Advertisement

Journal of High Energy Physics

, 2016:124 | Cite as

Entanglement entropy after selective measurements in quantum chains

  • Khadijeh Najafi
  • M.A. RajabpourEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We study bipartite post measurement entanglement entropy after selective measurements in quantum chains. We first study the quantity for the critical systems that can be described by conformal field theories. We find a connection between post measurement entanglement entropy and the Casimir energy of floating objects. Then we provide formulas for the post measurement entanglement entropy for open and finite temperature systems. We also comment on the Affleck-Ludwig boundary entropy in the context of the post measurement entanglement entropy. Finally, we also provide some formulas regarding modular hamiltonians and entanglement spectrum in the after measurement systems. After through discussion regarding CFT systems we also provide some predictions regarding massive field theories. We then discuss a generic method to calculate the post measurement entanglement entropy in the free fermion systems. Using the method we study the post measurement entanglement entropy in the XY spin chain. We check numerically the CFT and the massive field theory results in the transverse field Ising chain and the XX model. In particular, we study the post meaurement entanglement entropy in the infinite, periodic and open critical transverse field Ising chain and the critical XX model. The effect of the temperature and the gap is also discussed in these models.

Keywords

Boundary Quantum Field Theory Conformal and W Symmetry Conformal Field Theory Field Theories in Lower Dimensions 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    L. Bombelli et al., A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  2. [2]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  4. [4]
    C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 06 (2004) P06002 [hep-th/0405152].zbMATHGoogle Scholar
  6. [6]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  7. [7]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  9. [9]
    J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories, J. Phys. A 42 (2009) 504006 [arXiv:0906.2946] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  11. [11]
    L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517 [quant-ph/0703044] [INSPIRE].
  12. [12]
    K. Modi et al., The classical-quantum boundary for correlations: discord and related measures, Rev. Mod. Phys. 84 (2012) 1655.ADSCrossRefGoogle Scholar
  13. [13]
    J. Eisert, M. Cramer and M.B. Plenio, Area laws for the entanglement entropyA review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  15. [15]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory I, J. Stat. Mech. 21 (2009) P211001 [arXiv:0905.2069].Google Scholar
  16. [16]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 01 (2011) P01021 [arXiv:1011.5482] [INSPIRE].MathSciNetGoogle Scholar
  17. [17]
    A. Coser, L. Tagliacozzo and E. Tonni, On Rényi entropies of disjoint intervals in conformal field theory, J. Stat. Mech. 01 (2014) P01008 [arXiv:1309.2189].CrossRefGoogle Scholar
  18. [18]
    B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP 11 (2013) 164 [arXiv:1309.5453] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    B. Chen and J.-q. Wu, Holographic calculation for large interval Rényi entropy at high temperature, Phys. Rev. D 92 (2015) 106001 [arXiv:1506.03206] [INSPIRE].ADSGoogle Scholar
  20. [20]
    A. Coser, E. Tonni, P. Calabrese, Spin structures and entanglement of two disjoint intervals in conformal field theories, J. Stat. Mech. 05 (2016) 053109 [arXiv:1511.08328].MathSciNetCrossRefGoogle Scholar
  21. [21]
    P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP 05 (2016) 127 [arXiv:1601.06794] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    S. Furukawa, V. Pasquier and J. Shiraishi, Mutual information and compactification radius in a c = 1 critical phase in one dimension, Phys. Rev. Lett. 102 (2009) 170602 [arXiv:0809.5113] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    M. Caraglio and F. Gliozzi, Entanglement entropy and twist fields, JHEP 11 (2008) 076 [arXiv:0808.4094] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    V. Alba, L. Tagliacozzo and P. Calabrese, Entanglement entropy of two disjoint blocks in critical Ising models, Phys. Rev. B 81 (2010) 060411 [arXiv:0910.0706] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    M. Fagotti and P Calabrese, Entanglement entropy of two disjoint blocks in XY chains, J. Stat. Mech. 04 (2010) P04016 [arXiv:1003.1110].
  26. [26]
    V. Alba, L. Tagliacozzo and P. Calabrese, Entanglement entropy of two disjoint intervals in c = 1 theories, J. Stat. Mech. 06 (2011) P06012 [arXiv:1103.3166] [INSPIRE].MathSciNetGoogle Scholar
  27. [27]
    M.A. Rajabpour and F.Gliozzi, Entanglement entropy of two disjoint intervals from fusion algebra of twist fields, J. Stat. Mech. 02 (2012) P02016 [arXiv:1112.1225].
