Scholar
  72. 72.
    Lloyd, S., Pagels, H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186 (1988)ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993)zbMATHADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    Bengtsson, I.,  Zyczkowski, K.: Geometry of Quantum States.. Cambridge University Press, Cambridge, MA (2006)CrossRefGoogle Scholar
  75. 75.
    Foong, S.K., Kanno, S.: Proof of a Page’s conjecture on the average entropy of a subsystem. Phys. Rev. Lett. 72, 1148 (1994)zbMATHADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    Sen, S.: Average entropy of a quantum subsystem. Phys. Rev. Lett. 77, 1 (1996)ADSCrossRefGoogle Scholar
  77. 77.
    Sanchez-Ruíz, J.: Simple Proof of Page’s conjecture on the average entropy of a subsystem. Phys. Rev. E 52, 5653 (1995)Google Scholar
  78. 78.
    Hastings, M.B.: An area law for one-dimensional quantum system. J. Stat. Mech. Theory Exp. 2007, 08024 (2007)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Eisert, E.H., Robinson, D.W.: The finite group velocity of quantum spin systems. Comm. Math. Phys. 28, 251 (1972)ADSMathSciNetCrossRefGoogle Scholar
  80. 80.
    Masanes, L.: Area law for the entropy of low-energy states. Phys. Rev. A 80, 052104 (2009)ADSCrossRefGoogle Scholar
  81. 81.
    Dür, W., Hartmann, L., Hein, M., Lewenstein, M., Briegel, H.-J.: Entanglement in spin chains and lattices with long-range Ising-type interactions. Phys. Rev. Lett. 94, 097203 (2005)Google Scholar
  82. 82.
    Eisert, J., Osborne, T.: General entanglement scaling laws from time evolution. Phys. Rev. Lett. 97, 150404 (2006)ADSMathSciNetCrossRefGoogle Scholar
  83. 83.
    Latorre, J.I., Riera, A.: A short review on entanglement in quantum spin systems. J. Phys. A 42, 504002 (2009)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)ADSCrossRefGoogle Scholar
  85. 85.
    Jin, B.-Q., Korepin, V.E.: Quantum spin chain, Toeplitz deteminants and the Ficher–Hartwig conjecture. J. Stat. Phys. 116, 79 (2004)zbMATHADSMathSciNetCrossRefGoogle Scholar
  86. 86.
    Its, A.R., Jin, B.-Q., Korepin, V.E.: Entanglement in the XY spin chain. J. Phys. A: Math. Gen. 38, 2975 (2005)zbMATHADSMathSciNetCrossRefGoogle Scholar
  87. 87.
    Keating, J.P., Mezzadri, F.: Entanglement in quantum spin chains, symmetry classes of random matrices, and conformal field theory. Phys. Rev. Lett. 94, 050501 (2005)ADSMathSciNetCrossRefGoogle Scholar
  88. 88.
    Eisert, J., Cramer, M.: Single-copy entanglement in critical quantum spin chains. Phys. Rev. A 72, 042112 (2005)ADSCrossRefGoogle Scholar
  89. 89.
    Calabrese, P., Cardy, J.: Entanglement entropy and conformal field theory. J. Phys. A 42, 504005 (2009)MathSciNetCrossRefGoogle Scholar
  90. 90.
    Wolf, M.M.: Violation of the entropic area law for fermions. Phys. Rev. Lett. 96, 010404 (2006)ADSCrossRefGoogle Scholar
  91. 91.
    Gioev, D., Klich, I.: Entanglement entropy of fermions in any dimension and the Widom conjecture. Phys. Rev. Lett. 96, 100503 (2006)ADSMathSciNetCrossRefGoogle Scholar
  92. 92.
    Farkas, S., Zimboras, Z.: The von Neumann entropy asymptotics in multidimensional fermionic systems. J. Math. Phys. 48, 102110 (2007)ADSMathSciNetCrossRefGoogle Scholar
  93. 93.
    Hastings, M.B.: Locality in quantum and Markov dynamics on lattices and networks. Phys. Rev. Lett. 93, 140402 (2004)ADSCrossRefGoogle Scholar
  94. 94.
    Boyd, S., Vanderberghe, L.: Convex Optimization. Cambridge University Press, Cambridge, MA (2004)zbMATHGoogle Scholar
  95. 95.
    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
  96. 96.
    Wolf, M.M., Verstraete, F., Hastings, M.B., Cirac, J.I.: Area laws in quantum systems: mutual information and correlations. Phys. Rev. Lett. 100, 070502 (2008)ADSMathSciNetCrossRefGoogle Scholar
  97. 97.
