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A Covariant Holographic Entanglement Entropy Proposal

Veronika E. Hubeny, Mukund Rangamani, Tadashi Takayanagi

Published Jan 1, 2008• 977 citations
OpenAlex

Abstract

With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.

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With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.
A bstract We consider holographic entanglement entropy in AdS black hole backgrounds by using the limit of large number of dimensions. By dividing the geometry to two patches (with one patch covering the vicinity of the black hole horizon and another covering the other regions), we are able to obtain fully analytic expressions for the entropy when the entanglement region is a strip. We argue that apart from conformal field theories at finite temperature in high number of dimensions, our method works for nearly critical theories in 3+1 dimensions. In the case of extremal black holes, dual to quantum critical systems, the results take a particularly simple form. We also comment on the case of soliton geometries. Finally, we analyze entanglement entropy for wide strips, and propose a general formula for the first subleading term in the expansion of the entropy in (inverse) system size for generic entanglement regions.
We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Rényi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.
We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Rényi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.
Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
It was found, by studying black holes, that the compatibility of the theory of gravity and the laws of quantum mechanics demands that the universe must act like a hologram. That is, all the information inside a region of the universe should be encoded on a so-called holographic screen of one-lesser dimension. This is termed the holographic principle. In a generic spacetime, such as an expanding universe, holographic screens depend on the choice of an observer, which is consistent with the notion that the observer is part of the system in cosmology. In the first part of this dissertation, we study the observer-dependence of holographic screens. This will help us understand how the fundamental description of the universe depends on the choice of the observer. Furthermore, we study the dynamics of the holographic screens and their geometry in the first part of this dissertation.An example where the holographic principle is manifest is the AdS-CFT correspondence. This correspondence connects quantum information quantities, like entanglement and complexity, to geometric quantities, like area and volume, respectively. In the second and the third part of this dissertation, we use the AdS-CFT correspondence as a tool to calculate the entanglement entropy and the computational complexity of a quenched CFT state. Studying the time evolution of the entanglement entropy following a quantum quench teaches us how a CFT state thermalizes. On the other hand, studying the time evolution of the computational complexity allows us to check the validity of various recent conjectures involving black holes and complexity.
This thesis is devoted to a possible generalization of the entanglement entropy and its implication to the holography, especially AdS/CFT correspondence.The AdS/CFT correspondence is a way to define the quantum gravity in d + 1-dimensional asymptotically anti-de Sitter (AdS) spacetime by using the conformal field theory (CFT) in d-dimension.In this context, it is still unclear how we can realize the d + 1-dimensional full bulk geometries from the d-dimensional quantum field theories.The main object of this thesis is the minimal surface not necessarily anchored on the boundary, which has not been derived from the field theory analysis.First, we review how we can obtain minimal surfaces anchored on the boundary from the entanglement entropy.Then, we introduce a generalization of the entanglement entropy for mixed states and discuss its basic properties.Finally, we argue that we can extract the aforementioned minimal surfaces ending on the bulk from the generalized version of the entanglement entropy.Our explicit derivations include the geometries in three-dimensional AdS and BTZ black holes.First and foremost, I would like to express my deepest gratitude to Norihiro Iizuka, my supervisor.I could not reach here without his patient guidance and encouragement.