  28. [28]
    F.C. Alcaraz, M.I. Berganza and G. Sierra, Entanglement of low-energy excitations in conformal field theory, Phys. Rev. Lett. 106 (2011) 201601 [arXiv:1101.2881] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M. Berganza, F. Alcaraz and G. Sierra, Entanglement of excited states in critical spin chains, J. Stat. Mech. 01 (2012) P01016 [arXiv:1109.5673] [INSPIRE].Google Scholar
  30. [30]
    M.M. Sheikh-Jabbari and H. Yavartanoo, Excitation entanglement entropy in 2d conformal field theories, arXiv:1605.00341 [INSPIRE].
  31. [31]
    J. Lee et al., Quantum information and precision measurement, J. Mod. Opt. 47 (2000) 2151 [quant-ph/9904021].
  32. [32]
    G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    F. Verstraete, M. Popp and J.I. Cirac, Entanglement versus correlations in spin systems, Phys. Rev. Lett. 92 (2004) 027901 [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    F. Verstraete, M.A. Martin-Delgado and J.I. Cirac, Diverging entanglement length in gapped quantum spin systems, Phys. Rev. Lett. 92 (2004) 087201.ADSCrossRefGoogle Scholar
  35. [35]
    M. Popp, F. Verstraete, M.A. Martin-Delgado and J.I. Cirac, Localizable entanglement, Phys. Rev. A 71 (2005) 042306.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    S. O. Skrøvseth, S. D. Bartlett, Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality, Phys. Rev. A 80 (2009) 022316.ADSCrossRefGoogle Scholar
  37. [37]
    T.B. Wahl, D. Perez-Garcia and J.I. Cirac, Matrix product states with long-range localizable entanglement, Phys. Rev. A 86 (2012) 062314.ADSCrossRefGoogle Scholar
  38. [38]
    J.I. Cirac, Entanglement in many-body quantum systems, arXiv:1205.3742.
  39. [39]
    A. Bayat, P. Sodano and S. Bose, Negativity as the entanglement measure to probe the Kondo regime in the spin-chain Kondo model, Phys. Rev. B 81 (2010) 064429 [arXiv:0904.3341] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    A. Bayat, S. Bose, P. Sodano and H. Johannesson, Entanglement probe of two-impurity Kondo physics in a spin chain, Phys. Rev. Lett. 109 (2012) 066403 [arXiv:1201.6668] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: a field theoretical approach, J. Stat. Mech. 02 (2013) P02008 [arXiv:1210.5359].MathSciNetCrossRefGoogle Scholar
  43. [43]
    P. Calabrese, J. Cardy and E. Tonni, Finite temperature entanglement negativity in conformal field theory, J. Phys. A 48 (2015) 015006 [arXiv:1408.3043] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  44. [44]
    V. Alba, Entanglement negativity and conformal field theory: a Monte Carlo study, J. Stat. Mech. 05 (2013) P05013 [arXiv:1302.1110] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  45. [45]
    P. Calabrese, L. Tagliacozzo and E. Tonni, Entanglement negativity in the critical Ising chain, J. Stat. Mech. 05 (2013) P05002 [arXiv:1302.1113].MathSciNetCrossRefGoogle Scholar
  46. [46]
    C. Eltschka and J. Siewert, Negativity as an estimator of entanglement dimension, Phys. Rev. Lett. 111 (2013) 100503.ADSCrossRefGoogle Scholar
  47. [47]
    Y.A. Lee and G. Vidal, Entanglement negativity and topological order, Phys. Rev. A 88 (2013) 042318.ADSCrossRefGoogle Scholar
  48. [48]
    C. Chung et al., Entanglement negativity via the replica trick: a quantum Monte Carlo approach, Phys. Rev. B 90 (2014) 064401.ADSCrossRefGoogle Scholar
  49. [49]
    E. Bianchi and M. Smerlak, Entanglement entropy and negative energy in two dimensions, Phys. Rev. D 90 (2014) 041904 [arXiv:1404.0602] [INSPIRE].ADSGoogle Scholar
  50. [50]
    V. Eisler and Z. Zimboras, Evidence for large electric polarization from collinear magnetism in TmMnO3, New J. Phys. 16 (2014) 123020 [arXiv:0901.0787].ADSCrossRefGoogle Scholar
  51. [51]