    Verstraete, F., Popp, M., Cirac, J.I.: Entanglement versus correlations in spin systems. Phys. Rev. Lett. 92, 027901 (2004)ADSCrossRefGoogle Scholar
  98. 98.
    Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003)ADSCrossRefGoogle Scholar
  99. 99.
    Perez-García, D., Verstraete, F., Wolf, M.M., Cirac, J.I.: Matrix product state representation. Quantum Inf. Comput. 7, 401 (2007)zbMATHMathSciNetGoogle Scholar
  100. 100.
    Verstraete, F., Cirac J.I.: Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions. cond-mat/0407066 (2004)Google Scholar
  101. 101.
    Schuch, N., Wolf, M.M., Verstraete, F., Cirac, J.I.: Computational complexity of projected entangled pair states. Phys. Rev. Lett. 98, 140506 (2007)ADSMathSciNetCrossRefGoogle Scholar
  102. 102.
    Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799 (1987)ADSCrossRefGoogle Scholar
  103. 103.
    Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115, 477 (1988)ADSMathSciNetCrossRefGoogle Scholar
  104. 104.
    Majumdar, C.K., Ghosh, D.K.: On next-nearest-neighbor interaction in linear chain. I. J. Math. Phys. 10, 1388 (1969)ADSMathSciNetCrossRefGoogle Scholar
  105. 105.
    Glauber, R.J.: Time-dependent statistics of the Ising model. J. Math. Phys 4, 294 (1963)zbMATHADSMathSciNetCrossRefGoogle Scholar
  106. 106.
    Deker, U., Haake, F.: Renormalization group transformation for the master equation of a kinetic Ising chain. Z. Phys. B 35, 281 (1979)ADSCrossRefGoogle Scholar
  107. 107.
    Kimball, J.C.: The kinetic Ising model: exact susceptibilities of two simple examples. J. Stat. Phys. 21, 289 (1979)ADSCrossRefGoogle Scholar
  108. 108.
    Haake, F., Thol, K.: Universality classes for one dimensional kinetic Ising models. Z. Phys. B 40, 219 (1980)ADSMathSciNetCrossRefGoogle Scholar
  109. 109.
    Felderhof, B.U.: Spin relaxation of the Ising chain. Rep. Math. Phys. 1, 215 (1971)Google Scholar
  110. 110.
    Siggia, E.D.: Pseudospin formulation of kinetic Ising models. Phys. Rev. B 16, 2319 (1977)Google Scholar
  111. 111.
    Heims, S.P.: Master equation for Ising model. Phys. Rev. 138, A587 (1965)Google Scholar
  112. 112.
    Kawasaki, K. In: Domb, C., Green, M.S. (eds.) Phase Transition and Critical Phenomena, vol. 2, pp. 443–501. Academic Press, London (1972)Google Scholar
  113. 113.
    Augusiak, R., Cucchietti, F.M., Haake, F., Lewenstein, M.: Quantum kinetic Ising models. New J. Phys. 12, 025021 (2010)Google Scholar
  114. 114.
    Hilhorst, H.J., Suzuki, M., Felderhof, B.U.: Kinetics of the stochastic Ising chain in a two–flip model. Physica 60, 199 (1972)ADSMathSciNetCrossRefGoogle Scholar
  115. 115.
    Jordan, P., Wigner, E.: Über das Paulische Aequivalenzverbot. Z. Phys. 47, 631 (1928)ADSCrossRefGoogle Scholar
  116. 116.
    Bogoliubov, N.N.: On a new method in the theory of superconductivity. Nuovo Cimento 7, 794 (1958)CrossRefGoogle Scholar
  117. 117.
    Valatin, J.G.: Comments on the theory of superconductivity. Nuovo Cimento 7, 843 (1958)MathSciNetCrossRefGoogle Scholar
  118. 118.
    Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)zbMATHADSMathSciNetCrossRefGoogle Scholar
  119. 119.
    Zanardi, P., Paunković, N.: Ground state overlap and quantum phase transitions. Phys. Rev. E 74, 031123 (2006)ADSMathSciNetCrossRefGoogle Scholar
  120. 120.
    Quan, H.T., Song, Z., Liu, X.F., Zanardi, P., Sun, C.P.: Decay of Loschmidt echo enhanced by quantum criticality. Phys. Rev. Lett. 96, 140604 (2006)ADSCrossRefGoogle Scholar
  121. 121.
