Advertisement

Frontiers of Physics in China

, Volume 5, Issue 1, pp 1–25 | Cite as

Spin qubits for quantum simulations

  • Xin-hua Peng (彭新华)
  • Dieter Suter
Review Article

Abstract

The investigation of quantum mechanical systems mostly concentrates on single elementary particles. If we combine such particles into a composite quantum system, the number of degrees of freedom of the combined system grows exponentially with the number of particles. This is a major difficulty when we try to describe the dynamics of such a system, since the computational resources required for this task also grow exponentially. In the context of quantum information processing, this difficulty becomes the main source of power: in some situations, information processors based in quantum mechanics can process information exponentially faster than classical systems. From the perspective of a physicist, one of the most interesting applications of this type of information processing is the simulation of quantum systems. We call a quantum information processor that simulates other quantum systems a quantum simulator.

This review discusses a specific type of quantum simulator, based on nuclear spin qubits, and using nuclear magnetic resonance for processing. We review the basics of quantum information processing by nuclear magnetic resonance (NMR) as well as the fundamentals of quantum simulation and describe some simple applications that can readily be realized by today’s quantum computers. In particular, we discuss the simulation of quantum phase transitions: the qualitative changes that the ground states of some quantum mechanical systems exhibit when some parameters in their Hamiltonians change through some critical points. As specific examples, we consider quantum phase transitions where the relevant ground states are entangled. Chains of spins coupled by Heisenberg interactions represent an ideal system for studying these effects: depending on the type and strength of interactions, the ground states can be product states or they can be maximally entangled states representing different types of entanglement.

