[Paper Review] More on Holographic Volumes, Entanglement, and Complexity

Dean Carmi|arXiv (Cornell University)|Sep 29, 2017

Black Holes and Theoretical Physics57 references4 citations

TL;DR

This paper derives a universal formula for the shape dependence of holographic entanglement entropy via perturbations of the entangling surface, generalizing to codimension-p extremal surfaces. It computes universal terms in codim-one volumes under relevant deformations and explores bulk surface constructions related to holographic complexity, providing new insights into the geometric structure of entanglement and complexity in AdS/CFT.

ABSTRACT

Motivated by the holographic prescriptions for computing entanglement entropy and complexity, we study the properties of volumes/areas of bulk surfaces. We obtain a simple formula for the shape dependence of holographic entanglement entropy in terms of a certain integral over the entangling surface. This easily generalizes to any bulk codimension-$p$ extremal surface. We study additional properties of bulk codimension-$p$ extremal surfaces corresponding to strip/plane "entangling surfaces" in various geometries. We compute universal terms for codim-one volumes (conjectured to be dual to holographic subregion complexity) arising from performing relevant deformations. Finally, we describe several interesting bulk surface constructions which are presumably related to holographic complexity.

Motivation & Objective

• To derive a general formula for the shape dependence of holographic entanglement entropy in terms of an integral over the entangling surface.
• To study the behavior of codimension-p extremal surfaces in various bulk geometries, including thermal and confining backgrounds.
• To compute universal terms in codim-one volumes arising from relevant deformations, relevant to holographic subregion complexity.
• To explore bulk surface constructions that may be dual to quantum complexity in gauge-gravity duality.

Proposed method

• Uses perturbation theory on the entangling surface to derive a first-order change in entanglement entropy as an integral over the boundary entangling surface.
• Applies the same perturbative approach to codimension-p extremal surfaces, generalizing the Ryu-Takayanagi formula to arbitrary codimensions.
• Computes time evolution of codim-p surfaces in the thermofield double state to study dynamical complexity.
• Analyzes the bulk action on the intersection of the entanglement wedge and WDW patch to compute subregion complexity in global AdS.
• Performs explicit calculations in global AdS with a sphere entangling surface, deriving the divergence structure of the bulk action.
• Uses the replica method and boundary condition gluing to explore generalized complexity measures involving density matrix transposes.

Experimental results

Research questions

• RQ1 How does the entanglement entropy change under small deformations of the entangling surface, and can this be captured by a universal integral formula?
• RQ2 What are the universal terms in the volume of codim-one extremal surfaces after relevant deformations, and how do they relate to holographic complexity?
• RQ3 How do codimension-p extremal surfaces behave in thermal and confining geometries, and what universal features emerge?
• RQ4 What bulk surface constructions are relevant to the conjectured complexity=volume duality in AdS/CFT?

Key findings

• A simple formula for the shape dependence of holographic entanglement entropy is derived as an integral over the entangling surface, valid for any codimension-p extremal surface.
• The leading-order change in entanglement entropy due to surface deformations is shown to be proportional to the integral of the perturbation over the boundary surface, with a geometric kernel.
• Universal terms in codim-one volumes—conjectured to be dual to holographic subregion complexity—are computed after relevant deformations, revealing logarithmic and power-law divergences.
• In global AdS with a spherical entangling surface, the bulk action on the subregion complexity region exhibits a divergence structure that can be systematically expanded near the boundary.
• The time evolution of codim-p surfaces in the thermofield double state is computed, showing non-trivial dynamics relevant to complexity growth.
• The paper identifies a class of bulk surface constructions that may be dual to complexity, including those involving the entanglement wedge and WDW patch intersections.