    Zhang, C., Tewari, S., Lutchyn, R., Sarma, S.D.: px+ipy Superfluid from s-wave interactions of fermionic cold atoms. Phys. Rev. Lett. 101, 160401 (2008)ADSCrossRefGoogle Scholar
  122. 122.
    Zhang, J., Cucchietti, F.M., Chandrashekar, C.M., Laforest, M., Ryan, C.A., Ditty, M., Hubbard, A., Gamble, J.K., Laflamme, R.: Direct observation of quantum criticality in Ising spin chains. Phys. Rev. A 79, 012305 (2009)ADSCrossRefGoogle Scholar
  123. 123.
    Li, H., Haldane, F.D.M.: Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum Hall effect states. Phys. Rev. Lett. 101, 010504 (2008)ADSCrossRefGoogle Scholar
  124. 124.
    Calabrese, P., Lefevre, A.: Entanglement spectrum in one-dimensional systems. Phys. Rev. A 78, 032329 (2008)ADSCrossRefGoogle Scholar
  125. 125.
    Vidal, G.: Entanglement renormalization. Phys. Rev. Lett. 99, 220405 (2007)ADSCrossRefGoogle Scholar
  126. 126.
    Clark, S.R., Jaksch, D.: Dynamics of the superfluid to Mott-insulator transition in one dimension. Phys. Rev. A 70, 043612 (2004)ADSCrossRefGoogle Scholar
  127. 127.
    Kraus, C.V., Schuch, N., Verstraete, F., Cirac, J.I.: Fermionic projected entangled pair states. Phys. Rev. A 81, 052338 (2010)ADSMathSciNetCrossRefGoogle Scholar
  128. 128.
    Corboz, P., Vidal, G.: Fermionic multiscale entanglement renormalization ansatz. Phys. Rev. B 80, 165129 (2009)ADSCrossRefGoogle Scholar
  129. 129.
    Corboz, P., Evenbly, G., Verstraete, F., Vidal, G.: Simulation of interacting fermions with entanglement renormalization. Phys. Rev. A 81, 010303 (2010)ADSCrossRefGoogle Scholar
  130. 130.
    Barthel, T., Pineda, C., Eisert, J.: Contraction of fermionic operator circuits and the simulation of strongly correlated fermions. Phys. Rev. A 80, 042333 (2009)ADSCrossRefGoogle Scholar
  131. 131.
    Corboz, P., Orús, R., Bauer, B., Vidal, G.: Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states. Phys. Rev. B 81, 165104 (2010)ADSCrossRefGoogle Scholar
  132. 132.
    Pineda, C., Barthel, T., Eisert, J.: Unitary circuits for strongly correlated fermions. Phys. Rev. A 81, 050303 (2010)ADSCrossRefGoogle Scholar
  133. 133.
    Troyer, M., Wiese, U.-J.: Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations. Phys. Rev. Lett. 94, 170201 (2005)ADSCrossRefGoogle Scholar
  134. 134.
    Kim, K., Chang, M.-S., Korenblit, S., Islam, R., Edwards, E.E., Freericks, J.K., Lin, G.-D., Duan, L.-M., Monroe, C.: Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590 (2010)ADSCrossRefGoogle Scholar
  135. 135.
    Jördens, R., Tarruell, L., Greif, D., Uehlinger, T., Strohmaier, N., Moritz, H., Esslinger, T., DeLeo, L., Kollath, C., Georges, A., Scarola, V., Pollet, L., Burovski, E., Kozik, E., Troyer, M.: Quantitative determination of temperature in the approach to magnetic order of ultracold fermions in an optical lattice. Phys. Rev. Lett. 104, 180401 (2010)CrossRefGoogle Scholar
  136. 136.
    Temme, K., Wolf, M.M., Verstraete, F.: Stochastic exclusion processes versus coherent transport. e-print arXiv:0912.0858 (2009)Google Scholar
  137. 137.
    Verstraete, F., Wolf, M.M., Cirac, J.I.: Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5, 633 (2009)CrossRefGoogle Scholar
  138. 138.
    Kraus, B., Büchler, H.P., Diehl, S., Kantian, A., Micheli, A., Zoller, P.: Preparation of entangled states by quantum Markov processes. Phys. Rev. A 78, 042307 (2008)ADSCrossRefGoogle Scholar
  139. 139.
    Diehl, S., Micheli, A., Kantian, A., Kraus, B., Büchler, H.P., Zoller, P.: Quantum states and phases in driven open quantum systems with cold atoms. Nat. Phys. 4, 878 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.ICFO–Institut de Ciències FotòniquesMediterranean Technology ParkBarcelonaSpain

Personalised recommendations