Keywords

quantum simulation quantum computation quantum information quantum phase transition nuclear magnetic resonance (NMR) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. E. Moore, Electronics, 1965, 38: 114Google Scholar
  2. 2.
    P. S. Peercy, Nature, 2000, 406: 1023Google Scholar
  3. 3.
    L. B. Kish, Phys. Lett. A, 2002, 305: 144ADSGoogle Scholar
  4. 4.
    R. Landauer, Phys. Today, 1991, May: 23Google Scholar
  5. 5.
    R. P. Feynman, International Journal of Theoretical Physics, 1982, 21: 467MathSciNetADSGoogle Scholar
  6. 6.
    P. Benioff, J. Stat. Phys., 1982, 29: 515zbMATHMathSciNetADSGoogle Scholar
  7. 7.
    E. Bernstein and U. Vazirani, Quantum complexity theory, in: Proc. 25th ACM Symp. Theory Comp., 1993: 11Google Scholar
  8. 8.
    D. Coppersmith, arXiv: quant-ph/0201067, 1994Google Scholar
  9. 9.
    Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Piscataway, NJ: IEEE Press, 1994Google Scholar
  10. 10.
    S. Lloyd, Science, 1996, 273: 1073MathSciNetADSGoogle Scholar
  11. 11.
    R. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A, 2002, 65: 042323ADSGoogle Scholar
  12. 12.
    D. S. Abrams and S. Lloyd, Phys. Rev. Lett., 1997, 79: 2586ADSGoogle Scholar
  13. 13.
    D. S. Abrams and S. Lloyd, Phys. Rev. Lett., 1999, 83: 5162ADSGoogle Scholar
  14. 14.
    C. Zalka, Proc. R. Soc. Lond. A, 1998, 454: 313zbMATHADSGoogle Scholar
  15. 15.
    S. Wiesner, arXiv: quant-ph/9603028, 1996Google Scholar
  16. 16.
    B. M. Boghosian and W. Taylor, arXiv: quantph/9701016v2, 1997Google Scholar
  17. 17.
    L. A. Wu, M. S. Byrd, and D. A. Lidar, Phys. Rev. Lett., 2002, 89: 057904MathSciNetADSGoogle Scholar
  18. 18.
    G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A, 2001, 64: 022319ADSGoogle Scholar
  19. 19.
    H. Wang, S. Kais, A. Aspuru-Guzik, and M. R. Hoff-mann, Phys. Chem. Chem. Phys., 2008, 10: 5388Google Scholar
  20. 20.
    A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Science, 2005, 309: 1704ADSGoogle Scholar
  21. 21.
    D. A. Lidar and H. Wang, Phys. Rev. E, 1999, 59: 2429ADSGoogle Scholar
  22. 22.
    A. Y. Smirnov, S. Savel’ev, L. G. Mourokh, and F. Nori, Europhys. Lett., 2007, 80: 67008ADSGoogle Scholar
  23. 23.
    I. Kassal, S. P. Jordan, P. J. Love, M. Mohseni, and A. Aspuru-Guzik, Proc. Nat. Acad. Sci. USA, 2008, 105: 18681ADSGoogle Scholar
  24. 24.
    M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature, 2002, 415: 39ADSGoogle Scholar
  25. 25.
    D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett., 1998, 81: 3108ADSGoogle Scholar
  26. 26.
    J. J. García-Ripoll, E. Solano, and M. A. Martin-Delgado, Phys. Rev. B, 2008, 77: 024522ADSGoogle Scholar
  27. 27.
    C. H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T. F. Havel, and D. G. Cory, Phys. Rev. A, 1999, 61: 012302ADSGoogle Scholar
  28. 28.
    S. Somaroo, C. H. Tseng, T. F. Havel, R. Laflamme, and D. G. Cory, Phys. Rev. Lett., 1999, 82: 5381ADSGoogle Scholar
  29. 29.
    C. H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T. F. Havel, and D. G. Cory, Phys. Rev. A, 2000, 62: 032309ADSGoogle Scholar
  30. 30.
    A. K. Khitrin and B. M. Fung, Phys. Rev. A, 2001, 64: 032306ADSGoogle Scholar
  31. 31.
    C. Negrevergne, R. Somma, G. Ortiz, E. Knill, and R. Laflamme, Phys. Rev. A, 2005, 71: 032344ADSGoogle Scholar
  32. 32.
    U. Haeberlen and J. S. Waugh, Phys. Rev., 1968, 175: 453ADSGoogle Scholar
  33. 33.
    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2001Google Scholar
  34. 34.
