Gravity from entanglement for boundary subregions

Abstract

Abstract:We explore several aspects of the relation between gravity and entanglement in the context of AdS/CFT, in the simple setting of 3 bulk dimensions. Specifically, we consider small perturbations of the AdS metric and the CFT vacuum state and study what can be learnt about the metric perturbation from the Ryu-Takayanagi (RT) formula alone. It is well-known that, if the RT formula holds for all boundary spacelike segments, then the metric perturbation satisfies the linearized Einstein equations throughout the bulk. We generalize this result by showing that, if the RT formula holds for all spacelike segments contained in a certain boundary region, then the metric perturbation satisfies the linearized Einstein equations in a corresponding bulk region (in fact, it is completely determined in that region). We also argue that the same is true for small perturbations of the planar BTZ black hole and the CFT thermal state. We discuss the relation between our results and the ideas of subregion-subregion duality, and we point out that our argument also serves as a holographic proof of the linearized RT formula for boundary segments.

Modular Hamiltonian of a chiral fermion on the torus

“…Depending on the spin structure chosen on the circle, the Euclidean propagator can be either periodic or antiperiodic in x with period L, and it is antiperiodic in t with period β. Due to these quasiperiodicity properties, we may view G as a section of a line bundle over a torus of circumferences L and β. Inserting (1) in (7) one sees that the Euclidean propagator satisfies…”

“…Among other applications, the knowledge of modular Hamiltonians was essential for the proof of the averaged null energy condition [1], the derivation of quantum energy inequalities [2,3] and the formulation of a well-defined version of the Bekenstein bound [4]. Modular Hamiltonians also played a key role in applications to holography, the most notable case probably being the derivation of the linearized Einstein equations in the bulk from entanglement properties of the boundary Conformal Field Theory (CFT) [5][6][7][8].…”

Holographic Entanglement of Purification from Conformal Field Theories

“…Also in [17][18][19], connections between E W (ρ AB ) and quantities other than the EoP have been proposed. For other recent progresses on the HEoP, refer to [20][21][22][23][24][25][26][27][28][29][30][31][32] In this letter, we would like to present a direct comparison between the HEoP E W (ρ AB ) and the entanglement entropy S AÃ for a special class of purification |Ψ AÃBB , obtained by the path-integral optimization [33,34]. We will focus on several examples in AdS 3 /CFT 2 and will find both quantities always agree with each other in the regime of validity of our computations.…”

Entanglement Spectrum of Chiral Fermions on the Torus

“…In QFT, it is fundamental for the study of relative entropy [17,18] and its many applications to energy and information inequalities [19][20][21]. In the context of the AdS/CFT correspondence, it is instrumental in the program of reconstructing a gravitational bulk from the holographic data [22][23][24][25][26][27][28][29][30][31][32].…”