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.
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.
 L. Susskind, Some speculations about black hole entropy in string theory,.
 P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Experiment 2004 (2004) P06002.
 P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, Journal of Statistical Mechanics: Theory and Experiment 2005 (2005) P04010.
 P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory: a non-technical introduction, International Journal of Quantum Information 4 (2006) 429.
 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.
 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.
 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.
 H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Physics Letters B 600 (2004) 142.
 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.
 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.
 S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, Journal of High Energy Physics 2006 (2006) 045.
 V. E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, Journal of High Energy Physics 2007 (2007) 062.
 N. Engelhardt and A. C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, Journal of High Energy Physics 2015 (2015) 73.
 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.
 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.
 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.
 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.
 A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of hawking radiation, Review of Modern Physics 93 (2021) 035002.
 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.
 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.
 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.
 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.
 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.
 C. E. Shannon, A mathematical theory of communication, The Bell system technical journal 27 (1948) 379.
 C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, Concentrating partial entanglement by local operations, Physical Review A 53 (1996) 2046.
 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.
 R. Renner and S. Wolf, Smooth rényi entropy and applications, International Symposium on Information Theory, 2004. ISIT 2004. Proceedings. (2004) 233.
 L. Wang and R. Renner, One-shot classical-quantum capacity and hypothesis testing, Physical Review Letters 108 (2012) 200501.
 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.
 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.
 F. Dupuis, L. Kraemer, P. Faist, J. M. Renes and R. Renner, Generalized entropies, XVIIth International Congress on Mathematical Physics (2014) 134.
 M. Tomamichel, C. Schaffner, A. Smith and R. Renner, Leftover hashing against quantum side information, IEEE Transactions on Information Theory 57 (2011) 5524.
 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.
 X. Dong and H. Wang, Enhanced corrections near holographic entanglement transitions: a chaotic case study, Journal of High Energy Physics 2020 (2020) 1.
 P. Boes, J. Eisert, R. Gallego, M. P. Müller and H. Wilming, Von neumann entropy from unitarity, Physical Review Letters 122 (2019) 210402.
 H. Wilming, Entropy and reversible catalysis, Physical Review Letters 127 (2021) 260402.
 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.
 K. M. Audenaert, A sharp continuity estimate for the von neumann entropy, Journal of Physics A: Mathematical and Theoretical 40 (2007) 8127.
 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.
 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.
 N. Arkani-Hamed, J. Orgera and J. Polchinski, Euclidean wormholes in string theory, Journal of High Energy Physics 2007 (2007) 018.
 J. Pollack, M. Rozali, J. Sully and D. Wakeham, Eigenstate thermalization and disorder averaging in gravity, Physical Review Letters 125 (2020) 021601.
 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.
 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.
 J. Kudler-Flam, Relative entropy of random states and black holes, Physical Review Letters 126 (2021) 171603.
 V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mathematics of the USSR-Sbornik 1 (1967) 457.
 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.
 Jonah Kudler-Flam and Pratik Rath, "Large and small corrections to the JLMS Formula from replica wormholes", Journal of High Energy Physics 2022 8, 189 (2022).
 Jinzhao Wang, "Beyond islands: a free probabilistic approach", Journal of High Energy Physics 2023 10, 40 (2023).
 Keiichiro Furuya, Nima Lashkari, and Mudassir Moosa, "Renormalization group and approximate error correction", Physical Review D 106 10, 105007 (2022).
 Christopher Akers and Geoff Penington, "Quantum minimal surfaces from quantum error correction", SciPost Physics 12 5, 157 (2022).
The above citations are from Crossref's cited-by service (last updated successfully 2023-11-29 04:05:46) and SAO/NASA ADS (last updated successfully 2023-11-29 04:05:47). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.