The refined quantum extremal surface prescription from the asymptotic equipartition property

Jinzhao Wang

Institute for Theoretical Physics, ETH 8093 Zürich, Switzerland

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Information-theoretic ideas have provided numerous insights in the progress of fundamental physics, especially in our pursuit of quantum gravity. In particular, the holographic entanglement entropy is a very useful tool in studying AdS/CFT, and its efficacy is manifested in the recent black hole page curve calculation. On the other hand, the one-shot information-theoretic entropies, such as the smooth min/max-entropies, are less discussed in AdS/CFT. They are however more fundamental entropy measures from the quantum information perspective and should also play pivotal roles in holography. We combine the technical methods from both quantum information and quantum gravity to put this idea on firm grounds. In particular, we study the quantum extremal surface (QES) prescription that was recently revised to highlight the significance of one-shot entropies in characterizing the QES phase transition. Motivated by the asymptotic equipartition property (AEP), we derive the refined quantum extremal surface prescription for fixed-area states via a novel AEP replica trick, demonstrating the synergy between quantum information and quantum gravity. We further prove that, when restricted to pure bulk marginal states, such corrections do not occur for the higher Rényi entropies of a boundary subregion in fixed-area states, meaning they always have sharp QES transitions. Our path integral derivation suggests that the refinement applies beyond AdS/CFT, and we confirm it in a black hole toy model by showing that the Page curve, for a black hole in a superposition of two radiation stages, receives a large correction that is consistent with the refined QES prescription.

Information-theoretic ideas are more powerful than ever in helping us understand fundamental physics. In particular, the entanglement entropy has been a useful probe of correlations in quantum field theory and gravity. Its importance culminates in holography, where the entanglement entropy can be “geometrized” via the Ryu-Takayanagi formula, and the so inspired quantum extremal surface (QES) prescription recently serves as the key to solving the black hole information paradox. In quantum information theory, on the other hand, the von Neumann entropy is operationally understood as an “average” quantity, which only becomes relevant in the asymptotic regime where the law of large number kicks in. While it is considered a fundamental quantity in the gravity picture, it does not have this fundamental role in information theory. Instead, we think of the one-shot entropies as fundamental measures of information.

Motivated by the mismatch, we propose a novel replica trick, which is a method used to compute the entanglement entropy in field theory. Our novel replica trick can potentially be useful in other situations where the standard replica method faces the difficulty of analytic continuation. We demonstrate its efficacy in refining the phase transition mechanism responsible for resolving the black hole information paradox, manifesting the significance of the one-shot entropies. In particular, our path-integral derivation implies a revised Page curve that characterizes the entropy of the Hawking radiation. This correction is also confirmed in our work, showing the universal applicability of the refined QES prescription in gravity.

We combine both the ideas and techniques from quantum information and quantum gravity to propose a novel replica trick to compute entanglement entropies. We then derive a refined phase transition mechanism responsible for the black hole Page curve. We think both the method and result are of general interest to the physics community, and it is indeed a truly interdisciplinary work showing the synergy between quantum information and quantum gravity.

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[1] J. Bekenstein, Black holes and the second law, Lettere Al Nuovo Cimento (1971–1985) 4 (1972) 737.

[2] J. D. Bekenstein, Black holes and entropy, Physical Review D 7 (1973) 2333.

[3] J. D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Physical Review D 9 (1974) 3292.

[4] S. W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43 (1975) 199.

[5] S. W. Hawking, Breakdown of predictability in gravitational collapse, Physical Review D 14 (1976) 2460.

[6] W. G. Unruh, Notes on black-hole evaporation, Physical Review D 14 (1976) 870.

[7] J. J. Bisognano and E. H. Wichmann, On the duality condition for quantum fields, Journal of Mathematical Physics 17 (1976) 303.

[8] G. Hooft, On the quantum structure of a black hole, Nuclear Physics B 256 (1985) 727.

[9] L. Bombelli, R. K. Koul, J. Lee and R. D. Sorkin, Quantum source of entropy for black holes, Physical Review D 34 (1986) 373.