    M. Rangamani and M. Rota, Comments on entanglement negativity in holographic field theories, JHEP 10 (2014) 060.ADSCrossRefGoogle Scholar
  52. [52]
    C. De Nobili, A. Coser and E. Tonni, Entanglement entropy and negativity of disjoint intervals in CFT: Some numerical extrapolations, J. Stat. Mech. 06 (2015) P06021 [arXiv:1501.04311].MathSciNetCrossRefGoogle Scholar
  53. [53]
    E. Perlmutter, M. Rangamani and M. Rota, Central charges and the sign of entanglement in 4D conformal field theories, Phys. Rev. Lett. 115 (2015) 171601 [arXiv:1506.01679] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    A. Coser, E. Tonni, P. Calabrese, Towards the entanglement negativity of two disjoint intervals for a one dimensional free fermion, J. Stat. Mech. 03 (2016) 033116.MathSciNetCrossRefGoogle Scholar
  55. [55]
    V. Eisler and Z. Zimborás, Entanglement negativity in two-dimensional free lattice models, Phys. Rev. B 93 (2016) 115148.ADSCrossRefGoogle Scholar
  56. [56]
    C.P. Herzog and Y. Wang, Estimation for entanglement negativity of free fermions, J. Stat. Mech. 1607 (2016) 073102 [arXiv:1601.00678] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  57. [57]
    P.Y. Chang and X. Wen, Entanglement negativity in free-fermion systems: An overlap matrix approach, Phys. Rev. B 93 (2016) 195140.ADSCrossRefGoogle Scholar
  58. [58]
    B. Alkurtass et al., Entanglement structure of the two-channel Kondo model, Phys. Rev. B 93 (2016) 081106.ADSCrossRefGoogle Scholar
  59. [59]
    P. Chaturvedi, V. Malvimat and G. Sengupta, Entanglement negativity, holography and black holes, arXiv:1602.01147 [INSPIRE].
  60. [60]
    C. De Nobili, A. Coser and E. Tonni, Entanglement negativity in a two dimensional harmonic lattice: Area law and corner contributions, J. Stat. Mech. 1608 (2016) 083102 [arXiv:1604.02609] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  61. [61]
    K.-Y. Kim, C. Niu and D.-W. Pang, Universal corner contributions to entanglement negativity, Nucl. Phys. B 910 (2016) 528 [arXiv:1604.06891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    P. Ruggiero, V. Alba and P. Calabrese, Negativity spectrum of one-dimensional conformal field theories, Phys. Rev. B 94 (2016) 195121 [arXiv:1607.02992] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    M.A. Rajabpour, Post measurement bipartite entanglement entropy in conformal field theories, Phys. Rev. B 92 (2015) 075108 [arXiv:1501.07831] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    F.C. Alcaraz and M.A. Rajabpour, Universal behavior of the Shannon mutual information of critical quantum chains, Phys. Rev. Lett. 111 (2013) 017201 [arXiv:1305.1239] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    J.M. Stéphan, Shannon and Rényi mutual information in quantum critical spin chains, Phys. Rev. B 90 (2014) 045424.ADSCrossRefGoogle Scholar
  66. [66]
    F.C. Alcaraz and M.A. Rajabpour, Universal behavior of the Shannon and Rényi mutual information of quantum critical chains, Phys. Rev. B 90 (2014) 075132 [arXiv:1405.1074] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    F.C. Alcaraz and M.A. Rajabpour, Generalized mutual information of quantum critical chains, Phys. Rev. B 91 (2015) 155122 [arXiv:1501.02852] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    F.C. Alcaraz, Universal behavior of the Shannon mutual information in nonintegrable self-dual quantum chains, Phys. Rev. B 94 (2016) 115116.ADSCrossRefGoogle Scholar
  69. [69]
    J.M. Stéphan, Emptiness formation probability, Toeplitz determinants, and conformal field theory, J. Stat. Mech. 05 (2014) P05010 [arXiv:1303.5499].MathSciNetCrossRefGoogle Scholar
  70. [70]
