Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence
- 3.1k Downloads
- 102 Citations
Abstract
We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].
Keywords
AdS-CFT Correspondence Lattice Integrable ModelsNotes
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]A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
- [2]J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
- [3]S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [4]S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [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]M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].ADSMathSciNetGoogle Scholar
- [7]M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
- [8]M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [9]P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].ADSGoogle Scholar
- [10]A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [11]J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [12]N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].ADSCrossRefGoogle Scholar
- [13]G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91 (2003) 147902 [quant-ph/0301063].
- [14]F. Verstraete and J.I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066 [INSPIRE].
- [15]F. Verstraete, J. Cirac and V. Murg, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57 (2008) 143 [arXiv:0907.2796].ADSCrossRefGoogle Scholar
- [16]G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].ADSCrossRefGoogle Scholar
- [17]G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [INSPIRE].ADSCrossRefGoogle Scholar
- [18]G. Evenbly and G. Vidal, Entanglement renormalization in two spatial dimensions, Phys. Rev. Lett. 102 (2009) 180406 [arXiv:0811.0879] [INSPIRE].ADSCrossRefGoogle Scholar
- [19]G. Evenbly and G. Vidal, Algorithms for entanglement renormalization, Phys. Rev. B 79 (2009) 144108 [arXiv:0707.1454].ADSCrossRefGoogle Scholar
- [20]G. Evenbly and G. Vidal, Frustrated antiferromagnets with entanglement renormalization: ground state of the spin-1/2 Heisenberg model on a Kagome lattice, Phys. Rev. Lett. 104 (2010) 187203 [arXiv:0904.3383] [INSPIRE].ADSCrossRefGoogle Scholar
- [21]L. Susskind and E. Witten, The holographic bound in anti-de Sitter space, hep-th/9805114 [INSPIRE].
- [22]B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
- [23]B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
- [24]B. Yoshida, Information storage capacity of discrete spin systems, Annals Phys. 338 (2013) 134 [arXiv:1111.3275] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [25]J.I. Latorre and G. Sierra, Holographic codes, arXiv:1502.06618 [INSPIRE].
- [26]A.J. Ferris and D. Poulin, Tensor networks and quantum error correction, Phys. Rev. Lett. 113 (2014) 030501 [arXiv:1312.4578].ADSCrossRefGoogle Scholar
- [27]D. Bacon, S.T. Flammia, A.W. Harrow and J. Shi, Sparse quantum codes from quantum circuits, arXiv:1411.3334.
- [28]X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282 [INSPIRE].
- [29]W. Helwig, W. Cui, A. Riera, J.I. Latorre and H.-K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A 86 (2012) 052335 [arXiv:1204.2289] [INSPIRE].ADSCrossRefGoogle Scholar
- [30]W. Helwig, Absolutely maximally entangled qudit graph states, arXiv:1306.2879.
- [31]R. Cleve, D. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett. 83 (1999) 648 [quant-ph/9901025] [INSPIRE].ADSCrossRefGoogle Scholar
- [32]J. Weeks, KaleidoTile. A computer program for creating spherical, Euclidean and hyperbolic tilings, http://www.geometrygames.org/KaleidoTile.
- [33]C.H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity, Courier Corporation, (1998).Google Scholar
- [34]A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].ADSMathSciNetGoogle Scholar
- [35]I.A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography, JHEP 05 (2014) 053 [arXiv:1403.3426] [INSPIRE].ADSCrossRefGoogle Scholar
- [36]D. Kribs, R. Laflamme and D. Poulin, Unified and generalized approach to quantum error correction, Phys. Rev. Lett. 94 (2005) 180501 [quant-ph/0412076].ADSCrossRefGoogle Scholar
- [37]D.W. Kribs, R. Laflamme, D. Poulin and M. Lesosky, Operator quantum error correction, Quant. Inf. Comp. 6 (2006) 383 [quant-ph/0504189].
- [38]C. Bény, A. Kempf and D. Kribs, Quantum error correction of observables, Phys. Rev. A 76 (2007) 042303 [arXiv:0705.1574].ADSCrossRefGoogle Scholar
- [39]C. Bény, A. Kempf and D. Kribs, Generalization of quantum error correction via the Heisenberg picture, Phys. Rev. Lett. 98 (2007) 100502 [quant-ph/0608071].
- [40]M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].ADSCrossRefGoogle Scholar
- [41]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].ADSMathSciNetCrossRefMATHGoogle Scholar
- [42]B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [43]D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, arXiv:1412.8465 [INSPIRE].
- [44]D. Gottesman, An introduction to quantum error correction and fault-tolerant quantum computation, arXiv:0904.2557.
- [45]E. Mintun, J. Polchinski and V. Rosenhaus, Bulk-boundary duality, gauge invariance and quantum error correction, arXiv:1501.06577 [INSPIRE].
- [46]S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11 (2009) 043029 [arXiv:0810.1983].ADSCrossRefGoogle Scholar
- [47]F. Pastawski and B. Yoshida, Fault-tolerant logical gates in quantum error-correcting codes, Phys. Rev. A 91 (2015) 012305 [arXiv:1408.1720].ADSMathSciNetCrossRefGoogle Scholar
- [48]A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [49]D. Harlow, Jerusalem lectures on black holes and quantum information, arXiv:1409.1231 [INSPIRE].
- [50]J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].ADSMathSciNetMATHGoogle Scholar
- [51]S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
- [52]T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [53]L. Susskind, Computational complexity and black hole horizons, arXiv:1403.5695 [INSPIRE].
- [54]D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
- [55]D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, (1997), pg. 176 [quant-ph/9906129] [INSPIRE].
- [56]M. Grassl, T. Beth and M. Roetteler, On optimal quantum codes, Int. J. Quant. Inf. 2 (2004) 55 [quant-ph/0312164].
- [57]D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
- [58]S. Goldstein, J.L. Lebowitz, R. Tumulka and N. Zanghì, Canonical typicality, Phys. Rev. Lett. 96 (2006) 050403 [cond-mat/0511091].
- [59]J. Adler, Bootstrap percolation, Phys. A 171 (1991) 453.Google Scholar
- [60]D.A. Levin, Y. Peres and E.L. Wilmer, Markov chains and mixing times, American Mathematical Society, U.S.A. (2008).CrossRefGoogle Scholar
- [61]M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2000).MATHGoogle Scholar
- [62]N. Schuch, I. Cirac and D. Perez-Garcia, PEPS as ground states: degeneracy and topology, Ann. Phys. 325 (2010) 2153 [arXiv:1001.3807].ADSMathSciNetCrossRefMATHGoogle Scholar
- [63]M.B. ¸ahinoğlu et al., Characterizing topological order with matrix product operators, arXiv:1409.2150.
- [64]O. Buerschaper, Twisted injectivity in projected entangled pair states and the classification of quantum phases, Annals Phys. 351 (2014) 447 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [65]B. Yoshida and I.L. Chuang, Framework for classifying logical operators in stabilizer codes, Phys. Rev. A 81 (2010) 052302 [arXiv:1002.0085].ADSCrossRefGoogle Scholar
- [66]J. Haah and J. Preskill, Logical-operator tradeoff for local quantum codes, Phys. Rev. A 86 (2012) 032308 [arXiv:1011.3529].ADSCrossRefGoogle Scholar
- [67]P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar