Reected entropy in random tensor networks II a topological index from the canonical purication

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Reﬂected entropy in random tensor

networks II: a topological index from the

canonical puriﬁcation

Chris Akers,1Thomas Faulkner,2Simon Lin,2Pratik Rath3

1Center for Theoretical Physics,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

2Department of Physics, University of Illinois,

1110 W. Green St., Urbana, IL 61801-3080, USA

3Department of Physics, University of California, Santa Barbara, CA 93106, USA

cakers@mit.edu,tomf@illinois.edu,shanlin3@illinois.edu,rath@ucsb.edu

Abstract: In Ref. [1], we analyzed the reﬂected entropy (SR) in random tensor net-

works motivated by its proposed duality to the entanglement wedge cross section (EW)

in holographic theories, SR= 2EW

4G. In this paper, we discover further details of this du-

ality by analyzing a simple network consisting of a chain of two random tensors. This

setup models a multiboundary wormhole. We show that the reﬂected entanglement

spectrum is controlled by representation theory of the Temperley-Lieb (TL) algebra.

In the semiclassical limit motivated by holography, the spectrum takes the form of a

sum over superselection sectors associated to diﬀerent irreducible representations of the

TL algebra and labelled by a topological index k∈Z≥0. Each sector contributes to

the reﬂected entropy an amount 2kEW

4Gweighted by its probability. We provide a grav-

itational interpretation in terms of ﬁxed-area, higher-genus multiboundary wormholes

with genus 2k−1 initial value slices. These wormholes appear in the gravitational

description of the canonical puriﬁcation. We conﬁrm the reﬂected entropy holographic

duality away from phase transitions. We also ﬁnd important non-perturbative con-

tributions from the novel geometries with k≥2 near phase transitions, resolving the

discontinuous transition in SR. Along with analytic arguments, we provide numerical

evidence for our results. We comment on the connection between TL algebras, Type

II1von Neumann algebras and gravity.

arXiv:2210.15006v2 [hep-th] 2 Nov 2022

Contents

1 Introduction 1

2 Motivation: Canonical Puriﬁcation in Gravity 6

3 Setup 10

3.1 Replica Trick for the 2TN Model 11

3.2 Resolvent via the Temperley-Lieb Algebra 15

4 Reﬂected Entropy in 2TN 19

4.1 Finite χ20

4.2 Large χlimit 24

4.2.1 Even integer m24

4.2.2 Analytic continuation 29

4.3 Reﬂected spectrum and the eﬀective description 32

4.4 Corrections to the spectrum 36

4.4.1 Eigenvalue shifts 36

4.4.2 Fluctuations in each sector 37

4.5 Numerical results 40

5 Discussion 44

5.1 General RTNs and Multiboundary Wormholes 45

5.2 Junctions for the Cross Section 47

5.3 Emergent von Neumann algebras 48

A Multiboundary wormholes 51

B Temperley-Lieb algebra 53

B.1 Basic deﬁnitions 53

B.2 The standard module 57

C Finite χcorrections 62

C.1 Corrections to orthogonality condition 62

C.2 Corrections from subleading TL diagrams 65

C.3 explicit form for k= 0 and k= 1 71

– i –

D Proofs 74

D.1 Proof of Proposition 1 74

D.2 Proof of Proposition 2 77

D.3 Proof of Proposition 4 79

1 Introduction

The intriguing connection between geometry and entanglement in the context of holog-

raphy has resulted in big leaps in our understanding of quantum gravity. The Ryu-

Takayanagi (RT) formula [2–4] relating boundary entropy to the area of bulk extremal

surfaces is the hallmark of such an emergence of spacetime from entanglement. In the

pursuit of more such links, a proposal for the holographic dual to another geometric

object, the entanglement wedge cross section, was made in Ref. [5]. The proposed dual,

the reﬂected entropy, is a novel measure of correlation between bipartite mixed states,

or equivalently tripartite pure states.

The reﬂected entropy is deﬁned as

SR(A:B) = S(AA∗)|√ρAB i,(1.1)

where the state |√ρABi∈HAB ⊗ HA∗B∗is the canonical puriﬁcation of the density

matrix ρAB. The subsystems A∗, B∗are referred to as the reﬂected copies of the sub-

systems A, B respectively. The holographic proposal then states

SR(A:B) = 2EW (A:B)

4G,(1.2)

where EW (A:B) is the minimal cross section splitting the entanglement wedge of

AB (see Fig. (1)). In Eq. (1.2), we have ignored quantum corrections, as well as time

dependence (see Refs. [6, 7] for details). For simplicity, this paper will be limited

to discussing the static, classical proposal although there is no reason to suspect the

results do not generalize. 1This proposal has already been useful in demonstrating the

need for large amounts of tripartite entanglement in holographic states [11]. More so,

it can be thought of as a generalization of the RT formula with a boundary dual that

is rigorously well deﬁned even in the continuum limit [5]. Thus, it is interesting to ﬁnd

evidence for such a duality.