    K. Najafi and M.A. Rajabpour, Formation probabilities and Shannon information and their time evolution after quantum quench in the transverse-field XY chain, Phys. Rev. B 93 (2016) 125139.ADSCrossRefGoogle Scholar
  71. [71]
    M.A. Rajabpour, Formation probabilities in quantum critical chains and Casimir effect, Eur. Phys. Lett. 112 (2015) 66001.ADSCrossRefGoogle Scholar
  72. [72]
    M.A. Rajabpour, Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories, J. Stat. Mech. 12 (2016) 123101 [arXiv:1607.07016].MathSciNetCrossRefGoogle Scholar
  73. [73]
    M.A. Rajabpour, Entanglement entropy after a partial projective measurement in 1 + 1 dimensional conformal field theories: exact results, J. Stat. Mech. 06 (2016) 063109.MathSciNetCrossRefGoogle Scholar
  74. [74]
    J. Cardy, The entanglement gap in CFTs, talk given at Closing the entanglement gap: Quantum information, quantum matter, and quantum fields, June 1‒5, KITP, Santa Barbara,U.S.A. (2015).Google Scholar
  75. [75]
    B.B. Machta, S. L. Veatch and J.P. Sethna, Critical Casimir forces in cellular membranes, Phys. Rev. Lett. 109 (2012) 138101.ADSCrossRefGoogle Scholar
  76. [76]
    G. Bimonte, T. Emig and M. Kardar, Conformal field theory of critical Casimir interactions in 2D, Eur. Phys. Lett. 104 (2013) 21001 [arXiv:1307.3993].ADSCrossRefGoogle Scholar
  77. [77]
    M.A. Rajabpour, Fate of the area-law after partial measurement in quantum field theories, arXiv:1503.07771 [INSPIRE].
  78. [78]
    T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, EPR pairs, local projections and quantum teleportation in holography, JHEP 08 (2016) 077 [arXiv:1604.01772] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  79. [79]
    Y. Huang, Computing quantum discord is NP-complete, New J. Phys. 16 (2014) 033027.ADSMathSciNetCrossRefGoogle Scholar
  80. [80]
    K. Najafi and M.A. Rajabpour, A lower bound for localizable entanglement in quantum critical chains, in preparation.Google Scholar
  81. [81]
    J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
  82. [82]
    I. Affleck and A.W.W. Ludwig, Universal nonintegerground state degeneracyin critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    J. Cardy and P. Calabrese, Unusual corrections to scaling in entanglement entropy, J. Stat. Mech. 04 (2010) P04023 [arXiv:1002.4353].Google Scholar
  84. [84]
    Z. Nehari, Conformal mapping, McGraw-Hill, New York U.S.A. (1952).zbMATHGoogle Scholar
  85. [85]
    O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive QFT with a boundary: The Ising model, J. Statist. Phys. 134 (2009) 105 [arXiv:0810.0219] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    H.Q. Zhou et al., Entanglement and boundary critical phenomena, Phys. Rev. A 74 (2006) 050305(R).
  87. [87]
    N. Laflorencie, E.S. Sørensen, M.-S. Chang and I. Affleck, Boundary effects in the critical scaling of entanglement entropy in 1D systems, Phys. Rev. Lett. 96 (2006) 100603 [cond-mat/0512475] [INSPIRE].
  88. [88]
    D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402 [hep-th/0312197] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  89. [89]
    D.R. Green, M. Mulligan and D. Starr, Boundary entropy can increase under bulk RG flow, Nucl. Phys. B 798 (2008) 491 [arXiv:0710.4348] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  90. [90]
    T. Azeyanagi, A. Karch, T. Takayanagi and E.G. Thompson, Holographic calculation of boundary entropy, JHEP 03 (2008) 054 [arXiv:0712.1850] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  91. [91]
    J.M. Stéphan and J. Dubail, Logarithmic corrections to the free energy from sharp corners with angle 2π, J. Stat. Mech. 09 (2013) P09002 [arXiv:1303.3633].MathSciNetCrossRefGoogle Scholar
  92. [92]
    J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, arXiv:1608.01283 [INSPIRE].