    J. Stolze and D. Suter, Quantum Computing: A Short Course from Theory to Experiment, 2nd Ed., Berlin: Wiley-VCH, 2008zbMATHGoogle Scholar
  35. 35.
    D. Deutsch, Proc. R. Soc. Lond. A, 1989, 425: 1934MathSciNetGoogle Scholar
  36. 36.
    D. Deutsch, Proc. R. Soc. Lond. A, 1985, 400: 1934MathSciNetGoogle Scholar
  37. 37.
    L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys., 2004, 76: 1037ADSGoogle Scholar
  38. 38.
    R. Laflamme, E. Knill, D. Cory, E. Fortunato, T. Havel, C. Miquel, R. Martinez, C. Negrevergne, G. Ortiz, M. Pravia, et al., arXiv: quant-ph/0207172v1, 2002Google Scholar
  39. 39.
    J. A. Jones and E. Knill, J. Magn. Res., 1999, 141: 322ADSGoogle Scholar
  40. 40.
    N. Linden, B. Herve, R. J. Carbajo, and R. Freeman, Chem. Phys. Lett., 1999, 305: 28ADSGoogle Scholar
  41. 41.
    N. Sinha, T. S. Mahesh, K. V. Ramanathan, and A. Kumar, J. Chem. Phys., 2001, 114: 4415ADSGoogle Scholar
  42. 42.
    A. K. Khitrin and B. M. Fung, J. Chem. Phys., 2000, 112: 6963ADSGoogle Scholar
  43. 43.
    J. Du, M. Shi, J. Wu, X. Zhou, and R. Han, Phys. Rev. A, 2001, 63: 042302ADSGoogle Scholar
  44. 44.
    K. Dorai, Arvind, and A. Kumar, Phys. Rev. A, 2000, 61: 042306MathSciNetADSGoogle Scholar
  45. 45.
    T. S. Mahesh, N. Sinha, K. V. Ramanathan, and A. Kumar, Phys. Rev. A, 2002, 65: 022312ADSGoogle Scholar
  46. 46.
    J. Du, J. Wu, M. Shi, L. Han, X. Zhou, B. Ye, H. Weng, and R. Han, Chin. Phys. Lett., 2000, 17: 64ADSGoogle Scholar
  47. 47.
    K. V. R. M. Murali, N. Sinha, T. S. Mahesh, M. H. Levitt, K. V. Ramanathan, and A. Kumar, Phys. Rev. A, 2002, 66: 022313ADSGoogle Scholar
  48. 48.
    T. S. Mahesh, K. Dorai, Arvind, and A. Kumar, J. Magn. Res., 2001, 148: 95ADSGoogle Scholar
  49. 49.
    N. Linden, H. Barjat, and R. Freeman, Chem. Phys. Lett., 1998, 296: 61ADSGoogle Scholar
  50. 50.
    D. G. Cory, M. D. Price, and T. F. Havel, Physica D, 1998, 120: 82. In: Proceedings of the Fourth Workshop on Physics and ConsumptionGoogle Scholar
  51. 51.
    K. Dorai, Arvind, and A. Kumar, Phys. Rev. A, 2001, 63: 034101ADSGoogle Scholar
  52. 52.
    R. Das, T. S. Mahesh, and A. Kumar, J. Magn. Res., 2002, 159: 46, ISSN 1090-7807ADSGoogle Scholar
  53. 53.
    R. Das, T. S. Mahesh, and A. Kumar, Chem. Phys. Lett., 2003, 369: 8, ISSN 0009-2614ADSGoogle Scholar
  54. 54.
    D. G. Cory, A. F. Fahmy, and T. F. Havel, Proc. Nat. Acad. Sci. USA, 1997, 94: 1634ADSGoogle Scholar
  55. 55.
    N. A. Gershenfeld and I. L. Chuang, Science, 1997, 275: 350MathSciNetGoogle Scholar
  56. 56.
    E. Knill, I. Chuang, and R. Laflamme, Phys. Rev. A, 1998, 57: 3348MathSciNetADSGoogle Scholar
  57. 57.
    L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, Phys. Rev. Lett., 2000, 85: 5452ADSGoogle Scholar
  58. 58.
    X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, Phys. Rev. A, 2002, 65: 042315ADSGoogle Scholar
  59. 59.
    X. Peng, X. Zhu, X. Fang, M. Feng, K. Gao, X. Yang, and M. Liu, Chem. Phys. Lett., 2001, 340: 509ADSGoogle Scholar
  60. 60.
    Y. Sharf, T. F. Havel, and D. G. Cory, Phys. Rev. A, 2000, 62: 052314ADSGoogle Scholar
  61. 61.
    U. Sakaguchi, H. Ozawa, and T. Fukumi, Phys. Rev. A, 2000, 61: 042313MathSciNetADSGoogle Scholar
  62. 62.
    Z. L. Mádi, R. Brüschweiler, and R. R. Ernst, J. Chem. Phys., 1998, 109: 10603Google Scholar
  63. 63.
    X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, J. Chem. Phys., 2004, 120: 3579ADSGoogle Scholar
  64. 64.
    E. Knill, R. Laflamme, R. Martinez, and C.-H. Tseng, Nature, 2000, 404: 368ADSGoogle Scholar
