Advertisement

Quantum vs. classical information: operator negativity as a probe of scrambling

  • Jonah Kudler-FlamEmail author
  • Masahiro Nozaki
  • Shinsei Ryu
  • Mao Tian Tan
Open Access
Regular Article - Theoretical Physics
  • 15 Downloads

Abstract

We consider the logarithmic negativity and related quantities of time evolution operators. We study free fermion, compact boson, and holographic conformal field theories (CFTs) as well as numerical simulations of random unitary circuits and integrable and chaotic spin chains. The holographic behavior strongly deviates from known non- holographic CFT results and displays clear signatures of maximal scrambling. Intriguingly, the random unitary circuits display nearly identical behavior to the holographic channels. Generically, we find the “line-tension picture” to effectively capture the entanglement dynamics for chaotic systems and the “quasi-particle picture” for integrable systems. With this motivation, we propose an effective “line-tension” that captures the dynamics of the logarithmic negativity in chaotic systems in the spacetime scaling limit. We compare the negativity and mutual information leading us to find distinct dynamics of quantum and classical information. The “spurious entanglement” we observe may have implications on the “simulatability” of quantum systems on classical computers. Finally, we elucidate the connection between the operation of partially transposing a density matrix in conformal field theory and the entanglement wedge cross section in Anti-de Sitter space using geodesic Witten diagrams.

Keywords

Conformal Field Theory AdS-CFT Correspondence 

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]
    P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.4 (2005) P04010 [cond-mat/0503393].MathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Peres, Separability criterion for density matrices, Phys. Rev. Lett.77 (1996) 1413 [quant-ph/9604005] [INSPIRE].
  3. [3]
    M. Horodecki, P. Horodecki and R. Horodecki, On the necessary and sufficient conditions for separability of mixed quantum states, Phys. Lett.A 223 (1996) 1 [quant-ph/9605038] [INSPIRE].
  4. [4]
    J. Eisert and M.B. Plenio, A comparison of entanglement measures, J. Mod. Opt.46 (1999) 145 [quant-ph/9807034].
  5. [5]
    R. Simon, Peres-Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett.84 (2000) 2726 [quant-ph/9909044] [INSPIRE].
  6. [6]
    G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev.A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
  7. [7]
    M.B. Plenio, Logarithmic negativity: a full entanglement monotone that is not convex, Phys. Rev. Lett.95 (2005) 090503 [quant-ph/0505071] [INSPIRE].
  8. [8]
    M. Rangamani and M. Rota, Comments on entanglement negativity in holographic field theories, JHEP10 (2014) 060 [arXiv:1406.6989] [INSPIRE].
  9. [9]
    J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev.D 99 (2019) 106014 [arXiv:1808.00446] [INSPIRE].
  10. [10]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett.109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
  11. [11]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: A field theoretical approach, J. Stat. Mech.1302 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
  12. [12]
    P. Calabrese, L. Tagliacozzo and E. Tonni, Entanglement negativity in the critical Ising chain, J. Stat. Mech.5 (2013) P05002.MathSciNetCrossRefGoogle Scholar
  13. [13]
    V. Alba, Entanglement negativity and conformal field theory: a Monte Carlo study, J. Stat. Mech.5 (2013) P05013.Google Scholar
  14. [14]
    C.-M. Chung et al., Entanglement negativity via the replica trick: A quantum monte carlo approach, Phys. Rev.B 90 (2014) 064401.Google Scholar
  15. [15]
    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].
  16. [16]
    P. Ruggiero, V. Alba and P. Calabrese, Entanglement negativity in random spin chains, Phys. Rev.B 94 (2016) 035152 [arXiv:1605.00674] [INSPIRE].
  17. [17]
    O. Blondeau-Fournier, O.A. Castro-Alvaredo and B. Doyon, Universal scaling of the logarithmic negativity in massive quantum field theory, J. Phys.A 49 (2016) 125401.ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    V. Eisler and Z. Zimborás, Entanglement negativity in the harmonic chain out of equilibrium, New J. Phys.16 (2014) 123020 [arXiv:1406.5474].ADSCrossRefGoogle Scholar
  19. [19]
    A. Coser, E. Tonni and P. Calabrese, Entanglement negativity after a global quantum quench, J. Stat. Mech.12 2014 (2014) P12017.MathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Hoogeveen and B. Doyon, Entanglement negativity and entropy in non-equilibrium conformal field theory, Nucl. Phys.B 898 (2015) 78.Google Scholar
  21. [21]
    X. Wen, P.-Y. Chang and S. Ryu, Entanglement negativity after a local quantum quench in conformal field theories, Phys. Rev.B 92 (2015) 075109 [arXiv:1501.00568] [INSPIRE].
  22. [22]
    C. Castelnovo, Negativity and topological order in the toric code, Phys. Rev.A 88 (2013) 042319 [arXiv:1306.4990].
  23. [23]
    Y.A. Lee and G. Vidal, Entanglement negativity and topological order, Phys. Rev.A 88 (2013) 042318 [arXiv:1306.5711].
  24. [24]
    X. Wen, P.-Y. Chang and S. Ryu, Topological entanglement negativity in Chern-Simons theories, JHEP09 (2016) 012 [arXiv:1606.04118] [INSPIRE].
  25. [25]
    X. Wen, S. Matsuura and S. Ryu, Edge theory approach to topological entanglement entropy, mutual information and entanglement negativity in Chern-Simons theories, Phys. Rev.B 93 (2016) 245140 [arXiv:1603.08534] [INSPIRE].
  26. [26]
    C.G. West and T.C. Wei, Global and short-range entanglement properties in excited, many-body localized spin chains, arXiv:1809.04689.
  27. [27]
    C. De Nobili, A. Coser and E. Tonni, Entanglement entropy and negativity of disjoint intervals in CFT: some numerical extrapolations, J. Stat. Mech.1506 (2015) P06021 [arXiv:1501.04311] [INSPIRE].
  28. [28]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
  29. [29]
    D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett.115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
  30. [30]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
  31. [31]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev.D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
  32. [32]
    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
  33. [33]
    L. Nie, M. Nozaki, S. Ryu and M.T. Tan, Signature of quantum chaos in operator entanglement in 2d CFTs, J. Stat. Mech.1909 (2019) 093107 [arXiv:1812.00013] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  34. [34]
    P. Zanardi, Entanglement of quantum evolutions, Phys. Rev.A 63 (2001) 040304 [quant-ph/0010074] [INSPIRE].
  35. [35]
    T. Prosen and I. Pižorn, Operator space entanglement entropy in a transverse Ising chain, Phys. Rev.A 76 (2007) 032316 [arXiv:0706.2480].
  36. [36]
    J. Dubail, Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1 + 1d, J. Phys.A 50 (2017) 234001 [arXiv:1612.08630] [INSPIRE].
  37. [37]
    C. Sabín and G. García-Alcaine, A classification of entanglement in three-qubit systems, Eur. Phys. J.D 48 (2008) 435 [arXiv:0707.1780].
  38. [38]
    K. Umemoto and Y. Zhou, Entanglement of purification for multipartite states and its holographic dual, JHEP10 (2018) 152 [arXiv:1805.02625] [INSPIRE].
  39. [39]
    N. Bao and I.F. Halpern, Conditional and multipartite entanglements of purification and holography, Phys. Rev.D 99 (2019) 046010 [arXiv:1805.00476] [INSPIRE].
  40. [40]
    Y. Kusuki, J. Kudler-Flam and S. Ryu, Derivation of holographic negativity in AdS 3/CFT 2, Phys. Rev. Lett.123 (2019) 131603 [arXiv:1907.07824] [INSPIRE].
  41. [41]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
  42. [42]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].
  43. [43]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys.352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
  45. [45]
    C.A. Agón, J. De Boer and J.F. Pedraza, Geometric aspects of holographic bit threads, JHEP05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
  46. [46]
    D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, JHEP09 (2016) 068 [arXiv:1412.8465] [INSPIRE].
  47. [47]
    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
  48. [48]
    A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
  49. [49]
    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence, JHEP06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
  50. [50]
    T. Faulkner, M. Li and H. Wang, A modular toolkit for bulk reconstruction, JHEP04 (2019) 119 [arXiv:1806.10560] [INSPIRE].
  51. [51]
    T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys.14 (2018) 573 [arXiv:1708.09393] [INSPIRE].
  52. [52]
    P. Nguyen et al., Entanglement of purification: from spin chains to holography, JHEP01 (2018) 098 [arXiv:1709.07424] [INSPIRE].
  53. [53]
    K. Tamaoka, Entanglement wedge cross section from the dual density matrix, Phys. Rev. Lett.122 (2019) 141601 [arXiv:1809.09109] [INSPIRE].
  54. [54]
    S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, arXiv:1905.00577 [INSPIRE].
  55. [55]
    X. Dong, The gravity dual of Ŕenyi entropy, Nature Commun.7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
  56. [56]
    A. Nahum, J. Ruhman, S. Vijay and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev.X 7 (2017) 031016 [arXiv:1608.06950] [INSPIRE].
  57. [57]
    C. Jonay, D.A. Huse and A. Nahum, Coarse-grained dynamics of operator and state entanglement, arXiv:1803.00089 [INSPIRE].
  58. [58]
    M. Mezei, Membrane theory of entanglement dynamics from holography, Phys. Rev.D 98 (2018) 106025 [arXiv:1803.10244] [INSPIRE].
  59. [59]
    T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev.X 8 (2018) 031058 [arXiv:1710.09827] [INSPIRE].
  60. [60]
    C. von Keyserlingk, T. Rakovszky, F. Pollmann and S. Sondhi, Operator hydrodynamics, OTOCs and entanglement growth in systems without conservation laws, Phys. Rev.X 8 (2018) 021013 [arXiv:1705.08910] [INSPIRE].
  61. [61]
    A. Nahum, S. Vijay and J. Haah, Operator spreading in random unitary circuits, Phys. Rev.X 8 (2018) 021014 [arXiv:1705.08975] [INSPIRE].
  62. [62]
    V. Khemani, A. Vishwanath and D.A. Huse, Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws, Phys. Rev.X 8 (2018) 031057 [arXiv:1710.09835] [INSPIRE].
  63. [63]
    T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev.B 99 (2019) 174205 [arXiv:1804.09737] [INSPIRE].
  64. [64]
    T. Zhou and D.J. Luitz, Operator entanglement entropy of the time evolution operator in chaotic systems, Phys. Rev.B 95 (2017) 094206 [arXiv:1612.07327] [INSPIRE].
  65. [65]
    Y.-Z. You and Y. Gu, Entanglement features of Random hamiltonian dynamics, Phys. Rev.B 98 (2018) 014309 [arXiv:1803.10425] [INSPIRE].
  66. [66]
    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev.A 70 (2004) 052328 [quant-ph/0406196].
  67. [67]
    Y. Sekino and L. Susskind, Fast scramblers, JHEP10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
  68. [68]
    Z. Webb, The Clifford group forms a unitary 3-design, arXiv:1510.02769.
  69. [69]
    H. Zhu, Multiqubit Clifford groups are unitary 3-designs, arXiv:1510.02619.
  70. [70]
    T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
  71. [71]
    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev.D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
  72. [72]
    J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys.B 270 (1986) 186.Google Scholar
  73. [73]
    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].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    V. Eisler and Z. Zimborás, On the partial transpose of fermionic Gaussian states, New J. Phys.17 (2015) 053048 [arXiv:1502.01369].ADSCrossRefGoogle Scholar
  75. [75]
    A. Coser, E. Tonni and P. Calabrese, Partial transpose of two disjoint blocks in XY spin chains, J. Stat. Mech.1508 (2015) P08005 [arXiv:1503.09114] [INSPIRE].
  76. [76]
    A. Coser, E. Tonni and P. Calabrese, Towards the entanglement negativity of two disjoint intervals for a one dimensional free fermion, J. Stat. Mech.1603 (2016) 033116 [arXiv:1508.00811] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  77. [77]
    A. Coser, E. Tonni and P. Calabrese, Spin structures and entanglement of two disjoint intervals in conformal field theories, J. Stat. Mech.1605 (2016) 053109 [arXiv:1511.08328] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  78. [78]
