Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence

Abstract:We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

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“…Indeed the whole BTZ metric 11) can be rewritten into the metric 12) via the coordinate transformation…”

Section: Jhep11(2017)097mentioning

confidence: 99%

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

Caputa

Kundu

Miyaji

et al. 2017

J. High Energ. Phys.

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282

402

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“…In particular, since upon average of the random tensor located at each node gives 18) where I x corresponds to the identity , where F x corresponds to swapping the two copies of the tensors at V x , and D x the dimension of the Hilbert space at vertex x, which is equal to the product of dimensions of all the bonds that meet at x. The Renyi entropy of some boundary region A is then given by 19) JHEP08 (2016)086 r=r h r=r h Figure 10.…”

Section: A Comparison With the Random Tensor Constructionmentioning

confidence: 99%

“…To remedy the problem, we explore the 1 Interested readers are referred to [13,14] for a comprehensive review on tensor networks and its diverse applications in various branches of theoretical physics. Also we list out some references [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] where the connection between reconstruction of bulk spacetime and entanglement in the context of holography have been explored. This list is by no means complete and the readers are encouraged to check the citations of these papers and the references within.…”

Section: Jhep08(2016)086 1 Introductionmentioning

confidence: 99%

Exploring the tensor networks/AdS correspondence

Bhattacharyya

Gao

Hung

et al. 2016

J. High Energ. Phys.

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In this paper we study the recently proposed tensor networks/AdS correspondence. We found that the Coxeter group is a useful tool to describe tensor networks in a negatively curved space. Studying generic tensor network populated by perfect tensors, we find that the physical wave function generically do not admit any connected correlation functions of local operators. To remedy the problem, we assume that wavefunctions admitting such semi-classical gravitational interpretation are composed of tensors close to, but not exactly perfect tensors. Computing corrections to the connected two point correlation functions, we find that the leading contribution is given by structures related to geodesics connecting the operators inserted at the boundary physical dofs. Such considerations admit generalizations at least to three point functions. This is highly suggestive of the emergence of the analogues of Witten diagrams in the tensor network. The perturbations alone however do not give the right entanglement spectrum. Using the Coxeter construction, we also constructed the tensor network counterpart of the BTZ black hole, by orbifolding the discrete lattice on which the network resides. We found that the construction naturally reproduces some of the salient features of the BTZ black hole, such as the appearance of RT surfaces that could wrap the horizon, depending on the size of the entanglement region A.

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“…different efforts have been made to make the tensor network approach to holography more concrete [9][10][11][12][13][14][15]. To define a duality one needs to have a dual description of not only particular states such as the ground state, but also quantum operators, which requires a mapping between the two theories in the whole Hilbert space.…”

Section: Jhep01(2016)175mentioning

confidence: 99%

Bidirectional holographic codes and sub-AdS locality

Zhao

Hayden

2016

J. High Energ. Phys.

135

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Tensor networks implementing quantum error correcting codes have recently been used to construct toy models of holographic duality explicitly realizing some of the more puzzling features of the AdS/CFT correspondence. These models reproduce the Ryu-Takayanagi entropy formula for boundary intervals, and allow bulk operators to be mapped to the boundary in a redundant fashion. These exactly solvable, explicit models have provided valuable insight but nonetheless suffer from many deficiencies, some of which we attempt to address in this article. We propose a new class of tensor network models that subsume the earlier advances and, in addition, incorporate additional features of holographic duality, including: (1) a holographic interpretation of all boundary states, not just those in a "code" subspace, (2) a set of bulk states playing the role of "classical geometries" which reproduce the Ryu-Takayanagi formula for boundary intervals, (3) a bulk gauge symmetry analogous to diffeomorphism invariance in gravitational theories, (4) emergent bulk locality for sufficiently sparse excitations, and (5) the ability to describe geometry at sub-AdS resolutions or even flat space.

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Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence

Cited by 163 publications

References 25 publications

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

Exploring the tensor networks/AdS correspondence

Bidirectional holographic codes and sub-AdS locality

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