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arXiv:1907.12110 · Physical review. D/Physical review. D.

Surface/state correspondence and T T ¯ deformation

Bin Chen, Lin Chen, Cheng-Yong Zhang

Published May 11, 2020• 30 citations
arXivHTMLPDF

Abstract

The surface/state correspondence suggests that the bulk codimensional two surface could be dual to the quantum state in the holographic conformal field theory (CFT). Inspired by the cutoff-anti--de Sitter (AdS)/$T\overline{T}$-deformed-CFT correspondence, we propose that the quantum states of the two-dimensional $T\overline{T}$-deformed holographic CFT are dual to some particular surfaces in the ${\mathrm{AdS}}_{3}$ gravity. In particular, the time slice of the cutoff surface is dual to the ground state of the $T\overline{T}$-deformed CFT. We examine our proposal by studying the entanglement entropy and complexity. We find that the complexity of the ground state in the deformed theory is consistent with the one of a particular cMERA and the holographic complexity via CV or CA prescription.

DOI
10.1103/physrevd.101.106011

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The surface/state correspondence suggests that the bulk codimensional two surface could be dual to the quantum state in the holographic conformal field theory (CFT). Inspired by the cutoff-anti--de Sitter (AdS)/TT‾T\overline{T}TT-deformed-CFT correspondence, we propose that the quantum states of the two-dimensional TT‾T\overline{T}TT-deformed holographic CFT are dual to some particular surfaces in the AdS3{\mathrm{AdS}}_{3}AdS3​ gravity. In particular, the time slice of the cutoff surface is dual to the ground state of the TT‾T\overline{T}TT-deformed CFT. We examine our proposal by studying the entanglement entropy and complexity. We find that the complexity of the ground state in the deformed theory is consistent with the one of a particular cMERA and the holographic complexity via CV or CA prescription.
In recent years, the holographic duality between T T ¯ -deformed conformal field theory (CFT) and Anti-de Sitter (AdS) spacetime with finite radial cutoff has received significant attention. The study of T T ¯ deformation within the framework of de Sitter (dS)/CFT duality has also progressed. This paper shows that the trace flow equation in dS spacetime can be analytically extended from its AdS counterpart through double Wick rotations. Meanwhile, we generalize the replica method in both AdS and dS holography to derive a general expression for the entanglement entropy of arbitrary single spatial intervals within the T T ¯ -deformed framework. For both finite size and finite temperature systems, we obtain the analytical expression for the entanglement entropy after T T ¯ deformation. Finally, in dS/dS holography and half-dS holography, we find that the dual field theory exhibits nonlocality by analyzing the strong subadditivity and boosted strong subadditivity of entanglement entropy.
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.
Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT[over ¯]. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough et al. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the TT[over ¯] deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the TT[over ¯]-deformed CFT is the holographic dual of a finite region of spacetime.
In this paper, we discuss tensor network descriptions of AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT from two different viewpoints. First, we start with a Euclidean path-integral computation of ground state wave functions with a UV cutoff. We consider its efficient optimization by making its UV cutoff position dependent and define a quantum state at each length scale. We conjecture that this path integral corresponds to a time slice of anti--de Sitter (AdS) spacetime. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of AdS3{\mathrm{AdS}}_{3}AdS3​ in terms of the dual two-dimensional conformal field theory (CFT). Both approaches support a correspondence between the hyperbolic time slice H2{\mathrm{H}}_{2}H2​ in AdS3{\mathrm{AdS}}_{3}AdS3​ and a version of continuous multiscale entanglement renormalization ansatz. We also give a heuristic argument about why we can expect a sub-AdS scale bulk locality for holographic CFTs.
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.
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.