Universiteit Van Amsterdam

Master project

Tensor Networks As Probes ofNear Horizon Geometry.

Shustrov Yaroslav

supervised byDr. Jan de Boer

August 5, 2016

Contents

1 Introduction 1

2 General Concepts 32.1 AdS Space-time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Holographic shadow of BTZ black hole. . . . . . . . . . . . . . . . . . 62.3 Important concepts of Quantum Mechanics. . . . . . . . . . . . . . . 9

3 AdS/CFT 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Holographic Derivation of Entanglement Entropy from AdS/CFT . . 143.3 Bulk reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Global bulk reconstruction . . . . . . . . . . . . . . . . . . . . 213.3.2 AdS-Rindler reconstruction . . . . . . . . . . . . . . . . . . . 223.3.3 Paradoxes of AdS-Rindler reconstruction . . . . . . . . . . . . 233.3.4 AdS/CFT as quantum error correction code . . . . . . . . . . 24

4 Tensor Networks and AdS/CFT 274.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 MERA and Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Limitations on sub-AdS scale resolution of MERA. . . . . . . 304.3 Isometric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Perfect tensors and RT formula. . . . . . . . . . . . . . . . . . . . . . 324.5 Absence of sub-AdS scale resolution for TN formed by perfect tensors. 344.6 Tensor network formed by pluperfect tensors. . . . . . . . . . . . . . 36

4.6.1 Definition of pluperfect tensor. . . . . . . . . . . . . . . . . . . 364.6.2 Gauge invariance. . . . . . . . . . . . . . . . . . . . . . . . . . 384.6.3 Equivalence Of Two Definitions. . . . . . . . . . . . . . . . . . 394.6.4 Absence of local operators . . . . . . . . . . . . . . . . . . . . 404.6.5 Classical geometry states . . . . . . . . . . . . . . . . . . . . . 414.6.6 Emergence of Bulk Locality . . . . . . . . . . . . . . . . . . . 424.6.7 Low-energy subspace and gauge invariance. . . . . . . . . . . . 43

4.7 Random Tensor Networks . . . . . . . . . . . . . . . . . . . . . . . . 444.7.1 Definition of RTN . . . . . . . . . . . . . . . . . . . . . . . . . 444.7.2 Calculation of Second Renyi Entropy . . . . . . . . . . . . . . 464.7.3 Ryu-Takayanagi Formula For Direct-Product Bulk state. . . . 484.7.4 Ryu-Takayanagi formula With Bulk State Correction. . . . . . 494.7.5 Hawking-Page transition. . . . . . . . . . . . . . . . . . . . . . 504.7.6 Large D limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2

5 Conclusion 54

Appendices 56

A Cutoffs for Entropy expression. 57

B Two-point function from AdS/CFT 60

C Integral evaluation. 63

D Entropy of subsystem of random pure state. 64

3

Chapter 1

Introduction

One of the greatest puzzles of modern high energy physics is problem of unificationof General Relativity and Quantum Field Theory. It is also known as problem of“quantization of gravity”.

Many different approaches exist, however so called AdS/CFT correspondenceintroduced almost 20 years ago is considered to be one of the most notable amongthem. This proposal suggests equivalence between a (d+1)-dimensional theory ofquantum gravity in anti-de Sitter space and a conformal field theory in d dimensions.Although this statement still doesn’t have strict mathematical proof, huge numberof explicit calculations suggest validity of this proposal.

According to AdS/CFT correspondence bulk physics may be recovered solelyfrom information on the boundary. Bulk non-local quantum gravity is implied tohave local limit at low energies. It is important to understand how do bulk degreesof freedom organise themselves to produce local field theory in the aforementionedlimit.

Explicit realisation of this duality encounters number difficulties, many of whichhave been successfully resolved within past years. For instance certain paradoxesassociated with reconstruction of local bulk operators from part of the boundarywere resolved by means of so called quantum error correction code. In particularIt was shown that bulk operators may be reconstructed from multiple boundaryregions.

All observables in the bulk are implied to have interpretation in terms of bound-ary information. One of the most notable realisations of such statement is so calledRyu-Takayanagi proposal. It relates two concepts from seemingly disconnected the-ories. In particular this proposal states that entanglement entropy of boundaryregion is equal(up to constant) to the area of bulk minimal surface, bounding thisregion. First quantity is characteristic value from quantum mechanics while thenotion of minimal surfaces is used in general relativity.

In some particular cases[1] it may be shown that just knowing entanglemententropy of the boundary is enough to obtain explicitly bulk solution. This becameone of reasons to believe that bulk space-time is an emergent phenomena [2].

One of the most widely used probes of bulk geometry is minimal surface anchoredon the AdS boundary. For instance it appears in RT formula and AdS-Rindlerreconstruction. In case of AdS3 this minimal surface is reduced to geodesic. Thisgeodesic bounds bulk region which is known as causal wedge. One may show thatlinearized version of Einstein equations holds in this region[3].

1

However depending on amount of matter(entanglement) in the bulk, minimalsurfaces may fail to reach all internal regions. This happens not only in presenceof horizons in the bulk but also for regular geometries such as dense enough stars.The regions that can’t be reached by these minimal surfaces are called holographicshadows. It is very important to find interpretation of these bulk regions in termsof boundary information. For instance this information will shed light on nearhorizon geometry. Many crucial questions about in-falling observer may find explicitanswers. Although different kinds of probes were considered in the literature, allof them have holographic shadows of certain size(which may be as large as lAdS.).Purpose of our work is to propose new tool, namely tensor networks, as a toy modelof AdS/CFT which may be used to find interpretation of holographic shadow regionsin terms of boundary information.

Initially introduced in condensed matter physics tensor networks turned outto be also very useful in area of AdS/CFT. It was later realised by Swinger[26]that tensor network MERA(originally designed to represent ground state of certainhamiltonians) satisfies formula, analogous to RT formula for empty AdS. This wasfirst motivation to try tensor networks in general and MERA in particular as atoy model of AdS/CFT. A lot of progress has been done in this area within pastyears. Many nontrivial properties of AdS/CFT correspondence such as quantumerror correction properties found their reflection in different tensor networks.

There are two main directions in incorporation of tensor networks and hologra-phy. First one is to provide different modifications of MERA itself so they couldcapture more aspects of AdS/CFT. The second approach is to build other types oftensor networks from different types of tensors which basically define properties ofobtained object. Each of these approaches has its own advantages and disadvan-tages. However for purposes of our work only second one may be used, becauseonly this approach so far may introduce sub-AdS scale resolution(describe physicsat distances smaller than lAdS).

In this thesis we overview different kinds of tensor networks and argue whethereach of them may be used for the problem of interest. Although much work re-mains to be done our conclusion is that random tensor network seems to be themost promising one among considered. Firstly, it possesses sub-AdS scale resolu-tion. Secondly, RT formula can be proven for highly entangled bulk states. Theseproperties are needed in order to obtain TN representation of constant time slice ofBTZ black hole which captures nontrivially geometry of holographic shadow region.With this TN one may potentially find interpretation of aforementioned region interms of boundary information.

2

Chapter 2

General Concepts

2.1 AdS Space-time.

In this section Anti-de Sitter space-time will be introduced. In particular we willconsider different coordinate systems used to represent this space-time and connec-tions between them. Most of explicit calculations will be made for AdS3, because inour work we focus mainly on this particular case. Following section is based on [4].

Global coordinate systems.

AdS space-time is maximally symmetric solution of Einstein equations with neg-ative cosmological constant Λ = −d(d−1)

2l2AdS, where lAdS is radius of curvature and d is

dimension of space-time.

One can view AdS(p, q) as a hyperboloid, embedded into space-time Rp,q+1. Inparticular let us consider flat space-time with two spacial dimensions (X1, X2) andtwo time-like directions (X0, X3):

ds2 = −dX20 − dX2

3 + dX21 + dX2

2 (2.1)

Now to obtain AdS3 space-time we take locus of points at fixed time-like distancefrom the origin:

−X20 −X2

3 +X21 +X2

2 = −l2AdS (2.2)

The induced metric will indeed have constant negative curvature. One can alsocheck that it is maximally symmetric by listing all d(d+1)

2Killing vectors.

We can introduce change of coordinates:

X0 = lAdS cosh ρ cos τ

X3 = lAdS cosh ρ sin τ

X1 = lAdS sinhρ sin θ

X2 = lAdS sinhρ cos θ

(2.3)

and obtain new representation of metric:

3

ds2 = l2AdS(−cosh2ρdτ 2 + dρ2 + sinh2ρdθ2) (2.4)

where ρ ∈ R+ and ρ→∞ corresponds to the boundary of AdS space-time.τ ∈ [0, 2π]. To avoid closed time-like curves one should redefine τ in a way thatτ ∈ R.

Figure 2.1: AdS3 space-time. Constant time slice is given by hyperbolic disc. Imageform Wikipedia.

One can produce change of coordinates to obtain yet another global covering ofAdS space-time:

r ≡ lAdS · sinhρ

t ≡ lAdS · τWhich will lead to expression:

ds2 = −f(r)dt2 +1

f(r)dr2 + r2dθ2; f(r) = 1 +

r2

l2AdS

(2.5)

Let us define one more global covering(coordinate system, which covers the wholemanifold). We will also consider AdS3 case, but now its Euclidean version.

−X20 +X2

3 +X21 +X2

2 = −l2AdS (2.6)

ds2 = −dX20 + dX2

3 + dX21 + dX2

2

Change of variables may be introduced:

x =X1

X0 + lAdS

y =X2

X0 + lAdS

z =X3

X0 + lAdS

(2.7)

ρ2 ≡ x2 + y2 + z2 =X0 − lAdS

X0 + lAdS

Once put together one gets:

ds2 =4l2AdS

(1− ρ2)2(dx2 + dy2 + dz2) (2.8)

Or in spherical coordinates:

ds2 =4l2AdS(dr2 + r2(dθ2 + sin2θdφ2))

(1− r2)2(2.9)

4

Which for θ = Π2; 0 < r < 1 in known as Poincare disc metric.

Last of global coverings that we will introduce may be obtained by followingchange of coordinates in 2.8:

r = tanh(y

2

)(2.10)

New metric will look like:

ds2 = l2AdS(dy2 + sinh2ydΩ2) (2.11)

Incomplete coordinates patches.

There exist coordinate patches, which cover only parts of AdS space-time, yetwhich are widely used. We will also consider only AdS3 case:

−X20 −X2

3 +X21 +X2

2 = −l2AdS (2.12)

ds2 = −dX20 − dX2

3 + dX21 + dX2

2

After change of coordinates:X1 =

l2AdS

2r

(1 + r2

l4AdS(α2 + x2 − t2)

)X2 = r

lAdSt

X3 = rlAdS

x

X4 =l2AdS

2r

(1− r2

l4AdS(α2 − x2 + t2)

) (2.13)

we will obtain:

ds2 = − r2

l2AdS

dt2 +l2AdS

r2dr2 +

r2

l2AdS

dx2 (2.14)

where r ≥ 0, r →∞ corresponds to boundary of AdS.

If we introduce yet another coordinate transformation:

z ≡ 1

u · lAdS

we will obtain another wildly used coordinate patch:

ds2 =l2AdS(dz2 − dt2 + dx2)

z2(2.15)

where z =0 corresponds to conformal boundary. There are several notable propertiesof this metric. It is conformally equivalent to half space of Minkowski space-time. Orput it in different way we can see that in the limit z → 0 there is Minkowski space-time on the boundary. Any constant time slice of this metric gives hyperbolic half-plane also known as Poincare half-plane metric. (We also discuss certain propertiesof geodesics in these hyperbolic half-planes in appendix A.)

If one makes transformation to Euclidean time this metric becomes full coveringof Euclidean AdS space-time.

5

Figure 2.2: Green region represents patch 2.15. Full AdS(given by 2.4) is representedby cylinder. Image from [5].

2.2 Holographic shadow of BTZ black hole.

Holographic shadow[6] is the region in the bulk which can’t be reached by anyminimal surface anchored on the boundary. This type of regions exists both forbulk geometries with horizons and for regular ones. The example of latest typeof geometry is dense star in AdS space-time. Even though from point of view ofgeneral relativity nothing distinguishes holographic shadow region from rest of thebulk a lot of interesting things happen there. For instance it is unclear how toreconstruct bulk operators which lie inside this region. In case of BTZ black holeholographic shadow region surrounds horizon. Presence of this region is the reasonwhy it is difficult to get answers to number of questions about observer falling intoblack hole. This section gives an explicit derivation of this region. In particular weshow the connection between size of holographic shadow and radius of black hole.Following part of section summarises certain results from articles [6], [7] .

First let us give brief sketch of Ryu-Takayanagi(RT) proposal. Deeper analysisof certain aspects of this proposal will be given in next sections. For now we areinterested in case of BTZ black hole, which metric is given by[7]:

ds2 = −f(r)dt2 +dr2

f(r)+ r2dθ2, f(r) = r2 − r2

+ (2.16)

where r+ is radius of black hole.RT proposal connects entanglement entropy of boundary region with area of

minimal surface anchored on this region.Suppose we have constant time slice of BTZ bh. Let us consider simply connected

boundary region A. Now we take some bulk region R, which matches A on theboundary. This statement may be rewritten in following way:

∂R = EA ∪ A; ∂EA = ∂A

We need to change size of R in such a way so EA becomes minimal surface, ho-mologous to the boundary region A(can be continuously transformed to region A).Length(area in higher dim) of this minimal surface will be equal(up to constant) toentanglement entropy of boundary region A.

Depending on size of A the solution may be given by two different families ofbulk curves:

6

Figure 2.3: Left - connected family: EA is given by geodesic matching A; Right- disconnected family. EA is given by sum of geodesic and circumference of BH.Homology constraint on minimal surface is crucial.

