Abstract
At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedure over bulk and boundary modular time, and hence it can be somewhat intractable to evaluate in practice. We show that a much simpler channel, the Petz map, is sufficient for entanglement wedge reconstruction for any code space of fixed finite dimension — no twirling is required. Moreover, the error in the reconstruction will always be non-perturbatively small. From a quantum information perspective, we prove a general theorem extending the use of the Petz map as a general-purpose recovery channel to subsystem and operator algebra quantum error correction.
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References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [Adv. Theor. Math. Phys.2 (1998) 231] [hep-th/9711200] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence, JHEP06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett.117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle and M. Walter, Entanglement wedge reconstruction via universal recovery channels, Phys. Rev.X 9 (2019) 031011 [arXiv:1704.05839] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, extremality, euclidean variations and the equations of motion, JHEP01 (2018) 081 [arXiv:1705.08453] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
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].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
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].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
P. Hayden and G. Penington, Learning the alpha-bits of black holes, JHEP12 (2019) 007 [arXiv:1807.06041] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT and the fate of the BTZ singularity, AMS/IP Stud. Adv. Math.44 (2008) 85 [arXiv:0710.4334] [INSPIRE].
I.A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography, JHEP05 (2014) 053 [arXiv:1403.3426] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
M. Junge, R. Renner, D. Sutter, M.M. Wilde and A. Winter, Universal recovery maps and approximate sufficiency of quantum relative entropy, Ann. Henri Poincaré19 (2018) 2955.
M. Ohya and D. Petz, Quantum entropy and its use, Springer Science & Business Media, Berlin Germany (2004).
K. Li and A. Winter, Squashed entanglement,k-extendibility, quantum markov chains and recovery maps, Found. Phys.48 (2018) 910 [INSPIRE].
A.M. Alhambra, S. Wehner, M.M. Wilde and M.P. Woods, Work and reversibility in quantum thermodynamics, Phys. Rev.A 97 (2018) 062114 [arXiv:1506.08145].
A.M. Alhambra and M.P. Woods, Dynamical maps, quantum detailed balance and the Petz recovery map, Phys. Rev.A 96 (2017) 022118 [arXiv:1609.07496] [INSPIRE].
M. Lemm and M.M. Wilde, Information-theoretic limitations on approximate quantum cloning and broadcasting, Phys. Rev.A 96 (2017) 012304 [arXiv:1608.07569] [INSPIRE].
H. Barnum and E. Knill, Reversing quantum dynamics with near-optimal quantum and classical fidelity, J. Math. Phys.43 (2002) 2097 [quant-ph/0004088].
C.A. Fuchs and J. Van De Graaf, Cryptographic distinguishability measures for quantum-mechanical states, IEEE Trans. Inf. Theory45 (1999) 1216 [quant-ph/9712042].
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys.354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
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Chen, CF., Penington, G. & Salton, G. Entanglement wedge reconstruction using the Petz map. J. High Energ. Phys. 2020, 168 (2020). https://doi.org/10.1007/JHEP01(2020)168
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DOI: https://doi.org/10.1007/JHEP01(2020)168