    H. Shapourian, K. Shiozaki and S. Ryu, Partial time-reversal transformation and entanglement negativity in fermionic systems, Phys. Rev.B 95 (2017) 165101 [arXiv:1611.07536] [INSPIRE].
  79. [79]
    H. Shapourian and S. Ryu, Entanglement negativity of fermions: monotonicity, separability criterion and classification of few-mode states, Phys. Rev.A 99 (2019) 022310 [arXiv:1804.08637] [INSPIRE].
  80. [80]
    H. Shapourian and S. Ryu, Finite-temperature entanglement negativity of free fermions, J. Stat. Mech.1904 (2019) 043106 [arXiv:1807.09808] [INSPIRE].CrossRefGoogle Scholar
  81. [81]
    P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, Out-of-time-ordered correlators in (T 2 )n/n, Phys. Rev.D 96 (2017) 046020 [arXiv:1703.09939] [INSPIRE].
  82. [82]
    J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP04 (2003) 021 [hep-th/0106112] [INSPIRE].
  83. [83]
    K. Horodecki, M. Horodecki, P. Horodecki and J. Oppenheim, Locking entanglement with a single qubit, Phys. Rev. Lett.94 (2005) 200501 [quant-ph/0404096].
  84. [84]
    V. Alba and P. Calabrese, Quantum information dynamics in multipartite integrable systems, EPL126 (2019) 60001 [arXiv:1809.09119] [INSPIRE].
  85. [85]
    N. Bao and I.F. Halpern, Holographic inequalities and entanglement of purification, JHEP03 (2018) 006 [arXiv:1710.07643] [INSPIRE].
  86. [86]
    N. Bao, A. Chatwin-Davies and G.N. Remmen, Entanglement of purification and multiboundary wormhole geometries, JHEP02 (2019) 110 [arXiv:1811.01983] [INSPIRE].
  87. [87]
    J. Harper and M. Headrick, Bit threads and holographic entanglement of purification, JHEP08 (2019) 101 [arXiv:1906.05970] [INSPIRE].
  88. [88]
    T. Yu and J. H. Eberly, Sudden death of entanglement: classical noise effects, Opt. Commun.264 (2006) 393 [quant-ph/0602196].
  89. [89]
    H. Fujita, M. Nishida, M. Nozaki and Y. Sugimoto, Dynamics of logarithmic negativity and mutual information in smooth quenches, arXiv:1812.06258 [INSPIRE].
  90. [90]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev.E 50 (1994) 888 [cond-mat/9403051].
  91. [91]
    Y. Chen and G. Vidal, Entanglement contour, J. Stat. Mech.10 (2014) P10011 [arXiv:1406.1471].
  92. [92]
    J. Kudler-Flam, I. MacCormack and S. Ryu, Holographic entanglement contour, bit threads and the entanglement tsunami, J. Phys.A 52 (2019) 325401 [arXiv:1902.04654] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  93. [93]
    G. Di Giulio, R. Arias and E. Tonni, Entanglement hamiltonians in 1D free lattice models after a global quantum quench, J. Stat. Mech.1912 (2019) 123103 [arXiv:1905.01144] [INSPIRE].
  94. [94]
    C. De Nobili, On entanglement negativity in 1 + 1 and 2 + 1 dimensional quantum systems, Ph.D. thesis, SISSA, Italy (2016).Google Scholar
  95. [95]
    J. Kudler-Flam, H. Shapourian and S. Ryu, The negativity contour: a quasi-local measure of entanglement for mixed states, arXiv:1908.07540 [INSPIRE].
  96. [96]
    M. Kulaxizi, A. Parnachev and G. Policastro, Conformal blocks and negativity at large central charge, JHEP09 (2014) 010 [arXiv:1407.0324] [INSPIRE].
  97. [97]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys.104 (1986) 207 [INSPIRE].
  98. [98]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS geometry of conformal blocks, JHEP01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
  99. [99]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys.96 (1984) 419 [INSPIRE].
  100. [100]
    A. B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys.73 (1987) 1088.Google Scholar
  101. [101]
    L. Hadasz, Z. Jaskolski and M. Piatek, Classical geometry from the quantum Liouville theory, Nucl. Phys.B 724 (2005) 529 [hep-th/0504204] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  102. [102]
    D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
  103. [103]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys.B 477 (1996) 577 [hep-th/9506136] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  104. [104]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3gravity, JHEP12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
  105. [105]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys.B 241 (1984) 333 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  106. [106]
    H. Hirai, K. Tamaoka and T. Yokoya, Towards entanglement of purification for conformal field theories, PTEP2018 (2018) 063B03 [arXiv:1803.10539] [INSPIRE].
  107. [107]
    A. Prudenziati, A geodesic Witten diagram description of holographic entanglement entropy and its quantum corrections, JHEP06 (2019) 059 [arXiv:1902.10161] [INSPIRE].
  108. [108]
    P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP05 (2016) 127 [arXiv:1601.06794] [INSPIRE].
  109. [109]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro conformal blocks and thermality from classical background fields, JHEP11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
  110. [110]
    C.P. Herzog and Y. Wang, Estimation for entanglement negativity of free fermions, J. Stat. Mech.1607 (2016) 073102 [arXiv:1601.00678] [INSPIRE].

Copyright information

© The Author(s) 2020

Authors and Affiliations

  1. 1.Kadanoff Center for Theoretical PhysicsUniversity of ChicagoChicagoUSA
  2. 2.Berkeley Center for Theoretical PhysicsBerkeleyUSA
  3. 3.iTHEMS Program, RIKEN, WakoSaitamaJapan

Personalised recommendations