Equation for geodesics is given by[7]:(r, θ) : r = γ(θ, θ∞, r+) ≡ r+

(1− cosh2(r+θ)

cosh2(r+θ∞)

)−1/2(2.17)

In the above expression r+ is radius of BTZ black hole, θ∞ is parameter associatedto each geodesic(see Fig 2.3), θ is angle parameter along the geodesic. lAdS is set tobe equal to 1 and may be restored from dimensional analysis.

According to RT proposal in case of multiple families of minimal surfaces wepick one with the smallest length(area).

Hence if we start from small region A and then gradually increase its size thenentropy of region A will be given at first by family of connected minimal surfaces andstarting from some point by disconnected ones.This indicates that phase transitionbetween between two families takes place.

Figure 2.4: r+ = 0.2 At some point two geodesics from different families correspond-ing to orange region will have the same length. Image from [7]

This phase transition defines subregion in the bulk which can’t be reached by

7

means of minimal geodesics. Following notation from article [6] we will call thisregion holographic shadow.

Now based on approach introduced in this article we will evaluate size of holo-graphic shadow for given radius of black hole.

Boundary region is defined as θ∞; r → ∞. Length of geodesic in this metric isgiven by:

γA =

∫ds =

∫ √grr(r′)2 + gθθdθ =

∫ √(r′)2

(r2 − r+)2+ r2dθ, r′ ≡ dr

dθ(2.18)

Integrand is invariant under translations in θ direction. So we can consider it tobe some kind of Lagrangian. In this case r corresponds to spacial coordinate and θcorresponds to time. Then we can define hamiltonian corresponding to it:

H =−r2√

(r′)2

(r2−r+)2+ r2

In general H is some constant. We may fix it by considering some boundary con-dition. For instance we may note that geodesic reaches its minimum value r∗ (inradial direction) when r′ = 0.

This leads to equation:

dr

dθ=

r

r∗

√r2 − r2

∗

√r2 − r2

+ (2.19)

We may integrate it to obtain:

θ∞ =

∫ ∞r∗

drdθ

dr=

1

2r+

cosh−1(r2∗ + r2

+

r2∗ − r2

+

)(2.20)

Hencer∗ =

r+

tanh(θ∞r+)(2.21)

Now let us find θswitch at which phase transition takes place. It is defined by condi-tion:

γ(θswitch) = γ(π − θswitch) + 2πr+ (2.22)

where γ(θ) is length of geodesic defined by θ.

γ(θ∞) = 2

∫ ∞r∗

rdr√r2 − r2

∗√r2 − r2

+

(2.23)

Once we put these equations together we will obtain:

θswitch =π

2+

1

2r+

log(

cosh(πr+))

(2.24)

Hence:

rmin =r+

tanhπr+

+r+e

−πr+

sinh(πr+)(2.25)

And finally size of holographic shadow is given by:

4r0 ≡ rmin − r+ =2r+e

−πr+

sinh(πr+)(2.26)

8

One can see that size of holographical shadow indeed depends on radius of blackhole.

In particular :

r+ lAdS 4 r0 ∼ lAdS

r+ lAdS 4 r0 ∼ e− #r+lAdS

Figure 2.5: Size of holographic shadow rmin as a function of horizon radius r+ ofBTZ black hole. Image from [6]

One may also show [6] that holographical shadow also exists if we consider denseenough star in Ads.

There are other probes of geometry near BTZ bh. However for all of them shadowstill takes place.

2.3 Important concepts of Quantum Mechanics.

This section aims not to give strict definitions but rather to provide summary ofuseful notions and different connections between them, which will be used in futureinvestigations. It is mainly based on article[8].

Hilbert space is a complex-valued vector space which defines quantum system.One may say that Hilbert space is quantum analogue of phase space of classicalsystem. However we must stress out that the principal difference is that unlikephase space Hilbert space is vector space. That leads to the superposition principle,which states that superposition of two states of quantum system is also state ofthis system. It is obviously not true for classical system and exactly because ofaforementioned reason.

As was mentioned before quantum system is defined by its state. If one knowsquantum state together with its hamiltonian then one can predict evolution of systemby means of Schrodinger equation:

H |ψ(t)〉 = i~∂

∂t|ψ(t)〉 (2.27)

Quantum state may be either pure or mixed. Let us first introduce notion ofpure quantum state [8]. Pure state is a state that can be written as a ket vectorin Hilbert space. Let us consider Hilbert space which factorises into two subspaces:

9

H = H1⊗H2. Suppose |ai〉 is basis in H1, |bj〉 is basis in H2. The most generalpure state |ψ〉 ∈ H may be written as:

|ψ〉 =∑i,j

Ci,j(|ai〉 ⊗ |bj〉) =∑i,j

Ci,j |aibj〉 (2.28)

If one can rewrite state |ψ〉 as:

|ψ〉 =∑i,j

CAi |ai〉 ⊗ CB

j |bj〉 (2.29)

then we call this state separable(not entangled). This means that two parts of thisstate are uncorrelated. If the state can’t be written in aforementioned form then itis entangled.

In general the problem to say whether some pure state is separable or not isquite difficult. So called Schmidt decomposition is used to tackle this problem [9].

Let H1 and H2 be Hilbert spaces of dimensions n and m respectively. Sup-pose n ≥ m. Then for ∀w ∈ H1 ⊗ H2 there exist orthonormal sets u1, ...un ⊂H1, v1, ...vn ⊂ H2 such that:

w =m∑i=1

αiui ⊗ vi (2.30)

where αi are real, non-negative and uniquely determined by w.Not all states in Hilbert space may be represented as ket vectors. Suppose we

have system which is in state |ψi〉 with probability pi,∑pi = 1. We will call such

states mixed states. They may be represented by means of density matrix:

ρ =∑i

pi |ψi〉 〈ψi| (2.31)

where we imply that |ψi〉 is orthonormal basis.Suppose we have Hilbert space H = HA ⊗HB. |a〉 is basis in HA and |b〉 is

basis in HB correspondently. One may describe state of subsystem A(analogouslyof subsystem B) in the following way:

ρA ≡ Tr(ρ) =∑i

〈ai| ρ |ai〉 (2.32)

ρA is called reduced density matrix of subsystem A.Two mixed states may have the same density matrix. In this case it is impossible

to distinguish them. This may be seen from following consideration.Let us start from mixed state given by ρ(2.31). Suppose we have orthonormal

basis |βk〉. Then one may construct set of projectors Pj:

Pj = |βj〉 〈βj|

The probability to measure |βj〉 is given by:

Prj =∑i

pi 〈ψi| Pj |ψi〉 =∑i,k

pi 〈ψk| Pj |ψi〉 〈ψi|ψk〉 = Tr(Pjρ)

10

One of particular examples of mixed states is so called thermal state. In this caseprobability distribution pi is Boltzmann distribution:

pi = e−βEi , β =1

T

where T − temperature, Ei − energy of particular state |ψi〉.Unlike mixed states when system is in pure state(no matter entangled or sepa-

rable) it is in this state with probability 1. Thus pure state may also be representedby means of density matrix:

ρpure = |ψ〉 〈ψ|Measurement of a pure state will always yield results related to only one quan-

tum state, whereas with mixed state one cannot know beforehand what state willbe measured. We will always consider normalised states, so Tr(ρ) = 1. Densitymatrices gives us one more way to distinguish pure and mixed states[8]:

pure: ρ2 = ρ, Tr(ρ2) = 1

mixed: ρ2 6= ρ, Tr(ρ2) < 1

Tr(ρ2) is called purity of state.Note that closed system may be both in pure and mixed state. However if the

closes system was initially in pure state it can’t evolve to mixed state by meansof unitary process. To produce mixed state from pure one, one needs to introducesome temporary external interaction.

Mixed state can also be either separable or entangled. Mixed state is calledseparable [9] if it can be written as a convex combination of pure product states:

ρ =∑i

pi |ai〉 〈ai| ⊗ |bi〉 〈bi| =∑i

piρai ⊗ ρbi (2.33)

where |ai〉 and |bi〉 are(not necessary orthogonal) bases of Ha and Hb corre-spondingly. H = Ha ⊗Hb. Term ”convex” means 0 6 pi 6 1,

∑i

pi = 1.

Example of mixed separable state is:

ρ =1

2(|↑↑〉 〈↑↑|+ |↓↓〉 〈↓↓|)

If ρ is not separable then it is entangled.Now let us consider Hilbert space of dimension d with basis |ψi〉. One may

define mixed state which consists from mixture of states |ψi〉 all distributed with thesame probability pi = 1

d. We will call such state maximally mixed. Density matrix

of such state is given by:

ρmm =I

d(2.34)

There is also different concept of maximal entanglement. We will introduce thisconcept for pure states only. However the same thing may be done for mixed statesas well [10].

Suppose we have a bipartite pure state |ψ〉 ∈ HA ⊗ HB, dim(HA) 6 dim(HB).Then |ψAB〉 is maximally entangled iff[11]:

ρA =1

dim(HA)

11

It is said to be maximally entangled if reduced density matrices are maximallymixed.

Example of maximally entangled states is set of Bell states[12].Bell states are 4specific maximally entangled states of 2 qubits which form basis in two-qubit Hilbertspace:

∣∣φ+⟩

=1√2

(|0〉A ⊗ |0〉B + |1〉A ⊗ |1〉B)

∣∣φ−⟩ =1√2

(|0〉A ⊗ |0〉B − |1〉A ⊗ |1〉B) (2.35)

∣∣ψ+⟩

=1√2

(|0〉A ⊗ |1〉B + |1〉A ⊗ |0〉B)

∣∣ψ−⟩ =1√2

(|0〉A ⊗ |1〉B − |1〉A ⊗ |0〉B)

Another important quantity is entanglement entropy also known as von Neumannentropy. This quantity is a measure of entanglement between subsystem and rest ofthe system:

S(ρ) = −Tr(ρ log ρ) (2.36)

Since we can diagonalise density matrix(because it is unitary) we may rewrite theexpression in different way:

S(ρ) = −d∑

k=1

λk log λk (2.37)

where λk are nonnegative eigenvalues of ρ. Entropy gives yet another criterion todistinguish between pure and mixed states:

pure state: S(ρ) = 0

mixed state: S(ρ) > 0

If the quantum system is known to be in pure state, then entanglement entropy ofany subregion of this system is equal to entanglement entoropy of complementarypart of this subregion. However in the case when the whole system is in mixed statethese two entropies are not equal anymore.

12

Chapter 3

AdS/CFT

3.1 Introduction

Let us present short summary of basic aspects of AdS/CFT correspondence. Fol-lowing section is based on [13, 14, 25].

The origin of AdS/CFT correspondence lies in the fact that there exists dualitybetween description of some system in terms of closed and open strings. In particularMaldacena considered in his work[13] stack of D-3 branes[15]. At certain low energylimit the description of system in terms of closed strings is reduced to string theoryon AdS5×S5. At the same time description of initial system in terms of open stringscorresponds to some particular CFT on the boundary of aforementioned manifold,namely N=4 super Yang-Mills theory.

AdS/CFT correspondence states that two Hilbert spaces are equivalent to eachother[25]:

HCFT = HAdS-QGrav (3.1)

and all symmetries can be matched between two theories.Suppose we have field φ in bulk(string theory living of manifold M), which has

boundary value φ(0). Then one can define partition function in the bulk as:

Zstring[φ(0)] =

∫φ(0)

Dφexp(−S[φ]) (3.2)

AdS/CFT correspondence states that:

Zstring[φ(0)] = ZCFT[φ(0)] = 〈exp[

∫∂M

ddx√gφ(0)(x)O(x)]〉CFT (3.3)

where O(x) is some CFT operator on the boundary ∂M . Hence φ(0)(x) plays roleof source for correlation function of operator O(x). Now let us consider some limitof this correspondence, namely bulk side being in supergravity limit. In this caseat low energies Zstring[φ(0)] is mainly defined by saddle-point approximation. Henceone can write:

Zsugra[φ(0)] = exp(−S[φcl(φ(0))]) (3.4)

where φcl are fields satisfying low energy equations of motion with b.c.:φ(r, x)|r=0 = φ(0)(x)

13

Several examples of explicit calculations for AdS/CFT will be introduced in thisthesis. Notion of bulk reconstruction will be introduced in following sections andtwo-point functions on the boundary will be evaluated in appendix B.

3.2 Holographic Derivation of Entanglement En-

tropy from AdS/CFT

AdS/CFT correspondence is so far the best attempt to build precise theory of quan-tum gravity. It implies that any bulk observables should have interpretation in termsof boundary information. Possibly the most notable example of this principle is socalled Ryu-Takayanagi formula. It connects two concepts from seemingly differenttheories. It shows deep connection between quantum theory on the boundary andgravitational theory in the bulk. Although this formula was derived for arbitrarydimension case, we will focus of case of 3 bulk dimensions.

In this section RT proposal will be shown to hold for case of empty AdS. Inthe beginning we provide explicit calculation of length of geodesic matching someboundary region. After that we obtain expression for entropy of boundary regionA by means of so called ”Replica Trick”. Eventually equivalence of two expressionswill be shown. We base our derivations on [17].

Let us consider AdS3/CFT2. Anti de Sitter space is presented by metric

d2s = R2(− cosh ρ2dt2 + dρ2 + sinh ρ2dθ2) (3.5)

We will introduce cutoff at ρ = ρ0 because this metric is divergent at ρ→∞.

Figure 3.1: (a) AdS3, (b) slice of bulk at fixed t. The picture is taken from [17]

Let us consider subregion A of boundary at fixed t. It is defined by θ ∈ [0, 2πlL

]at fixed t.

