Petz reconstruction in random tensor networks
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Springer
Received: August 18, 2020 Accepted: September 13, 2020 Published: October 8, 2020
Hewei Frederic Jia and Mukund Rangamani Center for Quantum Mathematics and Physics (QMAP), Department of Physics, University of California, Davis, CA 95616 U.S.A.
E-mail: [email protected], [email protected] Abstract: We illustrate the ideas of bulk reconstruction in the context of random tensor network toy models of holography. Specifically, we demonstrate how the Petz reconstruction map works to obtain bulk operators from the boundary data by exploiting the replica trick. We also take the opportunity to comment on the differences between coarse-graining and random projections. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence, Random Systems ArXiv ePrint: 2006.12601
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)050
JHEP10(2020)050
Petz reconstruction in random tensor networks
Contents 1
2 Random tensor networks and replicas
2
3 Petz reconstruction of bulk states 3.1 The simplified Petz reconstruction 3.2 General recovery: 1-point functions 3.3 General recovery: higher point functions
5 6 8 9
4 Comments on holographic channels
1
11
Introduction
The emergence of bulk spacetime geometry from non-gravitational field theoretic degrees of freedom in the AdS/CFT correspondence can be understood by viewing the holographic map from the bulk to the boundary as a quantum error correcting code [1, 2]. The essential idea is that while the Hilbert space of the field theory is isomorphic to the full string theoretic quantum gravitational Hilbert space, semiclassical gravitational physics has access to a much smaller subspace of states. These ‘code subspace’ states corresponding to excitations of the vacuum (or other geometric states) by a few, O(ceff ), perturbative quanta, are to be viewed as the quantum message one wishes to encode into a bigger Hilbert space. 1 This encoding map can moreover be viewed as a noisy quantum channel. The question of recovering local bulk geometry can be rephrased in this framework as constructing a recovery map for this channel, one that allows us to reconstruct from field theory data, either the bulk state, or better yet (in the Heisenberg picture) local bulk operators. The latter are especially interesting given that the standard reconstruction of local bulk physics exploits only the bulk causal structure [3, 4] through the extrapolate dictionary [5, 6]. It however has become quite clear thanks to the holographic entanglement entropy proposals [7, 8] that one should be able to reconstruct operators in a larger domain of the bulk, the entanglement wedge [9–11]. While it has been argued that reconstructing operators in the entanglement wedge involves modular evolution [12, 13], an alternative viewpoint exploiting quantum recovery maps was presented in [14]. The idea as elaborated further in [15] is that the Petz map [16] and its twirled generalization [17], provide a universal general-purpose recovery maps which 1
We use ce
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