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Connecting Fisher information to bulk entanglement in holography

  • Souvik Banerjee
  • Johanna Erdmenger
  • Debajyoti Sarkar
Open Access
Regular Article - Theoretical Physics

Abstract

In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.

Keywords

AdS-CFT Correspondence Gauge-gravity correspondence Holography and condensed matter physics (AdS/CMT) 

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]
    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2010).Google Scholar
  2. [2]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
  3. [3]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
  4. [4]
    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
  5. [5]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
  7. [7]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
  10. [10]
    A. Uhlmann, The transition probability in the state space of a algebra, Annalen Phys. 42 (1985) 524 [INSPIRE].
  11. [11]
    N. Lashkari and M. Van Raamsdonk, Canonical energy is quantum Fisher information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  12. [12]
    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    D. Bak, M. Gutperle and S. Hirano, A dilatonic deformation of AdS 5 and its field theory dual, JHEP 05 (2003) 072 [hep-th/0304129] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M. Alishahiha, Holographic complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
  15. [15]
    D.D. Blanco et al., Relative entropy and holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M.J.S. Beach et al., Entanglement entropy from one-point functions in holographic states, JHEP 06 (2016) 085 [arXiv:1604.05308] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP 07 (2016) 114 [arXiv:1603.03057] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in conformal field theories of arbitrary dimensions, JHEP 02 (2017) 060 [arXiv:1611.02959] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    D.L. Jafferis et al., Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Hollands and R.M. Wald, Stability of black holes and black branes, Commun. Math. Phys. 321 (2013) 629 [arXiv:1201.0463] [INSPIRE].
  22. [22]
    J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Thermodynamical property of entanglement entropy for excited states, Phys. Rev. Lett. 110 (2013) 091602 [arXiv:1212.1164] [INSPIRE].
  23. [23]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    O. Ben-Ami and D. Carmi, On volumes of subregions in holography and complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].
  25. [25]
    E. Caceres, P.H. Nguyen and J.F. Pedraza, Holographic entanglement chemistry, Phys. Rev. D 95 (2017) 106015 [arXiv:1605.00595] [INSPIRE].
  26. [26]
    J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    N. Lashkari, Relative entropies in conformal field theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].
  28. [28]
    N. Lashkari, Modular Hamiltonian for excited states in conformal field theory, Phys. Rev. Lett. 117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].
  29. [29]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
  30. [30]
    S. Hollands, A. Ishibashi and D. Marolf, Comparison between various notions of conserved charges in asymptotically AdS-spacetimes, Class. Quant. Grav. 22 (2005) 2881 [hep-th/0503045] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, JHEP 09 (2016) 068 [arXiv:1412.8465] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    D. Carmi, R.C. Myers and P. Rath, Comments on holographic complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. Reynolds and S.F. Ross, Divergences in holographic complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].
  35. [35]
    A.R. Brown et al., Holographic complexity equals bulk action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A.R. Brown et al., Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
  37. [37]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070 [quant-ph/0502070].
  39. [39]
    J. J. Alvarez and C. Gomez, A comment on Fisher information and quantum algorithms, quant-ph/9910115 [quant-ph/9910115].
  40. [40]
    N. Lashkari et al., Gravitational positive energy theorems from information inequalities, PTEP 2016 (2016) 12C109 [arXiv:1605.01075] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Souvik Banerjee
    • 1
    • 2
  • Johanna Erdmenger
    • 2
    • 3
  • Debajyoti Sarkar
    • 3
    • 4
  1. 1.Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  2. 2.Institut für Theoretische Physik und AstrophysikJulius-Maximilians-Universität WürzburgWürzburgGermany
  3. 3.Max-Planck-Institut für Physik, Werner-Heisenberg-InstitutMunichGermany
  4. 4.Arnold Sommerfeld CenterLudwig-Maximilians-UniversityMunchenGermany

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