Ryu-Takayanagi proposal states that entropy of boundary region A is given upto constant by length of minimal surface(geodesic in case of AdS3) bounding thisregion:

SA =Length(γA)

4G(3.6)

14

where G stands for gravitational constant in the bulk(AdS).Let us first evaluate length of γA. It is possible to do these calculations using

metric mentioned above, however it’s much easier to consider Poincare covering.This metric does not cover Anti de Sitter space-time completely. At the same timeminimal geodesic fully belongs to this region.That’s why we can use it for this prob-lem. The calculation will be done in similar manner as the one in section 2.2.Poincare covering metric of AdS3 looks like:

ds2 = R2dz2 − dt2 + dx2

z2(3.7)

where R is radius of AdS and which at t fixed will give an expression:

ds2 = R2dz2 + dx2

z2(3.8)

Expression for length of static geodesic will look like:

γA =

∫ds =

∫ √gzz(

dz

dλ)2

+ gxx(dx

dλ)2

dλ =

∫R√

(z)2 + 1

zdx (3.9)

Boundary conditions in new coordinates will look like:

Oi : (z = a(cutoff), x = − l2

)

Of : (z = a(cutoff), x =l

2)

(3.10)

It is useful to take into account some specific properties of integrand. Remem-bering that z = z(x) one can see that the whole function is invariant under trans-formations x → x + 4, 4 - const. We can define integrant to be some kind of”Lagrangian” in analogy with classical mechanics:

S =

∫L(q, q, t)dt; t→ x, q → z, q → z. (3.11)

The integral is invariant under transformation x → x + 4, hence one can definecorresponding ”Hamiltonian”:

H =∂L

∂qq − L =

R

z√

(z)2 + 1(3.12)

which doesn’t depend on x.Corresponding ”equations of motion” will look like:

d

dx(Hz) +

R2

Hz3= 0 (3.13)

which may be solved by following expression:

z = z(x) =

√2R

Hl

( l24− x2

)(3.14)

Now let us plug this expression into integral for geodesic length and introduce newcoordinates:

15

x =l

2cosα z =

l

2sinα (3.15)

where α ∈ (ε, π − ε), ε = 2al

[19].Finally expression for entanglement entropy is given by:

SA =Length(γA)

4G(3)N

=c

3log( la

); c =

3R

2G(3)N

(3.16)

Replica Trick

Now let us obtain explicit expression for entanglement entropy of boundary re-gion A. We will use only information from boundary itself. This will be done bymeans of so called Replica trick [18]. If the expression we obtain will be equal toexpression form eqn. (3.16), that will mean that Ryu-Takayanagi proposal holds(atleast in this particular setup).

Suppose we are given 1+1 lattice QFT at zero temperature living on a line. Letlattice spacing be equal to a(UV cut-off). For simplicity we will consider subregionof the system to be given by single interval A = [u1, v1]. This setup captures majoraspects of the approach. More complicated cases can be seen in [17].

General expression for entanglement entropy for subregion A looks like:

SA = −Tr(ρA log ρA)

where ρA is reduced matrix.This expression is difficult to evaluate explicitly because log will lead to infinite

series. One can rewrite expression of entropy in different way.Let us consider an expression:

d

daxa =

d

daealnx = x lnx

hence we can state:

− lima→1

(d

daxa) = −x lnx

This result may be used do redefine entanglement entropy in new way[18]:

SA = − limn→1

(∂

∂nTr(ρnA)

)(3.17)

Now instead of infinite series we need to calculate Tr(ρnA) for some particular n.This formula works in principle for arbitrary real n. Now let us briefly sketch orstrategy[18]:

· evaluate Tr(ρnA) only for integer n > 1 (This case has clear physical interpre-tation, which will be shown within calculations)· make analytical continuation of this expression for all complex n· recover SA as a limit n→ 1

16

To calculate Tr(ρnA) we will first define ρ and then will arrive to expression forρA. In our calculations we define region A to be some spacial region. That is whyit is preferable to write ρ in basis of states that have explicit spacial dependence.

Let us start by reminding ourselves how to define density matrix in terms ofwave functions(which is used generally in QM). After that we will follow similarprocedure for our case. It will help us to obtain expression for density matrix ofour QFT state in basis which has explicit spacial dependence and hence is easier tooperate with.

One can define amplitude for point particle to evolve from initial to final statein following way:

〈f |i〉 =

xf (t=tf )∫x−(t=ti)

dxe−SE (3.18)

If we set tf = +∞ in Euclidean signature |f〉 ≡ |0〉. It takes place because evolutionoperator in euclidean signature decays exponentially when euclidean time goes toinfinity.

So ground state wave function may be defined as:

ψ0(x) ≡ 〈0|i〉 (3.19)

Conjugated wave function of ground state can be defined in the same way:

ψ∗0(x) =

x+∫−∞

dxe−SE (3.20)

Now we can use above equations to obtain explicit expression for density matrix ofground state:

ρ(x−, x+) = ψ∗0(x)ψ0(x) (3.21)

As was mentioned above one can use aforementioned procedure to define densitymatrix for ground state of QFT. However in this case we will have field dependentwave functionals:

ψ0(ϕ) =

tE=+∞∫ϕ=ϕ+,(t=+0)

Dϕe−SE (3.22)

Hence one can write density matrix of ground state in this basis [18]:

ρ(ϕ+, ϕ−) ≡ ψ∗0(x)ψ0(x) = (3.23)

=

+∞∫−∞

Dϕe−SE∏x

δ(ϕ(x,+0)− ϕ+)δ(ϕ(x,−0)− ϕ−)

where we took into account that:

t=+∞∫t=+0

Dϕ

t=−0∫t=−∞

Dϕ ≡t=+∞∫t=−∞

Dϕ (3.24)

17

Once traced out complementary of region A we will obtain:

ρA(ϕ+, ϕ−) = (Z1)−1

+∞∫−∞

Dϕe−SE∏x∈A

δ(ϕ(x,+0)− ϕ+)δ(ϕ(x,−0)− ϕ−) (3.25)

where Z1 is vacuum partition function on R2 needed for normalisation of ρA.This can be represented in this way:

Figure 3.2: ρA. The picture is taken from [19]

For our problem we need to obtain expression for TrρnA, which can be representedgraphically by means of gluing n aforementioned surfaces in particular way:

ϕ+(i − 1) → ϕ−(i);ϕ+(i) → ϕ−(i + 1)... For the case n = 3 this Riemanniansurface will look like:

Figure 3.3: Riemannian surface R3,1 where 3 stands for number of copies, 1 - numberof intervals.The picture is taken from [20]

So Tr(ρnA) may be represented in terms of path integral on n-sheeted Riemannsurface Rn:

Tr(ρnA) = (Z1)−n∫

(tE ,x)∈Rn

Dφe−S(φ) ≡ Zn(Z1)n

(3.26)

At this point we have QFT living on some complicated Riemannian surface Rn,1.Problem of evaluation of Tr(ρnA) in general is not solvable. However if we considerCFT instead(and this is exactly what we are interested in) then it is possible to getexplicit expression for trace of reduced density matrix. First step is to map Rn,1 toRc by means of conformal transformations, which are given by[18]:

18

ω = x+ it→ ζ =ω − uω − υ

ζ → z = ζ1/n = (ω − uω − υ

)1/n

We can write down expression for transformation of T (ω)[18]:

T (ω) = (dz

dω)2T (z) +

c

12z, ω; z, ω =

(z′′′z′ − 32z′′2)

z′2(3.27)

where c is central charge of conformal filed theory on a complex plane.Stress-energy tensor of CFT on a complex plane C possesses rotational and

translational invariance hence:

〈T (z)〉c = 0

Taking this into account we can write:

〈T (ω)〉Rn,1 =c

12z, ω =

c(1− (1/n)2)(υ − u)2

24(ω − u)2(ω − υ)2(3.28)

Now follows very important moment of our discussion. We claim that CFT on sur-face Rn,1 is equivalent to CFT on n disjoint complex planes C, yet in the presence ofsome primary operators φn and φ−n living at points ui and υi correspondingly(i ∈1, ..n). One can also generalise above procedure for greater number of disjoint in-tervals. These operators are known as twist fields. Generally speaking these arelocal fields which contain in themselves information about nontrivial topology(inour case) of points ui and υi of surface Rn,1. There are n pairs of them exactlybecause Rn,1 contains n complex surfaces glued together. More detailed definitionof them is given in [18]. At this stage conformal dimensions of these operators areyet to be defined. To do so let us first write down conformal Ward identity [18]:

〈T (z)φn(u)φ−n(υ)〉c = |υ − u|−24n−24n (3.29)

Equivalence of two descriptions mentioned above implies:

〈T (ω)〉Rn,1 = n · 〈T (z)φn(u)φ−n(υ)〉c (3.30)

Both φn and φ−n have the same conformal dimension due to the properties of trans-formation of correlation function of 2 dimensional CFT under conformal transfor-mations. Taking into account 3.28 we may deduce that they are given by:

4n = 4n =c

12

(n− 1

n

)(3.31)

where c is central charge of CFT on a complex plane C and n is number of theseplanes in Rn,1.

It may be shown[18] based on properties of twist fields that Tr(ρnA) behaves(up toconstant) under scale and conformal transformations exactly as two-point functionof twist fields. Hence:

ZA(n)

Zn∝ TrρnA = cn((υ − u)/a)−(c/6)(n−1/n) (3.32)

19

where UV cut-off parameter a is needed to make expression 3.32 dimensionless as itshould be[17].

We can now use this result to obtain expression for entropy. Yet in our initialproblem we have CFT with spacial dimension x of a final size L.

One may use property that TrρnA is transformed as two-point correlation func-tion. In this case we need to use transformation ω → z = L

2πlogω which will

transform each sheet of Rn,1 into cylinder. We need to orient branch cut perpendic-ular to the axis of the cylinder. This will be the right way to get expression whichwill correspond to subsystem of length l = (υ1−u1) of system with length of spatialdimension L. [18]

TrρnA = cn

( Lπa

sinπl

L

)−c(n−1/n)/6

(3.33)

Once we plug this result into formula for SA we will get:

SA =c

3log( Lπa

sinπl

L

)+ c

′

1 (3.34)

We will take c′n ≡

log cn1−n , and c1 = 1 because of consistency condition. [18]

This formula in case l L will indeed give the result obtained previously:

SA =c

3log( la

)(3.35)

where c - is central charge, l is length of region A. Quantity ’a’ defines cut-off.

20

3.3 Bulk reconstruction

3.3.1 Global bulk reconstruction

Let us consider procedure of reconstruction of local bulk fields in AdS from CFT[21]. We will perturbatively construct operators in CFT which obey bulk equationsof motion and satisfy boundary conditions dictated by ”extended dictionary” [22]:

limr→∞

r4φ(r, x) = O(x) (3.36)

The case of free bulk fields will be considered. In particular the case where inter-actions of fields are suppressed by 1

Nwill be of interest. At leading order of 1

Nfor

bulk scalar field we will have:

(−m2)Φ(r, x) = 0 (3.37)

Solution of this equation may be found by means of mode expansion:

Φ(r, x) =

∫dkakΦk(r, x) + h.c. (3.38)

where ak, a†k - annihilation/creation operators and integration is taken over momen-

tum.This expression on the boundary will look like:

Φ(x) =

∫dkakϕk(x) + h.c. (3.39)

wherelimr→∞

r4φk(r, x) = ϕk(x)

If ϕk(x) are orthogonal, then we can obtain ak from above equation:

ak =

∫dxϕ∗k(x)Φ(x) (3.40)

Putting everything together we will obtain:

Φ(r, x) =

∫dk[ ∫

dxϕ∗k(x)Φ(x)]Φk(r, x) + h.c. (3.41)

In the case when integrations over k and x are exchangeable we may obtain:

Φ(r, x) =

∫dxK(r, x, x)Φ(x) (3.42)

where

K(r, x, x) =

∫dkϕ∗k(x)Φk(r, x) + h.c. (3.43)

We may rewrite this expression in way, more often used in literature [21]:

Φ(r, x) =

∫dY K(r, x;Y )O(Y ) +O(1/N) (3.44)

21

Where Y defines conformal boundary Sd−1 × R, with R being time direction.K(r, x;Y ) is known as ”smearing function”, which obeys bulk equations of motion.It may be chosen [21] to have support (depend only on) in the region, where x andY are space-like separated. In case of AdS3 this region will look like:

Figure 3.4: Green region defines area of dependence of K for bulk field in the pointx. Image from [22]

3.3.2 AdS-Rindler reconstruction

This type of reconstruction realises the assumption that if one wishes to reconstructcertain bulk field placed just near the boundary one in principle doesn’t have touse information from the whole boundary, but rather only from some small region.To define this kind of reconstruction, we need to introduce notion of bulk causalwedge[22].

Let us consider Cauchy surface which will denote as∑

and boundary subregionA: A ∈

∑There exist number of points on the boundary with the property that all possible

time-like and null-like curves crossing those points will intersect region A as well. Wewill call union of these points as boundary domain of dependence of A and denoteit as D[A].

Let us consider some boundary region R and all bulk time-like and null-likecurves which start on R. We will call union of them bulk causal future J +[R]. Inthe same way one may define bulk causal past.

Now one can define causal wedge of CFT subregion A:

Wc[A] = J +[D[A]] ∩ J −[D[A]]

We will also introduce for future use so called causal surface of A, denoted by χA.It is defined as part of the intersection of boundaries of J +[D[A]], which does notintersect the conformal boundary on infinity.

22

Figure 3.5: AdS-Rindler wedge for AdS3 spacetime. Image from [22]

Once taken into account AdS-Rindler reconstruction claims [22] that ∀ bulk fieldsΦ(x, r) ∈ Wc[A] may be reconstructed by means of boundary operators O(Y ) ∈ D[A]

Φ(r, x) =

∫dY K(r, x;Y )O(Y ) +O(1/N) (3.45)

It is important to note that in this case K doesn’t exist as a function and should beconsidered as a distribution for integration against CFT expectation values [22].

3.3.3 Paradoxes of AdS-Rindler reconstruction

According to aforementioned procedure when one considers bulk field which is placednearer to the boundary, then one needs to use smaller boundary region for recon-struction. If we follow this logic naively we will run into several paradoxes.