[10] M. Srednicki, Entropy and area, Physical Review Letters 71 (1993) 666.

[11] L. Susskind, Some speculations about black hole entropy in string theory,.

[12] C. Callan and F. Wilczek, On geometric entropy, Physics Letters B 333 (1994) 55.

[13] C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nuclear Physics B 424 (1994) 443.

[14] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Experiment 2004 (2004) P06002.

[15] P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, Journal of Statistical Mechanics: Theory and Experiment 2005 (2005) P04010.

[16] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory: a non-technical introduction, International Journal of Quantum Information 4 (2006) 429.

[17] M. B. Hastings, Solving gapped hamiltonians locally, Physical Review b 73 (2006) 085115.

[18] A. Kitaev and J. Preskill, Topological entanglement entropy, Physical Review Letters 96 (2006) 110404.

[19] M. M. Wolf, F. Verstraete, M. B. Hastings and J. I. Cirac, Area laws in quantum systems: mutual information and correlations, Physical Review Letters 100 (2008) 070502.

[20] H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states, Physical Review Letters 101 (2008) 010504.

[21] M. Mézard, G. Parisi and M. A. Virasoro, Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications, vol. 9. World Scientific Publishing Company, 1987, 10.1142/​0271.

[22] A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, Journal of High Energy Physics 2013 (2013) 90.

[23] H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Physics Letters B 600 (2004) 142.

[24] H. Casini, M. Huerta, R. C. Myers and A. Yale, Mutual information and the f-theorem, Journal of High Energy Physics 2015 (2015) 1.

[25] H. Casini, E. Testé and G. Torroba, Markov property of the conformal field theory vacuum and the a-theorem, Physical Review Letters 118 (2017) 261602.

[26] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de sitter space/​conformal field theory correspondence, Physical Review Letters 96 (2006) 181602.

[27] S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, Journal of High Energy Physics 2006 (2006) 045.

[28] V. E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, Journal of High Energy Physics 2007 (2007) 062.

[29] M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Physical Review D 76 (2007) 106013.

[30] H. Casini, M. Huerta and R. C. Myers, Towards a derivation of holographic entanglement entropy, Journal of High Energy Physics 2011 (2011) 1.

[31] B. Swingle, Entanglement renormalization and holography, Physical Review D 86 (2012) 065007.

[32] T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, Journal of High Energy Physics 2013 (2013) 74.

[33] N. Engelhardt and A. C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, Journal of High Energy Physics 2015 (2015) 73.

[34] X. Dong, The gravity dual of rényi entropy, Nature Communications 7 (2016) 1.

[35] D. Harlow, The ryu–takayanagi formula from quantum error correction, Communications in Mathematical Physics 354 (2017) 865.

[36] X. Dong and A. Lewkowycz, Entropy, extremality, euclidean variations, and the equations of motion, Journal of High Energy Physics 2018 (2018) 1.

[37] X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, Journal of High Energy Physics 2019 (2019) 1.

[38] C. Akers and P. Rath, Holographic renyi entropy from quantum error correction, Journal of High Energy Physics 2019 (2019) 1.

[39] M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Springer Lecture Notes in Physics (2017) 35.

[40] T. Nishioka, Entanglement entropy: holography and renormalization group, Reviews of Modern Physics 90 (2018) 035007.

[41] M. Headrick, Lectures on entanglement entropy in field theory and holography, arXiv:1907.08126 (2019).

[42] X. Dong, D. Harlow and A. C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Physical Review Letter 117 (2016) 021601.

[43] T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, Journal of High Energy Physics 2017 (2017) 1.

[44] D. Harlow, TASI lectures on the emergence of the bulk in AdS/​CFT, arXiv:1802.01040 (2018).

[45] A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/​CFT, Journal of High Energy Physics 2015 (2015) 163.

[46] R. M. Wald, Black hole entropy is the noether charge, Physical Review D 48 (1993) R3427.

[47] V. Iyer and R. M. Wald, Some properties of the noether charge and a proposal for dynamical black hole entropy, Physical Review D 50 (1994) 846.

