Tensor networks from kinematic space

We discuss, from a quantum information perspective, recent proposals of Maldacena, Ryu, Takayanagi, van Raamsdonk, Swingle, and Susskind that spacetime is an emergent property of the quantum entanglement of an associated boundary quantum system. We review the idea that the informational principle of minimal complexity determines a dual holographic bulk spacetime from a minimal quantum circuit U preparing a given boundary state from a trivial reference state. We describe how this idea may be extended to determine the relationship between the fluctuations of the bulk holographic geometry and the fluctuations of the boundary low-energy subspace. In this way we obtain, for every quantum system, an Einstein-like equation of motion for what might be interpreted as a bulk gravity theory dual to the boundary system.

Section: Jhep12(2016)055mentioning

confidence: 99%

“…. , 99} and t is chosen uniformly in the interval [0,50]. To determine if a pair of such events was causally connected we calculated the commutator quantity…”

Section: Jhep12(2016)055mentioning

confidence: 99%

Holographic fluctuations and the principle of minimal complexity

Chemissany

Osborne

2016

“…This inspired many subsequent works that attempt to make further concrete connection between the AdS/CFT and the tensor network. It is for example observed that the MERA tensor network, that originally inspired Swingle's observation [3], is connected to the space of geodesics in AdS space, termed the kinematic space [4][5][6][7][8][9][10][11][12]. 1 More recently, [31] proposes a concrete construction of a tensor network such that the entanglement entropy of the physical degrees of freedom is given precisely by the size of a minimal surface that cuts through the bulk of the tensor network.…”

Section: Jhep08(2016)086 1 Introductionmentioning

confidence: 99%

Exploring the tensor networks/AdS correspondence

Bhattacharyya

Gao

Hung

et al. 2016

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.

“…space [7,8]. This proposal is inspired by the study of the hole entropy in the bulk from dual CFT data, which suggests a way to define the bulk geometry from differential entropy [11][12][13][14][15][16][17][18].…”

Section: Jhep01(2017)092mentioning

confidence: 99%

“…It has been widely suspected that the holographic entanglement entropy could play a pivotal role in constructing bulk spacetime and even bulk physics. There are several proposals to construct the bulk geometry from boundary CFT, mainly based on the concept of the tensor network [4][5][6][7][8][9][10]. Among them, one promising approach proposed by B. Czech et al is to view the MERA (Multi-scale Entanglement Renormalization Ansantz) tensor network as a discrete version of vacuum kinematic…”

Section: Introductionmentioning

confidence: 99%

Kinematic space and wormholes

Zhang

Chen

2017

Abstract:The kinematic space could play a key role in constructing the bulk geometry from dual CFT. In this paper, we study the kinematic space from geometric points of view, without resorting to differential entropy. We find that the kinematic space could be intrinsically defined in the embedding space. For each oriented geodesic in the Poincaré disk, there is a corresponding point in the kinematic space. This point is the tip of the causal diamond of the disk whose intersection with the Poincaré disk determines the geodesic. In this geometric construction, the causal structure in the kinematic space can be seen clearly. Moreover, we find that every transformation in the SL(2, R) leads to a geodesic in the kinematic space. In particular, for a hyperbolic transformation defining a BTZ black hole, it is a timelike geodesic in the kinematic space. We show that the horizon length of the static BTZ black hole could be computed by the geodesic length of corresponding points in the kinematic space. Furthermore, we discuss the fundamental regions in the kinematic space for the BTZ blackhole and multi-boundary wormholes.