For instance, suppose we have some bulk field φ(x) and a point Y on the bound-ary. One may pick boundary for reconstruction of φ(x) to be

∑/Y . In this case

bulk operator is mapped to boundary O(φ(x)) which has to commute with any local

operator O(Y ). Choice of Y is arbitrary. If we admit that φ(x) is always mapped tothe same boundary operator then we obtain that O(φ(x)) commutes with all localoperators on the boundary and hence by Schur’s lemma must be proportional tounity operator.

Le’t consider another example. Suppose we have bulk operator φ(x) which liesboth in causal wedge of boundary region A and B. In this case this operator maybe reconstructed from both Wc[A] and Wc[B]. But if we claim that correspondingboundary operator O(φ(x)) is unique then we must demand O(φ(x)) to have supportonly in Wc[A]

⋂Wc[B]. But we can have case as described in following image, when

x is located outside Wc[A]⋂Wc[B].

Finally we may consider setup of right part of image. In this case point x liesoutside Wc[A],Wc[B],Wc[C]. This means that φ(x) can be reconstructed from eitherA⋃B or B

⋃C or A

⋃C. Hence O(φ(x)) should have support only on 3 points of

the boundary. Suppose we now have 3 other regions of the same size but slightly

23

rotates with respect to initial ones. Then we will have to arrive to the conclusionthat φ(x) can’t be reconstructed at all.

All these paradoxes appear because we demand O(φ(x)) to be unique operatoron the boundary for particular φ(x). We may resolve them by claiming that theremay exist different ways to reconstruct the same bulk operator. This idea may beexplicitly realised by means of so called quantum error correction code.

Figure 3.6: Setups of three aforementioned paradoxes taking place for Ads-Rindlerreconstruction in AdS3 . Image from [22]

3.3.4 AdS/CFT as quantum error correction code

In this section we will consider the simplest example of quantum error correctioncode (QECC) and then we will point out its connection with AdS/CFT. More de-tailed analysis may be found in [22].

Suppose we have 1 qutrit(system that has 3 dimensional Hilbert space) living inthe bulk(which will correspond to logical Hilbert space) and 3 qutrits living on theboundary(and forming 27 dimensional physical Hilbert space).

Figure 3.7: Setup: 3 physical qutrits on the boundary, one logical qutrit in the bulk.

Now let us consider subspace(so called code subspace) of Hphysical of dimension3 formed by vectors |ψ〉. We may define encoding operator E:

Enc: Hlogical → Hphysical (3.46)

24

|0〉 → |0〉 = |000〉+ |111〉+ |222〉

|1〉 → |1〉 = |012〉+ |120〉+ |201〉 (3.47)

|2〉 → |2〉 = |021〉+ |102〉+ |210〉

Encoding map can be explicitly defined:

E =∑j

|j〉 〈j| (3.48)

This map is an isometry:E†E = Ilogical

In the case of interest states from code subspace satisfy important restriction. Anycollection of qutrits from code subspace either contain all information about logicalstate |ψ〉 or no information about it at all. In our setup it means(analogous conditionmay be written for any other subsystem):

TrAB

[ ∣∣∣ψ⟩⟨ψ∣∣∣ ] =1

3I (3.49)

In particular this condition means that for any state∣∣∣ψ⟩ reduced density matrix of

any of qutrits will be maximally mixed. Hence one can’t obtain information aboutthe state from any single qutrit. Now let us summarise scheme of QECC:

Enc: Hlogical → Hphysical

Noise: Hphysical → Hphysical

Decoder: Hphysical → Hlogical

Code can reconstruct logical state is following condition is satisfied:

Dec Noise Enc(|ψ〉 〈ψ|) = |ψ〉 〈ψ| (3.50)

Operator of noise may be in principle different. We are interested in case oferasure of one of qutrits(for instance C):

Noisec(ρ) = trc(ρ),where ρ = Enc(|ψ〉 〈ψ|)

Decoding operator which acts only on non-erased physical qudits:

Decc(ρ) = trA(UABρU†AB)

where for case of interest:UAB |a, b〉 = |a, b− a〉

Let us consider simple example:|ψ〉 = |1〉

Enc(|1〉 〈1|) = |1〉〈1| = (|012〉+ |120〉+ |201〉)(〈012|+ 〈120|+ 〈201|)

25

Noise(ρ) = (|01〉+ |12〉+ |20〉)(〈01|+ 〈12|+ 〈20|)Dec(Noise(ρ)) = K(|1〉+ |1〉+ |1〉)(〈1|+ 〈1|+ 〈1|)

K is just normalisation constant. We also took into account that (0− 2)mod3 = 1.We may immediately see similarity between aforementioned set up and 3d para-

dox of AdS-Rindler reconstruction. For instance in this case operators with supporton A

⋃B correspond to Decoder operator. Operator in the bulk corresponds to

logical qutrit. General idea is that this operator may be protected against erasureof one of boundary qutrits.

This simple example illustrates why it is natural to connect AdS/CFT andQECC. Depending on choice of logical qudits(systems with dim(H) = d) in the bulkwe obtain different corresponding code subspaces on the boundary. So AdS/CFTmay be viewed as many QECCs at once[22].

The fact of AdS-Rindler reconstruction that the same operator may be recon-structed by means of different causal wedges corresponds to freedom to choose dif-ferent code subspaces. Depending on position of this bulk operator it may be eitherwell protected against different kinds of erasures of boundary information or not.Now let us make a few final comments comments about this topic.

Figure 3.8: Operator in the centre(far from the boundary) is protected againsterasure of any of green regions. Operator near the boundary will be lost aftererasure of small red region. Image from [22]

Bulk reconstruction(both global and Rindler) relies on 1N

order approximation.The fundamental limitation on them appears when we have to take into accountbackreaction[22].

Procedure of definition of code subspace Hc as stated in [22] looks like this. Oneshould consider local bulk operators φi(x) which acting on bulk vacuum state(inregion of order of lAdS) produce certain set(with some subtleties[22] these statesmay define certain subspace of bulk Hilbert space):

|Ω〉 , φi(x) |Ω〉 , φi(x1)φj(x2) |Ω〉 , ...

These states will correspond to boundary code subspace Hc. One would like to knowhow large this code subspace can be in principal. To answer this question we needto realise that each φi(x) acting on the bulk state arises its energy. At some pointone will have to take into account backreaction. In this case we will not be ableto use AdS-Rindler reconstruction which relies on perturbation theory approxima-tion around a fixed background. As a result one can state that interpretation ofAdS/CFT as QECC also fails at high enough energies.

26

Chapter 4

Tensor Networks and AdS/CFT

4.1 Introduction

Tensor networks theory is relatively young and dynamically developing topic. Ithas many implications in areas like condensed matter and others. Within past fewyears however it was also realised that one can use TNs as toy models of AdS/CFTcorrespondence. The most advanced of them already capture a lot of non trivialaspects of holography. For instance certain types of TN allow representation oflogical operators on multiple boundary regions. This mimics so called AdS-Rindlerreconstruction. Quantum error-correcting properties may also be explicitly realisedin this set-up.

Although it is believed by many that TN may become powerful tool for betterunderstanding AdS/CFT correspondence there are still plenty open questions whichare needed to be solved. One of the most notable among them is the fact that sofar no TN can describe dynamical processes. It is only developed in a way to mapstates between boundary and bulk Hilbert spaces. Suppose one takes boundaryHamiltonian and maps it to bulk. Usually one will obtain highly nonlocal(in termsof bulk degrees of freedom) Hb. In low energy limit it is reduced to local Hamiltonianonly for very specific boundary Hamiltonians. It is not yet known how to find them.

We introduced this example to show that a lot of work remains to be done beforewe can truly say that TN approach realizes most properties of holography. Howeverwe will show in following chapters that question of holographic shadows can alreadybe addressed by formalism of tensor networks.

4.2 MERA and Geometry.

Tensor networks like multi-scale entanglement renormalization ansatz (MERA) wereinitially introduced as an advanced numerical approach to study strongly entangled

27

quantum many-body systems. However it was later realised by Swingle [26] thatMERA has some properties similar to AdS/CFT.

Since then a lot of work[27, 28] was done to modify MERA in such a way soit could capture essential points of AdS/CFT correspondence. Although plenty ofprogress has been done in this field, MERA is not of a much interest for purposesof current work because it doesn’t possess sub-Ads scale resolution. This propertyis essential for us because the largest size of holographic shadow is of order of lAdS.Yet we believe that this section plays its role from pedagogical point of view.

Let us first say few words about MERA in general.

As we already said MERA was initially used for variational estimation of theground state of the critical(gapless) systems in D dimensions. In particular we willconsider 1+1 dim case.

It is possible to view MERA in 2 different ways. One of them is so called

Figure 4.1: a) Example of MERA 1+1(k=2) b) Disentanglers which are used todecrease amount of entanglement from layer to layer c) Isometries which are usedfor coarse graining procedure. Picture from [29].

Down to up:

One starts from ”boundary state” and renormalises it from layer to layer bymeans of coarse-graining procedure. It is explicitly defined by so called rescalingfactor k (k=2 in our case) which tells how many sites will be coarse grained perblock. Once going from certain layer to upper one we obtain renormalised state inthe Hilbert space of smaller dimension. This state will still have some resemblanceto the previous one. However because of removed entanglement the new one is muchsimpler to deal with.

At the same time we can view this procedure in opposite direction. It may beshown[29] that one can use MERA to build an approximation of the ground stateof CFT on the boundary from simple initial input(upper layer state).

28

Now we would like to introduce concept of ”geometry” for MERA TN. At thispoint one should understand it as set of all possible cuts through TN. These cutswill play role of different curves in some real geometry. Certain ”length” will beassociated to each cut, depending on how many crossed lines it contains. As wasshown by Swinger minimal cuts in MERA behave like geodesics in empty AdS. Ourgoal will be to show that there is one in one correspondence between ”geometry”in MERA and geometry of AdS. To do so we will introduce two length scales ingiven tensor network. Then by matching ”lengths” of certain cuts with lengths ofcorresponding curves in AdS we will find explicit expressions for these two scalesin terms of radius of AdS. It will be shown that distance between nearest sites inMERA can’t be smaller than lAdS and no change of coordinates can fix it. Ourderivations are mainly based on [29].

Figure 4.2: Left image shows MERA TN(where disentanglers and isometries aresuppressed for convenience), right one shows time slice of Poincare AdS coordinates.Pictures are taken from [29].

Constant time slice of Poincare metric is given by:

ds2 =(dx2 + dz2)

z2L2AdS (4.1)

Lengths of curves are given by:

|γ1| = LAdS

∫ x0

0

√( dzdx

)2 + 1

zdx =

x0

z0

LAdS (4.2)

|γ2| = 2LAdS log(x0

a

)Now let us find lengths of appropriate curves in MERA:

We define z0 = kma (Can be seen from picture). Number of bonds between twosites is given by x0

kma

|γ1MERA| = L1(number of bonds) = L1x0

z0

|z0=kma (4.3)

|γ1MERA| = |γ1AdS|

29

LAdS = L1

Now let us find length of second curve:

|γ2MERA| = 2L2(number of vertical bonds) (4.4)

In this case number of vertical bonds between zero and m-th level is equal to m:

x0 = z0 = kma

m = logk

(z0

a

)= logk

(x0

a

)(4.5)

|γ2MERA| = |γ2AdS|

L2 = LAdS log k

4.2.1 Limitations on sub-AdS scale resolution of MERA.

To use tensor network as a probe of bulk physics we would like it to have sub-AdSscale resolution. It means that distance between nearest nodes should be smallerthan LAdS. Using arguments from [29] we will show that no allowed change ofcoordinates can lead to sub-AdS scale resolution.

In previous section we considered xMERA, zMERA to be trivially related to xAdS, zAdS.Suppose now that:

xMERA = f(xAdS)

zMERA = g(zAdS)

In new coordinates:f(a) - UV-most lattice spacing(in x direction)g(a) - UV cutoff in the holographic(z) direction.

Now let us compare γ1 in two coordinate systems. Because of transformationwe’ll get:

xMERA0 = f(xAdS0 )

zAdS0 = kma→ g(zAdS0 ) = kmg(a) (4.6)

km =g(zAdS0 )

g(a)

Number of nodes:x0

kma→ f(xAdS0 )

kmf(a)

30

|γ1AdS| =LAdSx

AdS0

zAdS0

|γ1MERA| = L1(number of nodes) = L1xAdS0

f(a)

g(a)

g(zAdS0 )=xAdS0 LAdSzAdS0

g(zAdS0 )∂γ1

∂xAdS0

= L1f′(xAdS0 )

g(a)

f(a)= LAdS

g(zAdS0 )

zAdS0

(4.7)

The middle expression depends only on x0 and the right one on z0. The factthat we can vary these 2 parameters independently and that these expressions areequal to each other gives restrictions on f(x) and g(x). We can only allow:

f(x) = εxx and g(z) = εzz where εx and εz are some constants.Now we can get:

|γ1MERA| = L1εxx

AdS0

εxa

εza

εzzAdS0

= LAdSxAdS0

zAdS0

(4.8)

L1 = LAdS

Now let us consider case γ2:

Number of bonds at zero level:xMERA0

f(a)

We can calculate number of bonds in geodesic γ2:

2 logk(xMERA0 /εxa)

γ2AdS = 2 log(xAdS0

a

)= 2LAdS

(xMERA0

εxa

)(4.9)

Hence L2 = L log k

So it was shown that no change of coordinates is able to give possibility to describesub-AdS scale. We shall consider different tensor network for that.

4.3 Isometric Tensors

We have shown in previous section that MERA TN(at least at level it is currently de-veloped) doesn’t have sub-AdS scale resolution. Before we introduce different kindsof tensor networks which may resolve this problem we need to introduce buildingblocks which will be used in future investigations. This review is mainly based onarticles [30] and [31].

Let us introduce concepts of isometry and perfect tensors:Definition: Map T is called isometry between Hilbert spaces HA and HB,

T : HA → HB if it preserves the inner product.

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There are some useful properties of isometry:

1. dim(A) ≤ dim(B).