[48] X. Dong, Holographic entanglement entropy for general higher derivative gravity, Journal of High Energy Physics 2014 (2014) 1.

[49] J. Camps, Generalized entropy and higher derivative gravity, Journal of High Energy Physics 2014 (2014) 70.

[50] A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, Journal of High Energy Physics 2019 (2019) 1.

[51] G. Penington, Entanglement wedge reconstruction and the information paradox, Journal of High Energy Physics 2020 (2020) 1.

[52] A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The page curve of hawking radiation from semiclassical geometry, Journal of High Energy Physics 2020 (2020) 1.

[53] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica wormholes and the entropy of hawking radiation, Journal of High Energy Physics 2020 (2020) 1.

[54] G. Penington, S. H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 (2019).

[55] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of hawking radiation, Review of Modern Physics 93 (2021) 035002.

[56] E. D’Hoker, X. Dong and C.-H. Wu, An alternative method for extracting the von neumann entropy from rényi entropies, Journal of High Energy Physics 2021 (2021) 1.

[57] R. P. Boas, Entire functions. Academic Press, 2011, 10.1016/​c2013-0-12422-1.

[58] P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, Journal of Statistical Mechanics: Theory and Experiment 2009 (2009) P11001.

[59] P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory: Ii, Journal of Statistical Mechanics: Theory and Experiment 2011 (2011) P01021.

[60] C. De Nobili, A. Coser and E. Tonni, Entanglement entropy and negativity of disjoint intervals in CFT: Some numerical extrapolations, Journal of Statistical Mechanics: Theory and Experiment 2015 (2015) P06021.

[61] P. Ruggiero, E. Tonni and P. Calabrese, Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks, Journal of Statistical Mechanics: Theory and Experiment 2018 (2018) 113101.

[62] C. E. Shannon, A mathematical theory of communication, The Bell system technical journal 27 (1948) 379.

[63] B. Schumacher, Quantum coding, Physical Review A 51 (1995) 2738.

[64] C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, Concentrating partial entanglement by local operations, Physical Review A 53 (1996) 2046.

[65] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Physical Review Letters 76 (1996) 722.

[66] B. Schumacher and M. D. Westmoreland, Sending classical information via noisy quantum channels, Physical Review A 56 (1997) 131.

[67] A. S. Holevo, The capacity of the quantum channel with general signal states, IEEE Transactions on Information Theory 44 (1998) 269.

[68] R. Renner and S. Wolf, Smooth rényi entropy and applications, International Symposium on Information Theory, 2004. ISIT 2004. Proceedings. (2004) 233.

[69] R. Renner, Security of quantum key distribution, International Journal of Quantum Information 6 (2008) 1.

[70] L. Wang and R. Renner, One-shot classical-quantum capacity and hypothesis testing, Physical Review Letters 108 (2012) 200501.

[71] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr and M. Tomamichel, On quantum rényi entropies: A new generalization and some properties, Journal of Mathematical Physics 54 (2013) 122203.

[72] M. M. Wilde, A. Winter and D. Yang, Strong converse for the classical capacity of entanglement-breaking and hadamard channels via a sandwiched rényi relative entropy, Communications in Mathematical Physics 331 (2014) 593.

[73] F. Dupuis, L. Kraemer, P. Faist, J. M. Renes and R. Renner, Generalized entropies, XVIIth International Congress on Mathematical Physics (2014) 134.

[74] M. Tomamichel, Quantum information processing with finite resources: mathematical foundations, vol. 5. Springer, 2015, 10.1007/​978-3-319-21891-5.

[75] M. Berta, Single-shot quantum state merging, arXiv:0912.4495 (2009).

[76] R. Konig, R. Renner and C. Schaffner, The operational meaning of min-and max-entropy, IEEE Transactions on Information theory 55 (2009) 4337.

[77] M. Tomamichel, C. Schaffner, A. Smith and R. Renner, Leftover hashing against quantum side information, IEEE Transactions on Information Theory 57 (2011) 5524.