1While time dependence is straightforward, quantum corrections would likely include subtleties

arising from corrections to the QES formula [8–10].

– 1 –

Figure 1:EW (A:B) is the minimal area surface that divides the entanglement wedge

of AB, bounded by the RT surface γAB, into regions homologous to subregions Aand

Brespectively.

The original argument for the proposal involved a two-parameter replica trick, fol-

lowed by an analytic continuation `a la Lewkowycz-Maldacena [12]. While the proposal

passes various sanity checks, it was noted in Ref. [13] that the replica trick argument

itself suﬀered from an order of limits issue. More so, the EW cross-section undergoes

a discontinuous transition when the entanglement wedge changes from disconnected to

connected. This raises the possibility of non-perturbative eﬀects becoming important

to resolve the phase transition. Thus, it is of interest to use solvable toy models to

better understand the above issues.

In Ref. [1], we used random tensor networks (RTNs) [14] as a playground to under-

stand the various subtleties associated with the replica trick argument. In particular,

for a tripartite state generated from a single random tensor, we were able to use analytic

and numerical techniques to extract the reﬂected spectrum, the entanglement spectrum

of ρAA∗. A crucial role was played by the addition of a novel saddle which dominated

in portions of parameter space and motivated a resolution to the order of limits issue.

Our analysis provided evidence for the validity of the proposal in Eq. (1.2). Since the

replica trick in RTNs involves a sum over permutations that is quite analogous to the

sum over topologies in the gravitational path integral, there is good reason to believe

that the analysis in RTNs is a faithful indicator of the calculation in gravity. More-

over, we were also able to solve the above problem in the West Coast model consisting

of Jackiw-Teitelboim gravity coupled to end-of-the-world branes [15], ﬁnding further

evidence for Eq. (1.2) along with novel features near phase transitions [16].

In this paper, we carry on with our analysis of reﬂected entropy in RTNs in the

hope of ﬁnding other undiscovered aspects of the replica trick. In particular, we will

focus on an RTN consisting of two random tensors, which we refer to as 2TN. 2TN can

– 2 –

Figure 2: (left) The 2TN tensor network considered in this section is built from

two random tensors T1and T2. The parameters are the boundary bond dimensions

χA, χB, χC1, χC2and the internal bond dimension χ. (right) The wormhole solution that

is modeled by 2TN. The external bond dimensions corresponds to the three horizon

areas and the internal bond dimension χcorresponds to the cross-section surface γW.

be interpreted as a model for a four-boundary wormhole as depicted in Fig. (2), where

the areas of the labelled surfaces are ﬁxed to a narrow window [17–19]. More generally,

we will provide heuristic arguments that the calculations in 2TN are also useful for

more general settings, e.g., the familiar setup of two intervals in vacuum AdS depicted

in Fig. (1). Since the bulk geometry is coarse-grained down to just two tensors, the

model cannot capture any of the local dynamics. However, it does capture general

topological aspects of the gravitational calculation which turn out to be the relevant

aspect for the reﬂected entropy, including near phase transition eﬀects.

In Sec. (2), we start by motivating the gravitational construction of novel, higher

genus saddles that contribute to the canonical puriﬁcation. We consider the gravita-

tional state corresponding to the four boundary wormhole depicted in Fig. (2), prepared

using a Euclidean path integral with ﬁxed area boundary conditions. As discussed in

Refs. [8–10, 15, 17–19], the replica trick for such ﬁxed-area states is simpliﬁed by the

fact that one can simply glue together multiple copies of the original bulk geometry

without having to solve for a new backreacted geometry. Thus, we have control over

the diﬀerent saddles contributing to the canonical puriﬁcation. By doing a replica trick

to construct the state |ρm/2

AB ifor even integer m[5], we ﬁnd saddles labelled by a topo-

logical index k∈Z>0. They correspond to geometries with initial data slices obtained

by gluing together 2kcopies of the shaded region (see Fig. (2)) of the connected en-

tanglement wedge of AB in the original state. Each such geometry contributes with

an amplitude √pkcomputed from the path integral. The canonically puriﬁed state

can then be obtained via analytic continuation to m= 1, and is approximately given

by a superposition over such geometries as shown in Fig. (3). Thus, we obtain a fam-

– 3 –

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摘要：

ReectedentropyinrandomtensornetworksII:atopologicalindexfromthecanonicalpuricationChrisAkers,1ThomasFaulkner,2SimonLin,2PratikRath31CenterforTheoreticalPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA2DepartmentofPhysics,UniversityofIllinois,1110W.GreenSt.,Urbana,IL61801-3080,USA3De...

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Reected entropy in random tensor networks II a topological index from the canonical purication

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