Simple example: Isometry preserves angles between vectors. If one wishes tofind isometry from 3d vector space to 2d one, he will face the fact that thereare no 3 vectors(apart from ~0) in 2d vector space orthogonal to each other.

2. Unitary transformation is particular case of isometry when dim(A) = dim(B).

3. If T isometry from HA to HB then:

(a) T †T = 1HA

(b) TT † - is projector from Hilbert space HB to subspace of dim(HA)

Suppose we have isometric tensor T. If some operator O is acting on its incominglegs one can ”push it through T” by means of following procedure:

TO = TOT †T = (TOT †)T ≡ O′T

Figure 4.3: Process of pushing operator O through isometric tensor T. Picture from[31]

4.4 Perfect tensors and RT formula.

Tensor networks formed by perfect tensors share many important properties withAdS/CFT correspondence. We will see in next sections that they do not possesssub-AdS scale resolution. However they may be generalised to other types of tensornetworks which will have this property. That is why it is important to understandexplicit construction of this type of TN.

Let us consider some tensor[31] Ta1...a2n . Its indices(legs) form Hilbert space ofdim D2n, where D is dim of each leg. We may split this Hilbert space in to twofractions, one of which containing |A| and other |Ac|.

Definition: We will call 2n legged tensor Ta1...a2n perfect if for any bipartition|A| 6 |Ac| the obtained tensor is proportional to isometric map from |A| to |Ac|.

Important property of this kind of tensors is that density matrix, formed fromany n legs of such tensor is maximally mixed.

Ryu-Takayanagi formula.Based on [31] we will derive RT formula for tensor network, built from identical

perfect tensors. This tensor network(TN) represents holographic state:

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|ψ〉 =∑a,b,i

PaiQbi |a〉 |b〉 ≡∑i

|Pi〉A ⊗ |Qi〉Ac a ∈ A, b ∈ Ac (4.10)

Note: In general sets |Pi〉 and |Qi〉 don’t have to be normalised or orthogo-nal.

Figure 4.4: Tensor network, split into two subregions. Picture from [31]

We want to get restriction on amount of entropy contained in the boundaryregion A. To do so we should obtain reduced density matrix of this region first:

ρA = trAc(|ψ〉 〈ψ|) =∑ij

〈Qj|Qi〉 |Pi〉 〈Pj| (4.11)

Its rank is defined by number of terms in the sum. Each leg i has dim ν (bonddim). Number of legs is defined by number of cuts |c| . Hence, rank of ρA is less orequal to ν |c|.

Maximal amount of entropy corresponds to maximally mixed density matrix. Itcan be proven that for such matrix S = logN , where N - rank of density matrix.

Taking into account this statement and the fact that rank of ρA can’t be greaterthan νc, one can see that the best bound on entropy of region A is obtained if weconsider cut c be minimal one(which will be denoted as γA):

SA ≤ |γA| log ν (4.12)

Note: It is worth to mention that we do not specify particular tensors at each site.We only claim that all of them are perfect and that each of them has the sameamount of legs. So far we haven’t specified properties of P and Q. Let us considerthem to be isometries from i to a and b correspondently.

It may be shown(taking into account one of properties of isometry) that in thiscase |Pi〉 and |Qi〉 are orthonormal sets.

From this one can see that in this case ρA is indeed maximally mixed densitymatrix. Hence RT formula holds exactly:

SA = |γA| log ν. (4.13)

So we have arrived to conclusion that in aforementioned setup entropy of bound-ary region is defined, up to constant, by ”length” of minimal cut bounding thisregion. This indeed resembles RT formula for AdS/CFT correspondence. It is im-portant to notice that 4.13 is proven only for single boundary regions. Discussionabout entropy of several disconnected intervals may be found in [31].

33

At last the theorem, which states when Pai is an isometry from |i〉 to |a〉 andQbi is isometry from |i〉 to |b〉 correspondently will be provided without prove:

Theorem: [31] Suppose that we have holographic state associated to a simply-connected planar tensor network of perfect tensors, whose graph has ”non-positivecurvature”. Then for any connected region A on the boundary, we have SA =|γA| log ν; in other words, the lattice RT formula holds.

Term ”simply-connected” means that tensor network doesn’t contain any graphloops or multiple edges. ”Non-positive curvature” is quite difficult to define ingeneral but what we really need is to ensure that distance functional from one siteof TN to another one doesn’t have interior local maximum.

4.5 Absence of sub-AdS scale resolution for TN

formed by perfect tensors.

Let us show why this particular type of TN can’t be used for purposes of our work.In particular we will show that distance between nearest sites can’t be smaller thanlAdS or equivalently that one should associate area of order l2AdS to each tensor inTN.

We will consider one of setups introduced in [31] and then argue that similararguments may be extended to different setups.

First let us make several assumptions. As was stated in previous chapters con-stant time slice of AdS3 is given by Poincare disc. It is defined by constant scalarcurvature:

R =2

−l2AdS(4.14)

Note: In Riemannian geometry the scalar curvature(Ricci scalar) is a certain cur-vature invariant of Riemannian manifold. This quantity assigns a real number toeach point of manifold. This number represents the amount by which the volumeof geodesic ball in curved Riemannian manifold deviates from that of the standardball in Euclidean space. In case of interest (2dim manifold) this value completelycharacterises the curvature of surface.

We will also need to introduce so called Gaussian curvature which is for ourpurposes may be viewed as some intrinsic measure of curvature which is case of 2dmanifold reduces to R/2. More deep introduction may be found here[32]. Very goodresource about hyperbolic geometry [33].

Now let us consider uniform tessellation of Poincare disc. Tessellation meansthat we take certain types of polygons(which in general may differ from each other)and fill our spacetime by them. In case of uniform tessellation all polygons are takento be identical.

34

First tessellation of interest is 5, 4. Here 5 stands for number of sides ofpolygon(pentagon in this case) and 4 stands for number of polygons meeting ateach vertex.

Now our goal is to evaluate area associated to each polygon. This may be doneby means of Gauss-Bonnet theorem, which applied to some region R states that:∫

R

KdA+

∫∂R

kgds = 2πχ(R) (4.15)

where K - is Gaussian curvature, kg is the geodesic curvature and χ(R) is the Eulercharacteristic of the residual region.

This theorem connects local notion of geometry of certain manifold with it’stopology.

In case of interest for certain polygon with m sides(remember that each side ispart of some spacial geodesic) this formula may be used to obtain area of polygon:

Am = a(m− 2)π − (a(1) + a(2)...+ a(m)) 1

−K(4.16)

where a(i) stands for inner angle of polygon(see image).

Figure 4.5: Uniform tessellation of hyperbolic plane 5,4. Image from internet

Let us consider pentagon tessellation 5, 4. In this case 4 polygons meet togetherin each vertex. Sum of corresponding angles is equal to 2π. Hence a(i) = π

2. Each

polygon is pentagon, so m=5. Once we plug everything to 4.16 we obtain:

A5 =π

2l2AdS (4.17)

We can vary tessellation in 2 different ways. First is to change number of sides ofpolygons, or change number of polygons meeting at each vertex. However it maybe shown by explicit calculation that in any setup area associated to polygon willbe of order of l2AdS. One possible way to obtain sub-AdS scale resolution will be tointroduce nonuniform tessellation. Even though this approach will not lead to anyimmediate problems(RT formula will still hold) if we use this new tessellation as wasdone in the article to build holographic code, then we will run into contradictions.

Let us sketch brief argument which explains why nonuniform tessellation leadsto problems and can’t be used to obtain TN with sub-AdS scale resolution. Fol-lowing logic of [31] we build tensor networks by means of putting one tensor perpolygon(pentagon) and connecting them together. Note that we use perfect tensors

35

defined above, all legs have the same dimensionality. This may be done at least intwo different ways. One way will be to pick tensors with number of legs equal tonumber of sides of corresponding polygon. In this case uncontracted legs will beonly those on the boundary of tensor network. In this way we obtain TN describingsingle boundary state. Second way will be to pick tensors which have one more legthen number of sides of corresponding polygon. As a result we will get tensor net-work which realises map from set of boundary legs to rest ones, which will be calledbulk legs. These two sets define 2 Hilbert spaces. We want obtained TN to mimicAdS/CFT correspondence, hence we can immediately conclude that dimensionalityof bulk Hilbert space can’t be greater than that of boundary Hilbert space. Butif we want to have sub-AdS scale resolution in this TN we will have to increasenumber of polygons, hence increase total number of bulk legs. It may be shown thatalmost instantly one will obtain bulk Hilbert space larger than that of boundary. Asa consequence any similarity between this TN and AdS/CFT correspondence willbe lost[31].

Figure 4.6: Left picture: Holographic code build on top of tessellation 5, 4 [31],red dots correspond to bulk legs. Right picture: Example of nonuniform tessellationwith sub-AdS scale resolution[30].

Possible way to resolve this contradiction would be to modify properties of build-ing blocks - tensors. This leads us to another type of tensor networks.

4.6 Tensor network formed by pluperfect tensors.

4.6.1 Definition of pluperfect tensor.

As was mentioned in previous section TN formed by perfect tensors can’t be used todescribe sub-AdS scale resolution because nonuniform tessellation will cause rapid

36

growth of dimension of bulk Hilbert space. Once it becomes larger than boundaryone we loose similarity between given TN and AdS/CFT. However we are goingto show that if we change properties of tensors we use, then it is possible to avoidthis problem. In case of tensor network formed by pluperfect tensors we start frombulk Hilbert space, which dimension is much larger than those of boundary. Yetit so happens that because of tuned properties of pluperfect tensors one can definephysical subspace(defined by so called gauge invariance property) in the bulk Hilbertspace. We will show that one can define isometry from this subspace to the boundaryHilbert space. This crucial property will give us sub-AdS scale resolution for thefirst time. However it will be shown that RT formula can be proven only for limitednumber of bulk states, namely for direct product states. That will be a problem,because bulk state corresponding to presence of BH is implied to be highly entangled.This will force us to move to yet another generalisation of this TN.

In this section notion of pluperfect tensor will be introduced. Then tensor net-work will built from them and some properties of this network will be presented.This section summarises results introduced in article [30].

In general pluperfect tensor should have odd number of indices. Without loss ofgenerality one may introduce all definitions for particular case 2n + 1 = 5. In thiscase pluperfect tensor will look like T Iαβγδ. In-plane indices α, β, γ, δ ∈ 1,2,...D.

Index I (logical degree of freedom) ∈ 1, 2...D4.

Pluperfect conditions.1. T realises unitary map from indices α, β, γ, δ to index I.

T IαβγδTJ∗αβγδ = δIJ (4.18)

2. There exists subset of indices of dim D2 among indices I, for which, for fixedI tensor T Iαβγδ is a perfect tensor. So, for fixed I ∈ 1, 2...D2 T Iαβγδ defines unitarytransformation from any two to other two inplane indices.

T IαβγδTI∗µνγδ =

1

D2δαµδβν , I ∈ 1, 2...D2 (4.19)

3. Let us consider I fixed ,I ∈ 1, 2...D2. For any inplane index α, T Iαβγδ is aunitary mapping from Iα to βγδ, up to normalisation.

T IαβγδTJ∗µβγδ =

1

DδIJδαµ I, J ∈ 1, 2...D2 (4.20)

Figure 4.7: Graphical representation of pluperfect tensor properties. [30]

37

4.6.2 Gauge invariance.

Let us consider pluperfect tensors T Iα1α2α3α4. We can form tensor network from them

by contracting in-plane legs. One may in principle obtain different tensor networkswith different geometries. Let us call V - number of bulk legs(or equivalently numberof sites), N - number of pairs of contracted legs, P - number of boundary legs.

Figure 4.8: . Examples of tensor networks with different geometries. Red arrowsrepresent bulk legs. [30]

Tensor network formed by pluperfect tensors realises map from boundary Hilbertspace H∂ to the bulk space HA:

M : H∂ → HA (4.21)

By construction one can see that M is an isometry. Hence:

dim(H∂) ≤ dim(HA) (4.22)

M †M = 1∂

It is important to notice that operator MM † is projector and defines subspace inHA of dimension H∂ . This subspace is also image of map from H∂ to HA. We willdenote it as HB. This subspace will have interpretation of physical bulk Hilbertspace.

It is formed by states satisfying condition:

MM † |ψ〉 = |ψ〉 (4.23)

One can see that this condition is always satisfied by states |ψ〉 = M |ϕ〉, where |ϕ〉is some boundary state.

We may rewrite 4.23 in local form by means of introduction of so called gaugeinvariance.

Let us consider contraction of two neighbour tensors in tensor network:

T Ixα1α2α3µT Jyα1α2α3µ

= T Ixα1α2α3νgνρg

∗ρµT

Jyα1α2α3µ

Where gνρ ∈ SU(D), gνρg∗ρµ = δνµ. Letters x, y define which site particular tensor

belongs to.

38

Operators gxiyj once ”pushed through” appropriate tensor(see section 4.3) Txi atsite xi define operator acting on bulk leg:

Wxi = Txi(gxiy1 ⊗ gxiy2 ⊗ gxiy3 ⊗ gxiy4)T †xi (4.24)

If we have N contracted lines in the bulk it will define SU(D)N gauge group.The physical bulk subspace definition may be written as:

Wx(gxy1 , gxy2 , gxy3 , gxy4) |ψ〉 = |ψ〉 , for ∀ gxy (4.25)

where yi are neighbour sites of x-site.Note: One can see using aforementioned definitions that state |ψ〉 = M |ϕ〉 will

indeed satisfy above expression.It is not obvious why 4.23 and 4.25 define the same condition. In next section

we will show that they are actually interchangeable.

4.6.3 Equivalence Of Two Definitions.

In this section we will show that two expressions for gauge invariance are indeedequivalent.

As a first step let us consider tensors T Iα1α2α3α4and T Jβ1β2β3β4 and define action

of gauge group on them.We will consider tn consisting from two sites, howevergeneralisation to greater number of sites is trivial. For that we need to find the mostgeneral state, which is invariant under action of operator W = T (g ⊗ g ⊗ g ⊗ g)T †.