[78] M. Tomamichel, R. Colbeck and R. Renner, A fully quantum asymptotic equipartition property, IEEE Transactions on information theory 55 (2009) 5840.

[79] C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, Journal of High Energy Physics 2021 (2021) 1.

[80] D. Marolf, S. Wang and Z. Wang, Probing phase transitions of holographic entanglement entropy with fixed area states, Journal of High Energy Physics 2020 (2020) 1.

[81] X. Dong and H. Wang, Enhanced corrections near holographic entanglement transitions: a chaotic case study, Journal of High Energy Physics 2020 (2020) 1.

[82] C. Akers, A. Levine and S. Leichenauer, Large breakdowns of entanglement wedge reconstruction, Physical Review D 100 (2019) 126006.

[83] P. Boes, J. Eisert, R. Gallego, M. P. Müller and H. Wilming, Von neumann entropy from unitarity, Physical Review Letters 122 (2019) 210402.

[84] H. Wilming, Entropy and reversible catalysis, Physical Review Letters 127 (2021) 260402.

[85] N. J. Beaudry and R. Renner, An intuitive proof of the data processing inequality, Quantum Information & Computation 12 (2012) 432.

[86] O. Fawzi and R. Renner, Quantum conditional mutual information and approximate markov chains, Communications in Mathematical Physics 340 (2015) 575.

[87] M. Berta, M. Christandl, R. Colbeck, J. M. Renes and R. Renner, The uncertainty principle in the presence of quantum memory, Nature Physics 6 (2010) 659.

[88] M. Tomamichel, R. Colbeck and R. Renner, Duality between smooth min-and max-entropies, IEEE Transactions on information theory 56 (2010) 4674.

[89] A. Vitanov, F. Dupuis, M. Tomamichel and R. Renner, Chain rules for smooth min-and max-entropies, IEEE Transactions on Information Theory 59 (2013) 2603.

[90] G. Gibbons and S. Hawking, Action integrals and partition functions in quantum gravity, Physical Review D 15 (1977) 2752.

[91] J. Hartle and S. Hawking, Path-integral derivation of black-hole radiance, Physical Review D 13 (1976) 2188.

[92] F. Furrer, J. Åberg and R. Renner, Min-and max-entropy in infinite dimensions, Communications in Mathematical Physics 306 (2011) 165.

[93] M. Berta, F. Furrer and V. B. Scholz, The smooth entropy formalism for von neumann algebras, Journal of Mathematical Physics 57 (2016) 015213.

[94] S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, arXiv:1905.00577 (2019).

[95] X. Dong, X.-L. Qi, Z. Shangnan and Z. Yang, Effective entropy of quantum fields coupled with gravity, Journal of High Energy Physics 2020 (2020) 1.

[96] N. Engelhardt, S. Fischetti and A. Maloney, Free energy from replica wormholes, Physical Review D 103 (2021) 046021.

[97] M. Fannes, A continuity property of the entropy density for spin lattice systems, Communications in Mathematical Physics 31 (1973) 291.

[98] K. M. Audenaert, A sharp continuity estimate for the von neumann entropy, Journal of Physics A: Mathematical and Theoretical 40 (2007) 8127.

[99] B. Czech, P. Hayden, N. Lashkari and B. Swingle, The information theoretic interpretation of the length of a curve, Journal of High Energy Physics 2015 (2015) 1.

[100] N. Bao, G. Penington, J. Sorce and A. C. Wall, Beyond toy models: distilling tensor networks in full AdS/​CFT, Journal of High Energy Physics 2019 (2019) 1.

[101] P. Hayden and G. Penington, Learning the alpha-bits of black holes, Journal of High Energy Physics 2019 (2019) 1.

[102] G. Kreweras, Sur les partitions non croisées d'un cycle, Discrete Mathematics 1 (1972) 333.

[103] P. Biane, Some properties of crossings and partitions, Discrete Mathematics 175 (1997) 41.

[104] J. Watrous, The theory of quantum information. Cambridge University Press, 2018, 10.1017/​9781316848142.

