Entanglement entropy and algebraic holography

Document Type : Letter


Department of Mathematics University of YORK YORK, YO10 5DD UK


In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the $d$-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where $G$ is the bulk Newton constant. (More precisely, RT showed this for d=1 and adduced evidence that it also holds in many examples in d>1.) We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to one 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S between the degrees of freedom of a given region with those of its complement takes the form S = A/4G (plus lower order terms) -- but only if the phrase `degrees of freedom' is replaced by `matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's `matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.



Article PDF

[1] J. M. Maldacena, ”The large-N limit of superconformal field theories and supergravity”, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)], arXiv:hep-th/9711200.
[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, ”Gauge theory correlators from non-critical string theory”, Phys. Lett. B 428, 105 (1998), arXiv:hep-th/9802109.
[3] E. Witten, ”Anti-de Sitter space and holography”, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv: hep-th/9802150.
[4] K.-H. Rehren, ”Algebraic holography”, Annales Henri Poincare 1, 607-623 (2000), arXiv:hep-th/9905179.
[5] K.-H. Rehren, ”Local quantum observables in the anti-de-Sitter-conformal QFT correspondence”, Phys. Lett. B 493, 383-388 (2000), arXiv:hep-th/9905179.
[6] K.-H. Rehren ”QFT lectures on AdS-CFT” (2004), arXiv:hep-th/0411086. [7] S. Ryu, T. Takayanagi, ”Holographic derivation of entanglement entropy from AdS/CFT”, Phys. Rev. Lett. 96, 181602 (2006), arXiv:hep-th/0603001.
[8] S. Ryu, T. Takayanagi, ”Aspects of holographic entanglement entropy”, J. High Energy Phys. 0608, 045 (2006), arXiv:hep-th/0605073.
[9] V.E. Hubeny, M. Rangamani, T. Takayanagi, ”A Covariant Holographic Entanglement Entropy Proposal”, J. High Energy Phys. 0707, 062 (2007), arXiv:0705.0016. (Note that arXiv:0705.0016v3 (2012) contains an erratum.)
[10] V.E. Hubeny, M. Rangamani, ”Causal Holographic Information”, J. High Energy Phys. 2012, 114 (2012), arXiv:1204.1698.
[11] V.E. Hubeny, M. Rangamani, E. Tonni, ”Global properties of causal wedges in asymptotically AdS spacetimes”, J. High Energy Phys. 1310, 059 (2013), arXiv:1306.4324.
[12] J.D. Brown, M. Henneaux, ”Central charges in the canonical realization of asymptotic symmetries”, Commun. Math. Phys. 104, 207 (1986).
[13] B. Kay, P. Larkin, ”Pre-holography”, Phys. Rev. 77, 121501(R) (2008), arXiv:0708.1283.
[14] L. Susskind, E. Witten, ”The holographic bound in anti-de Sitter space”, arXiv:hep-th/9805114.
[15] M. Bertola, J. Bros, U. Moschella and R. Schaeffer, ”A general construction of conformal field theories from scalar anti-de Sitter quantum field theories”, Nucl. Phys. B 587, 619 (2000), arXiv:hep-th/9908140.
[16] L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin (1986), ”Quantum source of entropy for black holes”, Phys. Rev. D, 34, 373 (1986).
[17] G. ’t Hooft, ”On the quantum structure of a black hole”, Nucl. Phys. B 256, 727-745 (1985).
[18] S. Mukohyama and W. Israel, ”Black holes, brick walls and the Boulware state”, Phys. Rev. D 58, 104005 (1988), arXiv:gr-qc/9806012.
[19] M. Arnsdorf and L. Smolin, ”The Maldacena conjecture and Rehren duality”, arXiv:hep-th/0106073.
[20] B.S. Kay and L. Ort´ız, ”Brick walls and AdS/CFT” Gen. Relativ. Gravit. 46, 1727 (2014), arXiv:1111.6429.
[21] B.S. Kay, ”Instability of enclosed horizons”, Gen. Relativ. Gravit. 47, 31 (2015), arXiv:1310.7395.
[22] B.S. Kay and U. Lupo, ”Non-existence of isometry-invariant Hadamard states for a Kruskal black hole in a box and for massless fields on 1+1 Minkowski spacetime with a uniformly accelerating mirror”, arXiv:1502.06582.
[23] B.S. Kay, ”Purification of KMS States”, Helv. Phys. Acta 58, 1030-1040 (1985).
[24] B.S. Kay, ”A uniqueness result for quasifree KMS states”, Helv. Phys. Acta 58, 1017-1029 (1985).
[25] E. Bianchi and R.C. Myers, ”On the architecture of spacetime geometry”, Class. Quantum Grav. 31, 214002-214014 (2014), arXiv:1212.5183.
[26] B.S. Kay, ”Entropy and quantum gravity”, Entropy 2015, 17, 8174-8186 (2015), arXiv:1504.00882.
[27] J.M. Maldacena, ”ternal black holes in anti-de Sitter”, J. High Energy Phys. 0304, 021 (2003), arXiv:hep-th/0106112.
[28] S.B. Giddings, ”Is string theory a theory of quantum gravity?”, Foundations of Physics 43, 115-139 (2013), arXiv:1105.6359.
[29] S.B. Giddings, ”Hilbert space structure in quantum gravity: an algebraic perspective”, J. High Energy Phys. 12, 099 (2015), arXiv:1503.08207
[30] S. Emelyanov, ”Holography versus Correspondence principle: eternal Schwarzschild-anti-de Sitter geometry”, Phys. Rev. D 95, 064044 (2017), arXiv:1507.03976
[31] L. McGough and H. Verlinde, ”Bekenstein-Hawking entropy as topological entanglement entropy”, J. High Energy Phys. 11, 208 (2013), arXiv:1308.2342.
[32] B.S. Kay, ”Modern foundations for thermodynamics and the stringy limit of black hole equilibria”, arXiv:1209.5085.
[33] B.S. Kay, ”More about the stringy limit of black hole equilibria”, arXiv:1209.5110.
Volume 1, Issue 1
November 2021
Pages 23-36
  • Receive Date: 09 September 2021
  • Revise Date: 09 October 2021
  • Accept Date: 10 October 2021