Let us write arbitrary state in bulk Hilbert space:

GIJ |ψI〉 |ψJ〉 = T Iα1α2α3α4T Jβ1β2β3β4 |α1〉 |α2〉 |α3〉 |α4〉 |β1〉 |β2〉 |β3〉 |β4〉 ≡ (4.26)

≡(...)∑

α1β1(...)

|α1〉 |α2〉 |α3〉 |α4〉 |β1〉 |β2〉 |β3〉 |β4〉

So far we’ve just introduced change of basis. We may rewrite action of W as actionof operators g on in-plane indices by means of ”pushing through” procedure. Nowthe condition will look like:

(...)∑α1β1(...)

|α1〉 |α2〉 |α3〉 |α4〉 |β1〉 |β2〉 |β3〉 |β4〉 = (4.27)

= gα1α1g∗β1β1

(...)∑α1β1(...)

|α1〉 |α2〉 |α3〉 |α4〉 |β1〉 |β2〉 |β3〉 |β4〉

This is not always true for ∀g ∈ SU(D). We need to impose constraint on∑(...)

α1β1(...)

39

Gauge invariance condition:

gα1α1g∗β1β1

(...)∑α1β1(...)

=

(...)∑α1β1(...)

(4.28)

which should be true for ∀g ∈ SU(D). Now based on Schur’s lemma we may claimthat:

(...)∑α1β1(...)

= F(...)(...) δα1β1 (4.29)

This means that initial bulk state is restricted to be exactly the one, that we wouldhave in tensor network:

GIJ |ψI〉 |ψJ〉 = T Iα1α2α3α4T Jα1β2β3β4

|α1〉 |α2〉 |α3〉 |α4〉 |β1〉 |β2〉 |β3〉 |β4〉 (4.30)

Now let us consider opposite direction. For simplicity we will work with tensornetwork formed by two sites x and y. However we can easily generalise argumentfrom bigger amount of sites.

One may introduce action of arbitrary operator SU(D) matrix on these two sitesin following way:

T IαβγδTJαµντ = gαπT

Iπβγδg

∗αεT

Jεµντ (4.31)

These operators may be ”pushed through” appropriate tensors and this will lead todifferent bulk states, depending on particular choice of g. However all these stateswill be mapped to the same boundary state |ϕ〉. If one has N internal lines thenaction of SU(D)N gauge group may be defined.

Physical state condition then may indeed be rewritten as:

Wx(gxy1 , gxy2 , gxy3 , gxy4) |ψ〉 = |ψ〉 , for ∀ gxy (4.32)

4.6.4 Absence of local operators

Let us consider operator O acting nontrivially on any internal site of tensor network.It may be mapped to operator M †OM acting on the boundary. We will prove thatthis boundary operator is trivial( ∝ I)

One may define operator gxa acting on any of boundary legs. It may be ”pushedthrough” tensor T Ixα1α2α3α4

to become Wx = Tx(1⊗ 1⊗ 1⊗ g)T †x .

40

Because of construction one can claim that:

[O,W ] = 0 (4.33)

where W = Ix1 ...Wx...IxVIt may be shown that in this particular setup Wx = Mx(1⊗ 1⊗ 1⊗ g)M †

x Oncewe plug this into (2.6) we may obtain:

[M †OM, gxa] = 0 (4.34)

This expression holds for ∀ gxa for all a.The only operator which commutes withall operators acting in H∂ must be (Schur’s lemma) proportional to I∂. Abovearguments may be extended for bulk operators acting nontrivially on any number ofinterior sites of TN. If we want to find operator which mapped to the boundary willnot be proportional to identity operator, we need to take bulk operator which hasnontrivial action on bulk site(s) with uncontracted inplane legs. All these operatorswill be explicitly nonlocal in terms of bulk degrees of freedom.

4.6.5 Classical geometry states

Let us consider bulk direct product state:∏x

|nx〉 where nx = 1, 2, ...D2 (4.35)

We will call such states classical geometry states. Taking into account secondproperty of pluperfect tensors one can see that T Iαβγρ is perfect tensor for fixed

I = 1, 2, ...D2. Hence corresponding boundary state M †∏x |nx〉 represents tensor

network formed by perfect tensors. Hence RT formula for single boundary regionmay be obtained in the same way as in section 3.4.

However it will be useful to prove it in slightly different way. Property 2 ofpluperfect tensor defines unitary map from any 2 in-plane indices to rest two ofthem for fixed I = 1, 2, ...D2. We can represent it as arrows drawing across tensornetwork. One may order them to obtain unitary map from set of input legs to setof output legs. However there are some restrictions on way to draw these arrows.For instance close loops are forbidden.

Now to derive RT formula let us consider single connected boundary regionA(A = A1

⋃A2) and geodesics γA. Suppose there exists such arrow drawing that

we have unitary map from γA⋃A1 to A2: It may be shown that in this case one may

also define isometry from γA to A. If it is also true for A as well then we arrive to thesame situation as in section 3.4. Hence one may conclude that SA = SA = |γA| logD.Such arrow drawing always exists [31].

41

Figure 4.9: Example of forbidden way of arrow drawing, which contains closedloop.[30]

Figure 4.10: Arrow drawing which defines unitary map from γA⋃A1 to A2 .[30]

Although it must be mentioned that arrow drawing fails to prove RT formulafor multiples disjoint intervals in certain cases [30].

4.6.6 Emergence of Bulk Locality

It was proven in previous sections that there are no nontirvial local bulk operatorsof BHC TN. However let us consider setup where certain local operator φx acts onclassical geometry bulk state. Then it is possible to show that one can map thisoperator isometrically to the boundary operator Ox.

Ox = M †φx ⊗∏y 6=x

PyM (4.36)

where Py =∏y

|ny=1〉 〈ny=1| is projector onto classical geometry state. Now let

us prove that∏y 6=x〈ny = 1|M defines isometry from site x to boundary. This may

be done by means of property 3 of pluperfect tensors. This property allows us todefine particular arrow drawing(as usual we are not allowed to draw closed loops):This arrow drawing shows us that we can define isometry from some bulk leg(whichis restricted by D2 dimensionality) to set of boundary legs. It may be seen fromimages that this can be done in different ways. This corresponds to AdS-Rindler

42

Figure 4.11: Arrow drawings which define unitary map from bulk leg⋃

geodesic⋃

certain boundary legs to set of other boundary legs. Meaning of black and whitecircles is the same as in previous section. Image from [30].

reconstruction. All bulk sites which can be reconstructed from the same boundaryregion form causal wedge of this boundary region [30].

This map may be generalised for case of multiple bulk sites. Suppose we have Pboundary legs. From dimensional argument we can’t map more than P/2 bulk siteson the boundary isometrically. Still this is not the only restriction. One should pickonly certain ”allowed” configurations. This algorithm is described in details in [30].

Suppose there exists a collection of n bulk sites x1, x2...xn which can be mappedisometrically to the boundary. Bulk operators acting on these sites will have cer-tain commutation relation. Corresponding boundary operators will have the samecommutation relations.

It is possible to estimate how many allowed configurations exist for particulartensor network for fixed n. The important property of BHC is that for small enoughn [30] almost all configurations are allowed, which means that at low energies onecan see appearance of local QFT.

4.6.7 Low-energy subspace and gauge invariance.

In this section we will consider particular example of square lattice. Still all resultswith small modifications will be the same for negatively curved tensor networks aswell.

Suppose we have L×L lattice and boundary legs being qutrits. As was stated inprevious sections to define isometry from bulk to boundary one should consider bulksubspace Hbulk such that dim(Hbulk) ≤ dim(H∂). It may be shown that there are 2global constraints on the boundary, based on stabilizer property [30]. So dimensionof boundary Hilbert space is equal to D4L−2. Hence the largest set of boundary legsmay contain at most 2L− 1 legs. Bulk legs are fixed to have dimension 2L− 1. Aswas mentioned in previous sections some of combinations are allowed and some arenot. For our argument we will consider combinations of first type.

Now we can define D2(2L−1) orthogonal states which have support on these 2L−1sites. It is possible to map them isometrically to the boundary. As a result one will

43

obtain D2(2L−1) orthogonal boundary states which will form basis in H∂. Thesestates then may be mapped back to bulk which will gives us D2(2L−1) states, whichare:

- gauge invariant- orthogonal to each other- not necessary direct product statesSo we have obtained orthogonal basis in physical Hilbert space. Suppose one

starts from different allowed configuration and follows the same procedure. Thenone will simply arrive to different basis in bulk physical Hilbert space.It is easy tosee that any other allowed configuration of smaller size may be written in terms ofaforementioned basis.

Let us summarise briefly the main idea of BHC. Tensor networks M definesisometry from boundary to bulk Hilbert space. Low energy subspace is needed todefine isometry in opposite direction.

4.7 Random Tensor Networks

4.7.1 Definition of RTN

In pervious chapters we have shown that even though BHC is able to describe physicsat sub-AdS scale it can’t be used(at least at this stage) as a tool to describe highlyentangled bulk states, because RT formula may be proven only for direct productbulk states for this TN. To overcome that problem we will consider yet anothertype of tensor network. Even though random tensor network is built in a differentway from what we did before it may be shown[34] that at the certain limit of in-plane bond dimension, random tensors used as building blocks for this TN inherit allproperties of pluperfect tensors. That would mean that this type of TN can also havesub-AdS scale resolution. More generally it may be shown[34] that RTN is reduced toBHC in this limit. This means that boundary theory may be mapped isometricallyto non-local theory in the bulk. As we have seen in previous sections this bulkHilbert space has ”low energy” subspace defined by gauge invariance property withemergent locality. So RTN may be used to define non-local QFT in the bulk withlocal limit. This makes it suitable candidate for description of quantum gravity.

In following sections we will show that RT formula may be proven for all sortsof bulk and boundary states. This will lead us to the conclusion that this type ofTN may be a good candidate to describe constant time slice of BTZ black hole andeventually use it to gain interpretation of holographic shadow region in terms ofboundary information.

44

Let us introduce certain definitions and tools for future use. The whole chapteris mainly based on article [34].

Let us consider some state |Vx〉 =∑µk Tµ1µ2...µn |µ1 ⊗ µ2 ⊗ ...⊗ µn〉 defined on

Hilbert spacen⊗k=1

Hk, where |µk〉 is basis in Hk. This state is completely defined

by Tµ1µ2...µn .

These kinds of vectors will be building blocks for tensor network. First, let usconsider direct product state of such vectors ⊗x |Vx〉, where x will correspond toparticular vertex.The only restriction we imply is that |Vx〉 are unit vectors chosenindependently at random from their respective Hilbert space [34].

Any leg of |Vx〉 is defined by HIlbert space Hk which may vary for different legs.We will define several different types of them for future use:

Hilbert space corresponding to leg connecting vertex x with vertex y will becalled Hxy with dimension Dxy. Dangling leg will correspond to Hx∂ with dimensionDx∂. Each bulk leg is considered to have dimension Db.

Now let us connect vertices of state ⊗x |Vx〉. In particular connection of two legsx⊗y corresponds to projection in Hilbert space Hxy⊗Hyx onto maximally entangled

state |xy〉 = 1√Dxy

Dxy∑µ=1

|µxy〉 ⊗ |µyx〉.

As a result one can write:∣∣∣ψ⟩ =( ⊗<xy>

〈xy|)(⊗

x

|Vx〉)

(4.37)

So far we obtained tensor network(TN) which maps boundary legs to bulk ones(holographiccode). One can equivalently view TN as a quantum state defined on Hilbert spaceHb ⊗ H∂, where H∂ =

⊗x∈∂

Hx∂ and Hb is bulk Hilbert space. Let us denote bulk

state by |φb〉. Then the boundary state(holographic state) may be written as:

|ψ〉 =(〈φb| ⊗

⊗<xy>

〈xy|)(⊗

x

|Vx〉)

(4.38)

For future purposes we will name |φb〉 ⊗⊗<xy>

|xy〉 as state of enlarged ”bulk Hilbert

space”.

One may incorporate possibility to have mixed states in the bulk(and bound-ary).Then boundary state will be given by:

ρ = trp

(ρp∏x

|Vx〉 〈Vx|)

(4.39)

ρp = ρb⊗<xy>

|xy〉 〈xy|

Subscript p may be understood as ”everything but boundary”.

45

4.7.2 Calculation of Second Renyi Entropy

In pervious section we introduced notion of RTN and in particular expressionfor density matrix of boundary state. Now let us consider some region A on theboundary and evaluate S2

A for it. We will eventually show that analogue of RTformula holds for second Renyi entropy (when bond dimension of tn is large enough),with the area of geodesic surface given by the graph metric of the network. Onecan also show that following arguments may be extended both for the case of higherentropies in large D limit and for von Neumann entropy, at least where min geodesicsare unique [34].