[105] J. Maldacena and L. Maoz, Wormholes in ads, Journal of High Energy Physics 2004 (2004) 053.

[106] N. Arkani-Hamed, J. Orgera and J. Polchinski, Euclidean wormholes in string theory, Journal of High Energy Physics 2007 (2007) 018.

[107] P. Saad, S. H. Shenker and D. Stanford, A semiclassical ramp in syk and in gravity, arXiv:1806.06840 (2018).

[108] P. Saad, S. H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 (2019).

[109] P. Saad, Late time correlation functions, baby universes, and ETH in JT gravity, arXiv:1910.10311 (2019).

[110] D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, arXiv:1907.03363 (2019).

[111] R. Bousso and M. Tomašević, Unitarity from a smooth horizon?, Physical Review D 102 (2020) 106019.

[112] R. Bousso and E. Wildenhain, Gravity/​Ensemble duality, Physical Review D 102 (2020) 066005.

[113] J. Pollack, M. Rozali, J. Sully and D. Wakeham, Eigenstate thermalization and disorder averaging in gravity, Physical Review Letters 125 (2020) 021601.

[114] D. Marolf and H. Maxfield, Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information, Journal of High Energy Physics 2020 (2020) 1.

[115] D. Marolf and H. Maxfield, Observations of hawking radiation: the page curve and baby universes, Journal of High Energy Physics 2021 (2021) 1.

[116] D. Stanford, More quantum noise from wormholes, arXiv:2008.08570 (2020).

[117] J. Cotler and K. Jensen, Ads3 gravity and random cft, Journal of High Energy Physics 2021 (2021) 1.

[118] H. Liu and S. Vardhan, Entanglement entropies of equilibrated pure states in quantum many-body systems and gravity, PRX Quantum 2 (2021) 010344.

[119] N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, Journal of High Energy Physics 2021 (2021) 1.

[120] Y. Chen, V. Gorbenko and J. Maldacena, Bra-ket wormholes in gravitationally prepared states, Journal of High Energy Physics 2021 (2021) 1.

[121] A. Belin and J. de Boer, Random statistics of ope coefficients and euclidean wormholes, Classical and Quantum Gravity 38 (2021) 164001.

[122] A. Maloney and E. Witten, Averaging over narain moduli space, Journal of High Energy Physics 2020 (2020) 1.

[123] A. Blommaert, Dissecting the ensemble in JT gravity, arXiv:2006.13971 (2020).

[124] S. B. Giddings and G. J. Turiaci, Wormhole calculus, replicas, and entropies, Journal of High Energy Physics 2020 (2020) 1.

[125] M. Van Raamsdonk, Comments on wormholes, ensembles, and cosmology, Journal of High Energy Physics 2021 (2021) 1.

[126] L. Eberhardt, Summing over geometries in string theory, Journal of High Energy Physics 2021 (2021) 1.

[127] P. Saad, S. H. Shenker, D. Stanford and S. Yao, Wormholes without averaging, arXiv:2103.16754 (2021).

[128] H. Verlinde, Deconstructing the wormhole: Factorization, entanglement and decoherence, arXiv:2105.02142 (2021).

[129] J. Kudler-Flam, Relative entropy of random states and black holes, Physical Review Letters 126 (2021) 171603.

[130] V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mathematics of the USSR-Sbornik 1 (1967) 457.

[131] D. N. Page, Average entropy of a subsystem, Physical Review Letters 71 (1993) 1291.

[132] P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, Journal of High Energy Physics 2016 (2016) 1.

[133] F. Dupuis, O. Fawzi and R. Renner, Entropy accumulation, Communications in Mathematical Physics 379 (2020) 867.

[134] F. Dupuis and O. Fawzi, Entropy accumulation with improved second-order term, IEEE Transactions on Information Theory 65 (2019) 7596.

[135] F. Dupuis, M. Berta, J. Wullschleger and R. Renner, One-shot decoupling, Communications in Mathematical Physics 328 (2014) 251.

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