Entanglement conservation, ER=EPR, and a new classical area theorem for wormholes

Abstract

We consider the question of entanglement conservation in the context of the ER=EPR correspondence equating quantum entanglement with wormholes. In quantum mechanics, the entanglement between a system and its complement is conserved under unitary operations that act independently on each; ER=EPR suggests that an analogous statement should hold for wormholes. We accordingly prove a new area theorem in general relativity: for a collection of dynamical wormholes and black holes in a spacetime satisfying the null curvature condition, the maximin area for a subset of the horizons (giving the largest area attained by the minimal cross section of the multi-wormhole throat separating the subset from its complement) is invariant under classical time evolution along the outermost apparent horizons. The evolution can be completely general, including horizon mergers and the addition of classical matter satisfying the null energy condition. This theorem is the gravitational dual of entanglement conservation and thus constitutes an explicit characterization of the ER=EPR duality in the classical limit.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    D. Dieks, Communication by EPR devices, Phys. Lett. A 92 (1982) 271 [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2010).

    Book  MATH  Google Scholar 

  3. [3]

    V. Coffman, J. Kundu and W.K. Wootters, Distributed entanglement, Phys. Rev. A 61 (2000) 052306 [quant-ph/9907047] [INSPIRE].

  4. [4]

    E.H. Lieb and M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14 (1973) 1938 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. [5]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].

  7. [7]

    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Entropy on a null surface for interacting quantum field theories and the Bousso bound, Phys. Rev. D 91 (2015) 084030 [arXiv:1406.4545] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  8. [8]

    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  11. [11]

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].

    ADS  Google Scholar 

  13. [13]

    N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully and M. Walter, The holographic entropy cone, JHEP 09 (2015) 130 [arXiv:1505.07839] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. [14]

    J.M. Bardeen, B. Carter and S. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  16. [16]

    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].

  18. [18]

    L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935) 777 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  21. [21]

    A. Einstein and N. Rosen, The particle problem in the general theory of relativity, Phys. Rev. 48 (1935) 73 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  22. [22]

    L. Susskind, Copenhagen vs. Everett, teleportation and ER=EPR, Fortsch. Phys. 64 (2016) 551 [arXiv:1604.02589] [INSPIRE].

  23. [23]

    N. Bao, J. Pollack and G.N. Remmen, Splitting spacetime and cloning qubits: linking no-go theorems across the ER=EPR duality, Fortsch. Phys. 63 (2015) 705 [arXiv:1506.08203] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    N. Bao, J. Pollack and G.N. Remmen, Wormhole and entanglement (non-)detection in the ER=EPR correspondence, JHEP 11 (2015) 126 [arXiv:1509.05426] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  25. [25]

    S.W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25 (1972) 152 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  26. [26]

    S. Hawking and G. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973).

    Book  MATH  Google Scholar 

  27. [27]

    A. Królak, Definitions of black holes without use of the boundary at infinity, Gen. Rel. Grav. 14 (1982) 793.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Proof of a quantum Bousso bound, Phys. Rev. D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].

    ADS  Google Scholar 

  29. [29]

    R. Bousso and N. Engelhardt, New area law in general relativity, Phys. Rev. Lett. 115 (2015) 081301 [arXiv:1504.07627] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  30. [30]

    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. [31]

    A.C. Wall, Maximin surfaces and the strong subadditivity of the covariant holographic entanglement entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. [32]

    V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary wormholes and holographic entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  33. [33]

    R.P. Geroch, Domain of dependence, J. Math. Phys. 11 (1970) 437 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].

    ADS  Article  Google Scholar 

  35. [35]

    S.M. Carroll, Spacetime and geometry: an introduction to general relativity, Addison Wesley, U.S.A. (2004).

    MATH  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Grant N. Remmen.

Additional information

ArXiv ePrint: 1604.08217

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Remmen, G.N., Bao, N. & Pollack, J. Entanglement conservation, ER=EPR, and a new classical area theorem for wormholes. J. High Energ. Phys. 2016, 48 (2016). https://doi.org/10.1007/JHEP07(2016)048

Download citation

Keywords

  • Classical Theories of Gravity
  • Black Holes
  • Models of Quantum Gravity
  • Gauge-gravity correspondence