  65. 65.
    B. M. Fung, Phys. Rev. A, 2001, 63: 022304MathSciNetADSGoogle Scholar
  66. 66.
    X. Peng, X. Zhu, X. Fang, M. Feng, X. Yang, M. Liu, and K. Gao, arXiv: quant-ph/0202010, 2002Google Scholar
  67. 67.
    W. S. Warren, Science, 1997, 277: 1688Google Scholar
  68. 68.
    D. Suter and T. S. Mahesh, J. Chem. Phys. 2008, 128: 052206ADSGoogle Scholar
  69. 69.
    G. L. Long, H. Y. Yan, and Y. Sun, J. Opt. B, 2001, 3: 376ADSGoogle Scholar
  70. 70.
    E. M. Fortunato, M. A. Pravia, N. Boulant, G. Teklemariam, T. F. Havel, and D. G. Cory, J. Chem. Phys., 2002, 116: 7599ADSGoogle Scholar
  71. 71.
    R. Das, T. S. Mahesh, and A. Kumar, Phys. Rev. A, 2003, 67: 062304ADSGoogle Scholar
  72. 72.
    E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science, 2001, 292: 472MathSciNetADSGoogle Scholar
  73. 73.
    A. Mizel, D. A. Lidar, and M. Mitchell, Phys. Rev. Lett., 2007, 99: 070502ADSGoogle Scholar
  74. 74.
    M. H. S. Amin, Phys. Rev. Lett., 2008, 100: 130503ADSGoogle Scholar
  75. 75.
    J. Roland and N. J. Cerf, Phys. Rev. A, 2002, 65: 042308ADSGoogle Scholar
  76. 76.
    M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, Phys. Rev. Lett., 2003, 90: 067903ADSGoogle Scholar
  77. 77.
    X. Peng, Z. Liao, N. Xu, G. Qin, X. Zhou, D. Suter, and J. Du, Phys. Rev. Lett., 2008, 101: 145501ADSGoogle Scholar
  78. 78.
    A. Mitra, A. Ghosh, R. Das, A. Patel, and A. Kumar, J. Magn. Res., 2005, 177: 285ADSGoogle Scholar
  79. 79.
    J. Roland and N. J. Cerf, Phys. Rev. A, 2005, 71: 032330MathSciNetADSGoogle Scholar
  80. 80.
    A. M. Childs, E. Farhi, and J. Preskill, Phys. Rev. A, 2001, 65: 012322ADSGoogle Scholar
  81. 81.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979zbMATHGoogle Scholar
  82. 82.
    M. Žnidariĉ and M. Horvat, Phys. Rev. A, 2006, 73: 022329ADSGoogle Scholar
  83. 83.
    T. Hogg, Phys. Rev. A, 2003, 67: 022314ADSGoogle Scholar
  84. 84.
    M. Žnidariĉ, Phys. Rev. A, 2005, 71: 062305ADSGoogle Scholar
  85. 85.
    S. Blanes, F. Casas, J. Oteo, and J. Ros, Physics Reports, 2009, 470: 151MathSciNetADSGoogle Scholar
  86. 86.
    M. Suzuki, Quantum Monte Carlo Methods in Condensed-Matter Physics, Singapore: World Scientific, 1993Google Scholar
  87. 87.
    W. K. Wootters, Phys. Rev. Lett., 1998, 80: 2245ADSGoogle Scholar
  88. 88.
    V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A, 2000, 61: 052306ADSGoogle Scholar
  89. 89.
    P. Rungta and C.M. Caves, Phys. Rev. A, 2003, 67: 012307ADSGoogle Scholar
  90. 90.
    P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, Phys. Rev. A, 2001, 64: 042315MathSciNetADSGoogle Scholar
  91. 91.
    W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A, 2000, 62: 062314MathSciNetADSGoogle Scholar
  92. 92.
    B. M. Terhal, Phys. Lett. A, 2000, 271: 319zbMATHMathSciNetADSGoogle Scholar
  93. 93.
    M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A, 1996, 223: 1zbMATHMathSciNetADSGoogle Scholar
  94. 94.
    M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Phys. Rev. A, 2000, 62: 052310ADSGoogle Scholar
  95. 95.
    A. Sanpera, D. Bruβ, and M. Lewenstein, Phys. Rev. A, 2001, 63: 050301ADSGoogle Scholar
  96. 96.
    S. Sachdev, Quantum Phase Transition, Cambridge: Cambrige: University Press, 1999Google Scholar
  97. 97.
    P. C. Canfield, Nature Phys., 2008, 4: 167ADSGoogle Scholar
  98. 98.
    H. M. Ronnow, R. Parthasarathy, J. Jensen, G. Aeppli, T. F. Rosenbaum, and D. F. McMorrow, Science, 2005, 308: 389ADSGoogle Scholar
  99. 99.
