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arXiv:1503.03542 · Progress of Theoretical and Experimental Physics

Surface/state correspondence as a generalized holography

Masamichi Miyaji, Tadashi Takayanagi

Published Jul 1, 2015• 184 citations
arXivHTMLPDF

Abstract

We propose a new duality relation between codimension-two space-like surfaces in gravitational theories and quantum states in dual Hilbert spaces. This surface|$/$|state correspondence largely generalizes the idea of holography such that we do not need to rely on the existence of boundaries in gravitational spacetimes. The present idea is motivated by the recent interpretation of anti-de Sitter space|$/$|conformal field theory (AdS|$/$|CFT) in terms of the tensor networks, so-called MERA (multi-scale entanglement renormalization ansatz). Moreover, we study this correspondence from the viewpoint of entanglement entropy and an information metric. The Cramer–Rao bound in quantum estimation theory implies that the quantum fluctuations of the radial coordinate of the AdS are highly suppressed in the large-|$N$| limit.

DOI
10.1093/ptep/ptv089

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We propose a new duality relation between codimension-two space-like surfaces in gravitational theories and quantum states in dual Hilbert spaces. This surface|///|state correspondence largely generalizes the idea of holography such that we do not need to rely on the existence of boundaries in gravitational spacetimes. The present idea is motivated by the recent interpretation of anti-de Sitter space|///|conformal field theory (AdS|///|CFT) in terms of the tensor networks, so-called MERA (multi-scale entanglement renormalization ansatz). Moreover, we study this correspondence from the viewpoint of entanglement entropy and an information metric. The Cramer–Rao bound in quantum estimation theory implies that the quantum fluctuations of the radial coordinate of the AdS are highly suppressed in the large-|NNN| limit.
The AdS/CFT correspondence, a remarkable duality between certain gravitational theories in anti-de Sitter (AdS) spacetime and quantum field theories with conformal symmetry (CFT), has had a profound effect on the development of theoretical physics in the past two decades. Recently, many connections of AdS/CFT to quantum information theory have been found, in particular by providing gravitationally dual descriptions of various entanglement measures. Understanding these manifestions of AdS/CFT --- or more generally, the conjectured holographic principle encompassing it --- requires the combination of tools from both high-energy theory and quantum information physics. In this cumulative thesis, the convergence between these two fields is approached from two fronts: First, by calculations within the dual gravitational theory, and second, using a tensor network ansatz to describe the quantum states suspected to possess such a gravitational description. In the first approach, using the gravitational side of AdS/CFT, entanglement entropies of complicated 2+1-dimensional excited CFTs are computed, thus showing how the holographic approach provides access to systems previously out of reach of practical methods, while introducing new numerical methods that this approach necessitates. The second approach is given by tensor networks, a highly successful ansatz for computing properties of one- and two-dimensional quantum systems. Efficiently computable classes of tensor networks are tested in their ability to represent simple holographic systems, successfully reproducing both hyperbolic geometrical features as well as critical boundary states. In addition, the general properties of tensor networks on regular hyperbolic tesselations are considered, leading to new connections to models not previously considered in the context of holography. This interplay of different approaches to quantum information holography showcases the richness of this new field and suggests that a wide range of physical phenomena is accessible via the new tools now at our disposal.
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.
We present a general procedure for constructing tensor networks for geometric states in the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. Given a state in a large-NNN CFT with a static, semiclassical gravitational dual, our procedure produces a tensor network for the boundary state whose internal geometry matches (a discretization of) the bulk spacetime geometry. By invoking the "holographic entanglement of purification" conjecture, our construction can be made to capture the structure of the bulk spacetime at sub-AdS scales.
We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of the large NNN limit where on the holographic side the gravity theory becomes classical and a non-fluctuating smooth spacetime description emerges. We show how a number of features of holographic duality in the large NNN limit emerge naturally from entanglement renormalization, including a classical spacetime generated by entanglement, a sparse spectrum of operator dimensions, and phase transitions in mutual information. We also address questions related to bulk locality below the AdS radius, holographic duals of weakly coupled large NNN theories, Fermi surfaces in holography, and the holographic interpretation of branching MERA. Some of our considerations are inspired by the idea of quantum expanders which are generalized quantum transformations that add a definite amount of entropy to most states. Since we identify entanglement with geometry, we thus argue that classical spacetime may be built from quantum expanders (or something like them).
We introduce a new quantity, called pseudo-entropy, as a generalization of entanglement entropy via postselection. We expect this quantity to provide a new class of order parameters in quantum many-body systems. In the anti--de Sitter space (AdS)/conformal field theory (CFT) correspondence, this quantity is dual to areas of minimal area surfaces in time-dependent Euclidean spaces which are asymptotically AdS. We call this geometric computation of pseudo-entropy via the AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT the holographic pseudo-entropy. We study its basic properties and classifications in qubit systems. In specific examples, we provide a quantum information theoretic meaning of this new quantity as an averaged number of Bell pairs when the post-selection is performed. We also present properties of the pseudo-entropy for random states. We then calculate the pseudo-entropy in the presence of local operator excitations for both the two dimensional free massless scalar CFT and two dimensional holographic CFTs. We find a general property in CFTs that the pseudo-entropy is highly reduced when the local operators get closer to the boundary of the subsystem. We also compute the holographic pseudo-entropy for a Janus solution, dual to an exactly marginal perturbation of a two dimensional CFT and find its agreement with a perturbative calculation in the dual CFT. We show the linearity property holds for holographic states, where the holographic pseudo-entropy coincides with a weak value of the area operator. Finally, we propose a mixed state generalization of pseudo-entropy and give its gravity dual.
Abstract In the context of the AdS/CFT correspondence, it is often convenient to regulate infinite quantities in asymptotically anti-de Sitter spacetimes by introducing a sharp cutoff at some finite, large value of a particular radial coordinate. This procedure is a priori coordinate dependent, and may not be well-motivated in full, covariant general relativity; however, the fact that physically meaningful quantities such as the entanglement entropy can be obtained by such a regulation procedure suggests some underlying covariance. In this paper, we provide a careful treatment of the radial cutoff procedure for computing holographic entanglement entropy in asymptotically anti-de Sitter spacetimes. We prove two results that are frequently assumed in the literature, but that have not been carefully addressed: (i) that the choice of a “globally minimal surface” among several extremal candidates is independent of the choice of regulator, and (ii) that finite CFT quantities such as the mutual information which involve “divergence-cancelling” sums of entanglement entropies are well-defined under the usual prescription for computing covariant holographic entanglement entropy. Our results imply that the “globally minimal surface” prescription for computing the holographic entanglement entropy is well-posed from the perspective of general relativity, and thus support the widely-held belief that this is the correct prescription for identifying the entanglement wedge of a boundary subregion in AdS/CFT. We also comment on the geometric source of state-dependent divergences in the holographic entanglement entropy, and identify precisely the regime of validity of the “vacuum subtraction” protocol for regulating infinite entanglement entropies in arbitrary states by comparing them to the entanglement entropies of identical regions in the vacuum. Our proofs make use of novel techniques for the covariant analysis of extremal surfaces, which are explained in detail and may find use more broadly in the study of holographic entanglement entropy.