  93. [93]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    H. Casini and M. Huerta, Entanglement entropy for the n-sphere, Phys. Lett. B 694 (2011) 167 [arXiv:1007.1813] [INSPIRE].ADSMathSciNetGoogle Scholar
  95. [95]
    H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. 26 (2009) 185005 [arXiv:0903.5284] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement temperature and entanglement entropy of excited states, JHEP 12 (2013) 020 [arXiv:1305.3291] [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys . Rev. A 78 (2008) 032329.ADSCrossRefGoogle Scholar
  99. [99]
    F. Loran, M.M. Sheikh-Jabbari and M. Vincon, Beyond logarithmic corrections to Cardy formula, JHEP 01 (2011) 110 [arXiv:1010.3561] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  100. [100]
    I. Peschel, On the entanglement entropy for an XY spin chain, J. Stat. Mech. 12 (2004) P12005 [cond-mat/0410416].
  101. [101]
    A.R. Its, B.Q. Jin and V.E. Korepin, Entanglement in XY spin chain, J. Phys. A 38 (2005) 2975 [quant-ph/0409027].
  102. [102]
    F. Franchini, A.R. Its and V.E. Korepin, Rényi entropy of the XY spin chain, J. Phys. A 41 (2008) 025302 [arXiv:0707.2534] [INSPIRE].ADSzbMATHGoogle Scholar
  103. [103]
    R. Weston, The entanglement entropy of solvable lattice models, J. Stat. Mech. 03 (2006) L03002 [math-ph/0601038].
  104. [104]
    F. Franchini, A.R. Its, B.Q. Jin and V.E. Korepin, Ellipses of constant entropy in the XY spin chain, J. Phys. A 40 (2007) 8467 [quant-ph/0609098] [INSPIRE].
  105. [105]
    A.R. Its, F. Mezzadri and M.Y. Mo, Entanglement entropy in quantum spin chains with finite range interaction, Commun. Math. Phys. 284 (2008) 117 [arXiv:0708.0161].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  106. [106]
    C.P. Herzog and M. Spillane, Tracing through scalar entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].ADSGoogle Scholar
  107. [107]
    M.C. Chung and I. Peschel, Density-matrix spectra for two-dimensional quantum systems, Phys. Rev. B 62 (2000) 4191.ADSCrossRefGoogle Scholar
  108. [108]
    I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36 (2003) L205.ADSMathSciNetzbMATHGoogle Scholar
  109. [109]
    T. Barthel and U. Schollwock, Dephasing and the steady state in quantum many-particle systems, Phys. Rev. Lett. 100 (2008) 100601.ADSCrossRefGoogle Scholar
  110. [110]
    E. Barouch and B. McCoy, Statistical mechanics of the XY model. II. Spin-correlation functions, Phys. Rev. A 3 (1971) 786.ADSCrossRefGoogle Scholar
  111. [111]
    M. Fagotti, Entanglement and correlations in exactly solvable models, Ph.D. thesis, ETD Universit à di Pisa, Pisa, Italy (2011).Google Scholar
  112. [112]
    J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  113. [113]
    E. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16 (1961) 407.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  114. [114]
    F.C. Alcaraz, M. Baake, U. Grimm and V. Rittenberg, Operator content of the XXZ chain, J. Phys. A 21 (1988) L117.ADSMathSciNetGoogle Scholar
  115. [115]
    U. Bilstein, The XX-model with boundaries: III. Magnetization profiles and boundary bound states, J. Phys. A 33 (2000) 4437.ADSMathSciNetzbMATHGoogle Scholar
  116. [116]
    J. Cardy, Measuring entanglement using quantum quenches, Phys. Rev. Lett. 106 (2011) 150404 [arXiv:1012.5116] [INSPIRE].ADSCrossRefGoogle Scholar
  117. [117]
    D.A. Abanin and E. Demler, Measuring entanglement entropy of a generic many-body system with a quantum switch, Phys. Rev. Lett. 109 (2012) 020504.ADSCrossRefGoogle Scholar
  118. [118]
    A. Daley, H. Pichler, J. Schachenmayer and P. Zoller, Measuring entanglement growth in quench dynamics of bosons in an optical lattice, Phys. Rev. Lett. 109 (2012) 020505.ADSCrossRefGoogle Scholar
  119. [119]
    L. Banchi, A. Bayat, S. Bose, Entanglement entropy scaling in solid-state spin arrays via capacitance measurements, arXiv:1608.03970.
  120. [120]
    R. Islam et al., Measuring entanglement entropy in a quantum many-body system, Nature 528 (2015) 77.ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of PhysicsGeorgetown UniversityNW, WashingtonU.S.A.
  2. 2.Instituto de FísicaUniversidade Federal FluminenseNiterói, RJBrazil

Personalised recommendations