    J. Custers, P. Gegenwart, H. Wilhelm, K. Neumaier, Y. Tokiwa, O. Trovarelli, C. Geibel, F. Steglich, C. Pépin, and P. Coleman, Nature, 2003, 424: 524ADSGoogle Scholar
  100. 100.
    A. Yeh, Y. A. Soh, J. Brooke, G. Aeppli, T. F. Rosenbaum, and S. M. Hayden, Nature, 2002, 419: 459ADSGoogle Scholar
  101. 101.
    T. Giamarchi, C. Ruegg, and O. Tchernyshyov, Nature Phys., 2008, 4: 198ADSGoogle Scholar
  102. 102.
    P. Gegenwart, Q. Si, and F. Steglich, Nature Phys., 2008, 4: 186ADSGoogle Scholar
  103. 103.
    S. Sachdev, Nature Phys., 2008, 4: 173ADSGoogle Scholar
  104. 104.
    D. M. Broun, Nature Phys., 2008, 4: 170ADSGoogle Scholar
  105. 105.
    Editorial, Nature Phys., 2008, 4: 157Google Scholar
  106. 106.
    A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature, 2002, 416: 608ADSGoogle Scholar
  107. 107.
    T. J. Osborne and M. A. Nielsen, Phys. Rev. A, 2002, 66: 032110MathSciNetADSGoogle Scholar
  108. 108.
    M. C. Arnesen, S. Bose, and V. Vedral, Phys. Rev. Lett., 2001, 87: 017901ADSGoogle Scholar
  109. 109.
    R. Somma, G. Ortiz, H. Barnum, E. Knill, and L. Viola, Phys. Rev. A, 2004, 70: 042311MathSciNetADSGoogle Scholar
  110. 110.
    S. J. Gu, S. S. Deng, Y. Q. Li, and H. Q. Lin, Phys. Rev. Lett., 2004, 93: 086402ADSGoogle Scholar
  111. 111.
    F. Gebbhard, The Mott Metal-Insulator Transition: Models and Methods, New York: Springer-Verlag, 1997Google Scholar
  112. 112.
    R. B. Laughlin, Phys. Rev. Lett., 1983, 50: 1395ADSGoogle Scholar
  113. 113.
    L. Zhou, H. S. Song, Y. Q. Guo, and C. Li, Phys. Rev. A, 2003, 68: 024301ADSGoogle Scholar
  114. 114.
    X. Wang, Phys. Rev. A, 2002, 66: 034302ADSGoogle Scholar
  115. 115.
    G. Lagmago Kamta and A. F. Starace, Phys. Rev. Lett., 2002, 88: 107901ADSGoogle Scholar
  116. 116.
    R. J. Baxter and F. Y. Wu, Phys. Rev. Lett., 1973, 31: 1294ADSGoogle Scholar
  117. 117.
    F. Igloi, J. Phys. A: Math. Gen., 1987, 20: 5319MathSciNetADSGoogle Scholar
  118. 118.
    P. Lou, W. C. Wu, and M. C. Chang, Phys. Rev. B, 2004, 70: 064405ADSGoogle Scholar
  119. 119.
    P. Suranyi, Phys. Rev. Lett., 1976, 37: 725ADSGoogle Scholar
  120. 120.
    C. D’Cruz and J. K. Pachos, Phys. Rev. A, 2005, 72: 043608ADSGoogle Scholar
  121. 121.
    D. I. Tsomokos, J. J. García-Ripoll, N. R. Cooper, and J. K. Pachos, Phys. Rev. A, 2008, 77: 012106ADSGoogle Scholar
  122. 122.
    J. K. Pachos and E. Rico, Phys. Rev. A, 2004, 70: 053620ADSGoogle Scholar
  123. 123.
    J. K. Pachos and M. B. Plenio, Phys. Rev. Lett., 2004, 93: 056402ADSGoogle Scholar
  124. 124.
    H. P. Buchler, A. Micheli, and P. Zoller, Nature Phys., 2007, 3: 726ADSGoogle Scholar
  125. 125.
    J. C. Anglès d’Auriac and F. Iglói, Phys. Rev. E, 1998, 58: 241ADSGoogle Scholar
  126. 126.
    K. A. Penson, R. Jullien, and P. Pfeuty, Phys. Rev. B, 1982, 26: 6334ADSGoogle Scholar
  127. 127.
    K. A. Penson, J. M. Debierre, and L. Turban, Phys. Rev. B, 1988, 37: 7884ADSGoogle Scholar
  128. 128.
    L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Nature, 2001, 414: 883ADSGoogle Scholar
  129. 129.
    J. Zhang, et al., Phys. Rev. Lett., 2008, 100Google Scholar
  130. 130.
    N. Linden, E. Kupce, and R. Freeman, Chem. Phys. Lett., 1999, 311: 321ADSGoogle Scholar
  131. 131.
    I. L. Chuang, N. Gershenfeld, M. G. Kubinec, and D. W. Leung, Proc. R. Soc. Lond. A, 1998, 454: 447zbMATHADSGoogle Scholar
  132. 132.
    G. Teklemariam, E. M. Fortunato, M. A. Pravia, T. F. Havel, and D. G. Cory, Phys. Rev. Lett., 2001, 86: 5845ADSGoogle Scholar
