Discrete gravity on random tensor network and holographic Rényi entropy

125

222

We use the Einstein-Hilbert gravitational path integral to investigate gravitational entanglement at leading order O(1/G). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies S n are independent of n at this order. The n-dependence of S n in more general states then arises from sums over the RT/HRT-area, which are generally dominated by different values of this area for each n. This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of algebraic centers which arise in the quantum errorcorrecting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.

Section: Introductionsupporting

confidence: 57%

Flat entanglement spectra in fixed-area states of quantum gravity

Dong

Harlow

Marolf

2019

125

222

“…Our results imply that not only will the STN have a nonflat spectrum, but computing Renyi entropies will be qualitatively similar to the AdS/CFT prescription involving an extremal brane. One should also compare our results to those of [37]. They obtained the correct Renyi entropies in the case of AdS 3 by using the specific form of the gravitational action.…”

Section: Tensor Networksupporting

confidence: 52%

Holographic Renyi entropy from quantum error correction

Akers

Rath

2019

113

147

We study Renyi entropies S n in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing S n ≡ n 2 ∂ n ( n−1 n S n ). We find that general operator algebra codes have a similar, more general prescription. Notably, for the AdS/CFT code to match the specific cosmic brane prescription, the code must have maximal entanglement within eigenspaces of the area operator. This gives us an improved definition of the area operator, and establishes a stronger connection between the Ryu-Takayanagi area term and the edge modes in lattice gauge theory. We also propose a new interpretation of existing holographic tensor networks as area eigenstates instead of smooth geometries. This interpretation would explain why tensor networks have historically had trouble modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a method to construct holographic networks with the correct spectrum.

“…In this interpretation, a holographic boundary state should be described not as a single tensor network that encodes the quantum fluctuations in the geometry, but as a weighted quantum superposition of networks, each of which describes a (very slightly different) non-fluctuating bulk geometry. The idea that fluctuations over different geometries correspond to taking a quantum superposition of different tensor networks has been previously discussed in [29,30,51,52].…”

Section: Superpositions Of Tensor Network: Sweeping Away the "Cobwebs"mentioning

confidence: 99%

Beyond toy models: distilling tensor networks in full AdS/CFT

Bao

Penington

Sorce

et al. 2019

117

126

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For "tree" tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the "holographic entanglement of purification" conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.