Second Renyi entropy is defined by:

e−S2A =

tr(ρ2A)

(trρ)2(4.40)

where ρ is generally not normalised.Now let us define swap operator FA acting on 2 copies of Hilbert space:

FA(|nA〉1 |mA〉1 ⊗ |n′A〉2 |m

′A〉2) = |n′A〉1 |mA〉1 ⊗ |nA〉2 |m

′A〉2 (4.41)

Let us consider simple example of action of swap operator to clarify notation:

FA(C↑↑ |↑〉A |↑〉B + C↓↓ |↓〉A |↓〉B)⊗ (D↑↓ |↑〉A′ |↓〉B′ ) = (4.42)

= C↑↑ |↑〉A |↑〉B ⊗D↑↓ |↑〉A′ |↓〉B′ + C↓↓ |↑〉A |↓〉B ⊗D↑↓ |↓〉A′ |↓〉B′

It may be shown that:tr(ρ2

A)

(trρ)2=

tr[(ρ⊗ ρ)FA]

tr(ρ⊗ ρ)(4.43)

Let us define:Z1 = tr[(ρ⊗ ρ)FA]

Z0 = tr(ρ⊗ ρ)

So now we have:

e−S2A =

Z1

Z0

(4.44)

where Z1 and Z0 are functions of random state |Vx〉, where x defines particularvertex. Now let us average over all states in the single-vertex Hilbert space:

S2(A) = −logZ1 + δZ1

Z0 + δZ0

= − logZ1

Z0

+∞∑n=1

(−1)n−1

n

(δZn0

Zn0

− δZn1

Zn1

)(4.45)

It may be shown[34] that if we consider large enough Dxy, then:

S2(A) ' − logZ1

Z0

(4.46)

Now let us consider expression for Z1:

Z1 = tr[(ρp ⊗ ρp)FA

∏x

|Vx〉 〈Vx| ⊗ |Vx〉 〈Vx|]

(4.47)

46

In appendix C it is shown that:

|Vx〉 〈Vx| ⊗ |Vx〉 〈Vx| =Ix + FxD2x +Dx

(4.48)

where Fx is a swap operator acting at vertex x. Dx =∏

yn.n.xDxy, product of alldimensions, corresponding to all legs adjacent to x. To clarify structure of expressionZ1 lets’s consider case of 2 sites:

Z1 = tr[(ρp ⊗ ρp)FA

∏x

(Ix + Fx)

D2x +Dx

]= (4.49)

= tr[(ρp ⊗ ρp)FA

(IxIy + IxFy + FxIy + FxFy)

(D2x +Dx)(D2

y +Dy)

]=

=1∏

x(D2x +Dx)

tr[(ρp ⊗ ρp)FA

∏x with sx=−1

Fx]

We can see that Z1 consists from 2N terms(N = 2, for this case).This motivates us to redefine Z1:

Z1 =∑sx

e−A[sx]

e−A[sx] ≡ 1∏x(D

2x +Dx)

tr[(ρp ⊗ ρp)FA

∏x with sx=−1

Fx]

(4.50)

Z1 can be viewed as partition function of sx spins:We redefine problem of evaluation of S2

A in following way:Let us consider sx , on the lattice, which has exactly the same structure as

our tensor network. sx = ±1 corresponds to Ix, Fx respectively.One may define boundary pinning field:

hx =

+1 , x ∈ A-1 , x ∈ A

We can then rewrite A[sx] as:

A[sx] = S2(sx = -1; ρp)−∑x∈∂

1

2logDx∂(3 + hxsx) +

∑x

log(D2x +Dx

)(4.51)

If one takes into account previously defined expression for ρp we can simplify ex-pression aforementioned expression even more:

A[sx] = −∑<xy>

1

2logDxy(sxsy − 1)−

∑x∈∂

1

2logDx∂(hxsx − 1)+ (4.52)

+S2(sx = -1; ρb) + const

where last term defines entanglement between bulk legs protruding from sites wheresx = −1 and rest of bulk. First two terms define Ising model sx in the presenceof pining field.

47

One can analogously derive expression for Z0. In this case hx = 1 for the wholeboundary.

Now let us define F1 = − logZ1, F0 = − logZ0, where F1, F0 are correspondingfree energies of the Ising model with appropriate boundary conditions.

Then S2(A) ' F1−F0 has interpretation of ”energy cost” for flipping the bound-ary pinning field in region A to (-1) while keeping it equal to (+1) in region A.

4.7.3 Ryu-Takayanagi Formula For Direct-Product Bulk state.

Suppose we have bulk state which is direct product state: ρb =∏x

|φx〉 〈φx|. In this

case one can contract bulk leg at each site with corresponding tensor. It will leadto new tensor at each site with smaller number of legs. We had random tensors ateach site at the beginning. Resulting tensors will still be random after contractionwith bulk states. So now one has holographic state(Tensor network without anybulk legs).

In this setup (3.49) will be reduced to:

A[sx] = −1

2

[ ∑<xy>

logDxy(sxsy − 1) +∑x∈∂

logDx∂(hxsx − 1)]

(4.53)

where we have omitted constant terms. In the large D limit the Ising model is in thelow temperature limit, hence partition function is dominated by the lowest energyconfiguration.

to calculate Z1 we need to take into account b.c: h(x) = −1 inside boundaryregion A and h(x) = 1 elsewhere. Energy in this case is proportional to length ofdomain wall in the bulk, separating spins up and spins down.

To evaluate Z0 we need to follow the same procedure, however one must considerdifferent boundary conditions: hx = 1 everywhere on the boundary. It may be shownthat in this case F0 = 0. Eventually we obtain expression of second Renyi entropyfor large D limit(we assumed that minimal geodesic is unique):

S2(A) = F1 − F0 ' logD min∑boundA

∣∣∣∑∣∣∣ ≡ logD|γA| (4.54)

It may also be shown[34] that aforementioned arguments may be extended to vonNeumann entropy at large D limit.

We may compare obtained expression for entropy with continuous Riemann man-ifold. Let us denote length between 2 sites as lg. Then for two expressions for entropyto be consistent one must have:

∣∣γCA ∣∣ = lg|γA|, where∣∣γCA ∣∣ - length of geodesic in

continuous case, |γA| - counts number of cuts. Therefore:

S(A) = l−1g logD

∣∣γCA ∣∣ (4.55)

48

Figure 4.12: Minimal energy configuration corresponds to domain wall positionedalong minimal geodesic, bounding region A. Image from[34]

One can see that l−1g logD corresponds to gravitational coupling constant 1

4GN.

This proof holds also for multiple disjoint intervals. It is so because of propertiesof Ising model[34]. In case when minimal surface is uniques formula for entropy isapplicable even for manifold with zero and positive curvature[34].

4.7.4 Ryu-Takayanagi formula With Bulk State Correction.

Now let us consider case when bulk state is not direct product state anymore. Wewill still consider large D limit, hence Ising model is still defined by the lowest energyspin configuration. However in new setup total expression for entropy is now definedby two terms: logD|

∑| area low energy and energy cost from bulk entropy.

So second Renyi entropy is given by:

S2(A) ' logD|γA|+ S2(EA; ρb) (4.56)

where EA is entanglement wedge - bulk area bounded by γA.

Figure 4.13: This image explicitly shows origin of second term in above expression.

49

So second term can shift position of domain wall(if bulk entanglement is largeenough) and it also provides additional contribution to general expression.

4.7.5 Hawking-Page transition.

As was mentioned in previous section position of minimal surface is influenced byterm S2(EA; ρb). When value of Db reaches certain value it may change position ofminimal surface qualitatively. In this section we will address this problem in moredetails.

Suppose that bulk state is given by:

|ψ〉bulk =(⊗|~x|>b

|ψ~x〉)⊗∣∣ψ|~x|<b⟩ (4.57)

where state inside region |~x| < b is a random pure state.It may be shown[35] thatrandom pure state is nearly maximally entangled and thus may be used as a toymodel of a thermal state. Rest of of bulk state |~x| > b is given by direct product ofrandom states at different sites.

We can change amount of entanglement in this state by means of varying param-eter Db. It will lead to phase transition in behaviour of minimal geodesics similarto one described by Hawking-Page transition in black hole formation[36].

Let us consider tensor network defined as uniform triangulation of hyperbolicdisc.

Suppose we have region A on the boundary. Entropy of this region is given byexpression:

S2(A) ' logD|γA|+ S2(sx = −1; ρb) (4.58)

Now let us consider in more details second term of above expression. It depends onvolume (number of sites) of spin-down domain in the disk region. In given setup allstates play symmetric role. As shown in appendix D, after averaging over randomstates entropy of bulk region with N sites is given by:

S2(N) = log( DNT

b + 1

DNb +DNT−N

b

)(4.59)

where NT is total number of sites in region |~x| < b, N is number of sitesin (sx = −1 ∪ |~x| < b). Suppose average distance between sites is lg. Totalexpression for second Renyi entropy os subregion A is given by:

S2[M↓] = logD · l−1g |∂M↓|+ log

( DVT /l

2g

b + 1

D|M↓|/l2gb +D

(VT−|M↓|)/l2gb

)(4.60)

50

where M↓ is a spin-down bulk region bounding region A (|M↓| its volume). |∂M↓|is size(number of cutes) of region M↓(without A). VT = NT l

2g is ”volume” of bulk

region |~x| < b.Now let us place our tensor network on Poincare disk model of hyperbolic space.

Its metric is given by:

ds2 = 4l2AdS

(dr2 + r2dθ2)

(1− r2)2(4.61)

We introduce cutoff at r = 1 − ε . From now on let us put lAdS = 1. Now one candefine region A as r = 1, θ ∈ [−ϕ, ϕ], ϕ ≤ π

2. We do not need to consider ϕ larger

than π2

because pure boundary state guaranties that SA = SA.

Figure 4.14: Orange region corresponds to random entangled bulk state. Blackcurve defines geodesic.[34]

Entangled region is defined as |~r| ≤ b. b = tanh(1/2), which means that inproper distance b = lAdS. Suppose geodesic is given by some curve r(θ). Now onecan rewrite expression for entropy in following way:

S2(ϕ) = minr(θ)

ϕ∫−ϕ

dθ2l−1g logD

1− r2(θ)

√(r′(θ))2 + r2(θ) + log

DVT /l

2g

b + 1

DVr(θ)/l2gb +D

(VT−Vr(θ))/l2gb

(4.62)

Vr(θ) =

ϕ∫−ϕ

dθ

b∫minb,r(θ)

dr4r2

(1− r2)2, VT =

2π∫0

dθ

b∫0

4r2

(1− r2)2=

4πb2

1− b2

In our setup we fix l−1g logD and gradually increase value of l−2

g logDb.There appears to be 3 different phases: perturbed AdS, small black hole and

maximal black hole. They are defined by different behaviour of minimal surfaces.Explicit calculations are given in section 3.3 and appendix A of [34]. We will

provide here just final results, which we will need for future investigations.Similar analysis may be done for mixed bulk state:

ρb = (⊗|x|>b

ρpurex )⊗ (

⊗|x|<b

ρmixx ) (4.63)

In a conclusion we would like to state that this model has some limitations. Inparticular structure of bulk state used in above section doesn’t reflect backreaction.

51

Figure 4.15: Numerical evaluation of minimal surfaces. Orange region correspondsto random pure state. l−1

g logD = 10, b = tanh(1/2) a) l−2g logDb = 1, perturbed

AdS phase b) l−2g logDb = 5 small black hole phase c) l−2

g logDb = 15, maximalblack hole phase.[34]

Figure 4.16: Phase diagram of 3 possible regimes of system.[34]

For instance number of geodesics which do not intersect entangled region areequivalent to those of empty AdS, which is not true.

Most importantly for us this model doesn’t reflect existence of holographicshadow of larger than size of region b.

If we consider star(black hole) of radius Rh ≥ lAdS then the model is consistentbecause in this case size of holographic shadow is exponentially small. Howeverfor star(black hole) which size is smaller than lAdS one has to modify bulk state toobtain consistent picture.

4.7.6 Large D limit.

Notion of large D limit was used in section 3.5.2. Let us define it in more rigorousway. For detailed derivation of following conditions see section 7.1 of [34].

52

Large D limit was used to justify assumption:

Sn(A) =logZ

(n)1

logZ(n)0

(4.64)

Within our procedure we mapped unnrumalized Zn1,0 to Ising partition functions

with inverse temperature: β = logD. One can see that large D limit means that weneed to consider Ising model at zero temperature. Hence it is only defined by thelowest energy configuration of spins. It may be shown that in this setup Sn(A) isapproximated by RT formula. Otherwise we can claim that the following:·) Let us consider bulk system of volume V(V = N · l2g, where N - number of sites

in this region)∀δ > 0 one can define critical bond :

Dc = αδ−2eC2nV (4.65)

where α,C2n are some constants, which do not depend on V. Then we may provethat following expression holds:∣∣Sn(A)− SRTn (A)

∣∣ < δ, with probability P (δ) = 1− Dc

D(4.66)

where

SRTn (A) ≡ logD|γA|+ Sn(EA; ρb)

·) Condition 3.62 puts strong constrains on range of D one can use. However it ispossible to show that under certain plausible physical assumptions[34] on free energyof statistical model one can obtain:

Dc = α′δ−2V 2/42n (4.67)

where α′,42n are some constants which do not depend on V.

53

Chapter 5

Conclusion

In this thesis we considered a problem of interpretation of holographic shadow re-gions in terms of boundary information. Holographic shadow is bulk region whichcan’t be penetrated by any minimal surface matching some boundary region.Forinstance such region surrounds BTZ black hole and may be as large as radius ofAdS space-time depending on size of black hole. Understanding of physics insidethis region is important because it will shed some light on questions about in-fallingobserver.

Bulk geometry probes other than minimal surfaces were considered in literature[6]. However each of them experiences holographic shadow region of certain size.

We considered different types of tensor networks as toy models of AdS/CFT cor-respondence. Depending on internal structure they may reflect different nontrivialaspects of holographic duality between bulk and boundary Hilbert spaces. In par-ticular we were interested in tensor networks which admit sub-AdS scale resolution.

First candidate was so called Bidirectional Holographic code. This is first tensornetwork which has sub-AdS scale resolution. However its main disadvantage is thatRT formula may be proven only for direct product bulk states. Hence one can’tdescribe highly entangled bulk states as BTZ black hole by means of this tensornetwork.

However different kind of tensor network namely Random Tensor Network wasrecently introduced. This tool has many advantages over previous one. First, RTformula holds for bulk and boundary states which can be both highly entangledand/or mixed. Second, this TN inherits all properties of BHC at certain limit of in-plane bond dimension. That means that this TN also has sub-AdS scale resolution.

Although authors of this approach consider toy model of bulk state correspondingto BTZ black hole, they still get too rough picture which can’t reflect all aspectsneeded for description of holographic shadows.

One possible way to resolve this problem may be to consider thermal boundarystate(Thermal CFT is known to be dual to BTZ black hole) and then map it thebulk. Then one can see whether obtained state reflects BTZ black hole geometrywith needed accuracy.