  133. 133.
    A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, Nature Phys., 2008, 4: 757ADSGoogle Scholar
  134. 134.
    F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A, 2002, 65: 052112MathSciNetADSGoogle Scholar
  135. 135.
    O. Osenda, Z. Huang, and S. Kais, Phys. Rev. A, 2003, 67: 062321ADSGoogle Scholar
  136. 136.
    Z. Y. Sun, K. L. Yao, W. Yao, D. H. Zhang, and Z. L. Liu, Phys. Rev. B, 2008, 77: 014416ADSGoogle Scholar
  137. 137.
    V. Subrahmanyam, Phys. Rev. A, 2004, 69: 022311ADSGoogle Scholar
  138. 138.
    F. C. Alcaraz, A. Saguia, and M. S. Sarandy, Phys. Rev. A, 2004, 70: 032333ADSGoogle Scholar
  139. 139.
    X. Wang, Phys. Rev. A, 2001, 64: 012313ADSGoogle Scholar
  140. 140.
    J. Zhao, I. Peschel, and X. Wang, Phys. Rev. B, 2006, 73: 024405ADSGoogle Scholar
  141. 141.
    A. Kopp and K. L. Hur, Phys. Rev. Lett., 2007, 98: 220401ADSGoogle Scholar
  142. 142.
    X. Jia, A. R. Subramaniam, I. A. Gruzberg, and S. Chakravarty, Phys. Rev. B, 2008, 77: 014208ADSGoogle Scholar
  143. 143.
    L. Cincio, J. Dziarmaga, M. M. Rams, and W. H. Zurek, Phys. Rev. A, 2007, 75: 052321ADSGoogle Scholar
  144. 144.
    C. Wellard and R. Orùs, Phys. Rev. A, 2004, 70: 062318ADSGoogle Scholar
  145. 145.
    J. I. Latorre, E. Rico, and G. Vidal, Quant. Inf. Comput., 2004, 4: 48zbMATHMathSciNetGoogle Scholar
  146. 146.
    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett., 2003, 90: 227902ADSGoogle Scholar
  147. 147.
    M. F. Yang, Phys. Rev. A, 2005, 71: 030302ADSGoogle Scholar
  148. 148.
    T. R. de Oliveira, G. Rigolin, and M. C. de Oliveira, Phys. Rev. A, 2006, 73: 010305(R)Google Scholar
  149. 149.
    D. A. Meyer and N. R. Wallach, Journal of Mathematical Physics, 2002, 43: 4273zbMATHMathSciNetADSGoogle Scholar
  150. 150.
    X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, Phys. Rev. A, 2005, 72: 052109ADSGoogle Scholar
  151. 151.
    G. Schaller, Phys. Rev. A, 2008, 78: 032328MathSciNetADSGoogle Scholar
  152. 152.
    R. Schützhold and G. Schaller, Phys. Rev. A, 2006, 74: 060304ADSGoogle Scholar
  153. 153.
    J. I. Latorre and R. Orùs, Phys. Rev. A, 2004, 69: 062302ADSGoogle Scholar
  154. 154.
    T. Caneva, R. Fazio, and G. E. Santoro, arXiv: 0706. 1832v1, 2007Google Scholar
  155. 155.
    Quantum information processing and communication: strategic report on current status, visions and goals for research in EuropeGoogle Scholar
  156. 156.
    H. G. Krojanski and D. Suter, Phys. Rev. Lett., 2004, 93: 090501ADSGoogle Scholar
  157. 157.
    H. G. Krojanski and D. Suter, Phys. Rev. Lett., 2006, 97: 150503ADSGoogle Scholar
  158. 158.
    H. G. Krojanski and D. Suter, Phys. Rev. A, 2006, 74: 062319ADSGoogle Scholar
  159. 159.
    M. Lovric, H. G. Krojanski, and D. Suter, Phys. Rev. A, 2007, 75: 042305ADSGoogle Scholar
  160. 160.
    R. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, H. H. F. Schmidt-Kaler, I. Chuang, and R. Blatt, Nature (London), 2003, 421: 48ADSGoogle Scholar
  161. 161.
    P. Chen, C. Piermarocchi, and L. J. Sham, Phys. Rev. Lett., 2001, 87: 067401ADSGoogle Scholar
  162. 162.
    E. Collin, G. Ithier, A. Aassime, P. Joyez, D. Vion, and D. Esteve, Phys. Rev. Lett., 2004, 93: 157005ADSGoogle Scholar

Copyright information

© Higher Education Press and Springer Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern PhysicsUniversity of Science and Technology of ChinaHefei, AnhuiChina
  2. 2.Fakultät PhysikTechnische Universität DortmundDortmundGermany

Personalised recommendations