Relative entropy equals bulk relative entropy

A puzzling aspect of the AdS/CFT correspondence is that a single bulk operator can be mapped to multiple different boundary operators, or precursors. By improving upon a recent model of Mintun, Polchinski, and Rosenhaus, we demonstrate explicitly how this ambiguity arises in a simple model of the field theory. In particular, we show how gauge invariance in the boundary theory manifests as a freedom in the smearing function used in the bulk-boundary mapping, and explicitly show how this freedom can be used to localize the precursor in different spatial regions. We also show how the ambiguity can be understood in terms of quantum error correction, by appealing to the entanglement present in the CFT. The concordance of these two approaches suggests that gauge invariance and entanglement in the boundary field theory are intimately connected to the reconstruction of local operators in the dual spacetime.

Section: Discussionmentioning

confidence: 99%

Precursors, gauge invariance, and quantum error correction in AdS/CFT

Freivogel

Jefferson

Kabir

2016

“…1 Relative entropy has also started to play important roles in black hole physics, holography and quantum field theory; see e.g. [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Relative entropy and the RG flow

Casini

Testé

Torroba

2017

We consider the relative entropy between vacuum states of two different theories: a conformal field theory (CFT), and the CFT perturbed by a relevant operator. By restricting both states to the null Cauchy surface in the causal domain of a sphere, we make the relative entropy equal to the difference of entanglement entropies. As a result, this difference has the positivity and monotonicity properties of relative entropy. From this it follows a simple alternative proof of the c-theorem in d = 2 space-time dimensions and, for d > 2, the proof that the coefficient of the area term in the entanglement entropy decreases along the renormalization group (RG) flow between fixed points. We comment on the regimes of convergence of relative entropy, depending on the space-time dimensions and the conformal dimension ∆ of the perturbation that triggers the RG flow.

“…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

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.