Once obtained this toy model may become appropriate tool to find interpretationof holographic shadow region in terms of boundary information. In particular it willbe interesting to investigate non-minimal geodesics which can penetrate holographicshadow in terms of this model. May be they can have some interpretation in termsof boundary region which will become more clear in terms of new approach.

54

AcknowledgementsFirst of all I would like to thank my supervisor Prof. Dr. Jan de Boer. He pro-

vided me with very interesting problem for thesis which covers most of my interestsin High Energy Physics. Within this year I have learnt a lot not only about subjectitself but also about different ways to approach new material, organise my thoughtsand ask precise questions. After each meeting I started to think in a different wayabout things I thought I understood before, which helped me a lot within processof work.

I would also want to thank my examiner Erik Verlinde for number of fruitfulconversations and hints which helped me to find answers to many questions.

I had great pleasure to communicate with Fernando Pastawski, Zhao Yang andSepehr Ghazi Nezami who helped me to understand many subtle points which werenot directly addressed in number of articles.

Special thanks to numerous fellow students and PhD students who helped me toresolve many tricky questions and to organise my knowledge about the subject.

The last but not least I would like to mention all people from physics stackexchange forum who gave me great hints and answers to my numerous questions.

55

Appendices

56

Appendix A

Cutoffs for Entropy expression.

We consider Poincare coordinates

ds2 =dz2 + dx2

z2. (A.1)

There are 2 ways to define cutoff of coordinate z to obtain length of geodesic:

Figure A.1: Geodesic in Poincare coordinates and two possible ways to introducecutoff.

Let us consider case I:Expression for length of static geodesic will look like:

γA =

∫ds =

∫ √gzz(

dz

dλ)2

+ gxx(dx

dλ)2

dλ =

∫R√

(z)2 + 1

zdx (A.2)

It is useful to take into account some specific properties of integrand. Rememberingthat z = z(x) one can see that the whole function is invariant under transformations

x→ x + a, a - const. We can defineR√

(z)2+1

z= L(z, z, x) in analogy with classical

mechanics:S =

∫L(q, q, t)dt, where t→ x, q → z, q → z.

57

Invariance under transformation x → x + a will mean existence of conserved”energy”. We can also define equations of motion with respect to this lagrangian:

H =∂L

∂q− L =

R

z√

(z)2 + 1(A.3)

which doesn’t depend on x.Now we can use this definition to find x as a function of z. The final result will

look like:

x = −√R2 − (zH)2

H+K (A.4)

In this expression we have two constants H and K, which are yet to be defined bymeans of boundary conditions we imply on geodesic.

In this particular case (−√

l2

4− ε2, ε), (

√l2

4− ε2, ε). So we have K = 0, H = 2R

l

Once we plug it into expression for length of geodesic we obtain:

γAI =Rl

2

∫dx

l2

4− x2

= 2RLog( l +

√l2 − 4ε2

2ε

)(A.5)

Now let us consider case II:We obtain for boundary conditions: (− l

2, ε), ( l

2, ε)

K = 0, ε2 +l2

4=R2

H2, z2 + x2 = ε2 +

l2

4(A.6)

For length of geodesic one obtains:

γAII =R2

H

∫dx

z2= 2R

√ε2 +

l2

4

∫ 0

l2

dx

ε2 + l2

4− x2

= 2RLog( l +

√l2 + 4ε2

2ε

)(A.7)

Consistency checks of obtained expressions.

Now we would naturally like to find some ways to see which expression is thecorrect one. The very first thing to do would be to see whether both expressionssatisfy SSA inequalities:

SA+B + SB+C ≥ SA+B+C + SB (A.8)

SA+B + SB+C ≥ SA + SC

This expressions should hold for arbitrary non intersecting intervals.One can convince himself that they hold for both of our expressions.

58

Anther way would be to use expression from [37]

d

dl

(ld

dlSfinite

)≤ 0. (A.9)

However this inequality was derived [38] for essentially Lorentz invariant theory.At the same time both of our expressions explicitly depend on cutoff, so they arenot Lorentz invariant. We could set ε→ 0 to recover Lor. invar, but then this checkwill give to us trivial answer.

According to our knowledge there isn’t a simple check in 1+1 case to distinguishbetween these two expressions for entropy. To obtain some unambiguous quantityone may calculate mutual information or relative entropy, which do not depend onregularisation scheme.

These quantities may be used in higher dimensions to find out the right approachfor regularisation procedure.

59

Appendix B

Two-point function fromAdS/CFT

Central point of AdS/CFT correspondence is equivalence between boundary theoryand bulk one. Bulk observables should be defined by boundary correlators at anygiven time. In this appendix particular realisation of this idea is considered. We willmainly follow [23]. A lot of relevant information about subject of current appendixmay be found in [24].

Our goal is to evaluate two-point function on the boundary CFT based onAdS/CFT prescription.

At low energies bulk theory is approximated by classical supergravity. In oursetup we consider massless bulk field φ(x0, xi) with boundary condition:

φ(x0, xi) = φ0(xi)|x0=0 (B.1)

Boundary value of field φ(x0, xi) corresponds to source for CFT operator O(x)[23].Euclidean AdSd+1 may be written in Poincare coordinates in following way:

ds2 =d∑i=0

dx2i

x20

(B.2)

AdSd+1 boundary is given by Rd at x0 = 0 and single point P at x0 =∞(becausemetric vanishes in this direction).

According to AdS/CFT:

ZSUGRA = ZCFT (B.3)

where

ZSUGRA =

∫φ(0)

dφe−I[φ] (B.4)

which is in general very complicated expression to handle. However in classical limitit is reduced to:

ZCLSUGRA = e−Icl[φ] (B.5)

where Icl[φ] is action of bulk field evaluated ”on shell” and satisfying aforementionedb.c. Once put everything together we can write[23]:

e−Icl[φ] = 〈exp

∫d~xφ0(x)O(x)〉CFT (B.6)

60

From this expression one can obtain 2-point function for this operator:

〈O(x1)O(x2)〉 =1

Z[0]

δ

δφ0(x1)

δ

δφ0(x2)

∣∣∣φ0(x)=0

ZCFT = (B.7)

=δ

δφ0(x1)

δ

δφ0(x2)e−Icl[φ]

∣∣∣φ0(x)=0

Our goal now is to obtain solution of bulk equations of motion for φ(xi, x0) withaforementioned boundary conditions. To do so we will follow procedure considered in[23]. The strategy is to ”reconstruct” bulk field from its boundary value. Uniquenessof such reconstruction is justified in the same article. For now we will just take it forgranted. In particular we will need to evaluate so called smearing function K(whichis also known as ”bulk-to-boundary propagator”)

Action of massless bulk field is given by:

IAdS =

∫dd+1x

√g|dφ|2 (B.8)

So equation of motion for this bulk field is just Laplace equation in AdS with itsboundary value defining corresponding b.c. To be more precise one should in prin-ciple add constant term(due to presence of curvature) in this equation of motion.However as was argued in [23] this does not introduce any principal changes, so wewon’t include it as well. Let us sketch general approach to these kinds of differentialequations.

Consider homogeneous equation a manifold M subject to Dirichlet b.c:

Lxu(x) = 0

u(x)|∂M = ν

where Lx is some differential operator. ν doesn’t have to be constant. (We giveformulation for flat spacetime, however generalisation to curved case is straightfor-ward.) General solution of above equation may be found in terms of so called KernelK(x, x′):

u(x) =

∫∂M

K(x, x′)ν(x′)dx′

whereLxK(x, x′) = 0

limx→x′

K(x, x′) = δ(x− x′)

Second condition follows from Dirichlet b.c. for u(x)[24].Now we can use above approach to solve equation of motion for bulk fields. Let us

summarise approach used in [23]. First one solves Laplace equation for K. Then oneneeds to show that this solution is singular at some points on the boundary(that willindicate presence of source.) Finally it is important to check what kind of sourcesone obtained and if they are physical(in this setup it will be point source.)

We consider Poincare covering of bulk. Both boundary conditions and bulkmetric are invariant under translations in xi direction, hence K should be functiononly of x0:

d

dx0

x−d+10

d

dx0

K(x0) = 0 (B.9)

61

There are two solutions of this equation one which vanishes at x0 = 0(normalisablemode) and another one proportional to constant(non normalisable). We will considerfirst part [23]:

K(x0) = cxd0 (B.10)

where c is some constant. This function grows when x0 goes to infinity(in pointP). We will show that this function is a delta function. To do so let us applytransformations which map point P to some finite point:

xi →xi

x20 +

∑dj=1 x

2j

, i = 0, ..., d (B.11)

K(x) = cxd0

(x20 +

∑dj=1 x

2j)d

One can see that this function satisfies several properties:

1.∫K(x)dx1...dxd doesn’t depend on x

2. K - positive

3. K = 0 at x0 → 0, except x1 = ... = xd = 0

We conclude from this that K is delta function at x0 → 0.Now we can construct solution of field φ in bulk with appropriate b.c.:

φ(x0, xi) = c

∫d~x′

xd0(x2

0 + |~x− ~x′|2)dφ0(x′i) (B.12)

For x0 → 0 we can obtain:

∂φ

∂x0

∼ dcxd−10

∫d~x′

φ0(x′)

|~x− ~x′|2d+O(xd+1

0 ) (B.13)

To obtain expression for I(φ) one needs to introduce cutoff because metric is diver-gent at x0 → 0.

After integrating by parts:

I(φ) = limε→0

∫Tε

d~x√hφ(~n~5)φ (B.14)

where Tε is surface x0 = ε, h is induced metric on this surface, n is unit normalvector to Tε. One has

√h = x−d0 , ~n~5φ = x0( ∂φ

∂x0)

Finally:

Icl[φ] =cd

2

∫d~xd~x′

φ0(~x)φ0(~x′)

|~x− ~x′|2d(B.15)

By taking 2 functional derivatives one can obtain well known from CFT 2-pointfunction:

〈O(x), O(x′)〉 =1

|~x− ~x′|2d(B.16)

where O(x) has conformal dimension d.

62

Appendix C

Integral evaluation.

In this appendix we will obtain explicit expression needed for calculation of secondRenyi entropy of boundary region of random tensor network. We need to prove that:

|Vx〉 〈Vx| ⊗ |Vx〉 〈Vx| =Ix + FxD2x +Dx

(C.1)

where the imply |Vx〉 to be random unit vector: 〈Vx| |Vx〉 = 1.We may rewrite left site do B.1 expression in following way:

I =

∫πdψiδ(

∑|ψi|2 − 1)ψiψ

∗jψkψ

∗l

a2d+2

∫πdψiδ(

∑|ψi|2 − 1)ψiψ

∗jψkψ

∗l =

∫πdψiδ(

∑|ψi|2 − a)ψiψ

∗jψkψ

∗l

Now let us multiply both sides by e−a and integrate over∞∫0

da

(2d+ 2)!

∫πdψiδ(

∑|ψi|2 − 1)ψiψ

∗jψkψ

∗l =

∞∫0

dae−a∫πdψiδ(

∑|ψi|2 − a)ψiψ

∗jψkψ

∗l

Hence we may rewrite initial integral in following form:

I =1

(2d+ 2)!

∞∫0

dae−a∫πdψiδ(

∑|ψi|2 − a)ψiψ

∗jψkψ

∗l =

1

(2d+ 2)!

∫πdψie

−∑|ψi|2ψiψ

∗jψkψ

∗l

which is now of Gaussian integral.General expression for such kinds of integrals is:∫dxk1 ...dxk2Nexp

(−1

2

n∑i,j=1

Aijxixj

)dnx =

√(2π)n

detA

1

2NN !

∑σ∈S2N

(A)−1kσ(1)kσ(2)...(A)−1

kσ(2N−1)kσ(2N)

In our case it is equal to:C(δijδkl + δilδjk)

where first term corresponds to 1 ⊗ 1 and second term corresponds to (1 ⊗ 1)Fx.Constant C is defined by traces of these operators and is equal to 1

D2x+Dx

. Eventuallywe obtain that:

I =Ix + FxD2x +Dx

(C.2)

63

Appendix D

Entropy of subsystem of randompure state.

In this appendix we will derive formula for section ”Hawking-Page transition”:

S2(N) = log( DNT

b + 1

DNb +DNT−N

b

)(D.1)

To do so we will use [39].General expression for second Renyi entropy is given by:

e−S#2 =

tr(ρ2#)

(trρ)2(D.2)

In our setup we consider normalised state. Hence our problem is to evaluate expres-sion tr(ρ2

#). Let us remind how do we define overall state:uiA - is homogeneously distributed unit vector in nK-dimensional Hilbert space.

Matrix ρ# is obtained by tracing out K-labels:

ρ#ij = uiAu∗jA (D.3)

We need to evaluate HMG〈tr(ρ2#)〉 , where ”HMG” stands for homogeneous averaging[39].

Now let us rewrite this expression:

HMG〈tr(ρ2#)〉 = 〈uiAu∗jAujBu∗iB〉 (D.4)

One can write[39](and in analogy with previous appendix) that:

〈ab∗cd∗〉 = C(1ab1cd + 1ad1bc) (D.5)

where we need to fix constant C. To do so let us remember that u is unit vector.Hence:

〈uau∗a〉 = 〈1〉 = 1

On the other hand:〈uau∗b〉 = C ′1ab

Suppose N is dimension of nK-dm Hilbert space. Hence:

〈uau∗a〉 = C ′N

64

So one obtains:

〈uau∗b〉 =1

N1ab

Now we can write that

〈ab∗〉 =1

N1ab (D.6)

This equation may be used to write[39]:

〈ab∗cc∗〉 = 〈ab∗〉 =1

N1ab (D.7)

Yet from C.5 we also know that this equation is equal to:

C(1ab1cc + 1ac1bc) = C(N + 1)1ab (D.8)

Hence:

C =1

N(N + 1)(D.9)

So now once can write:

〈ab∗cd∗〉 =1

N(N + 1)(1ab1cd + 1ad1bc) (D.10)

Overal dimension is N= Kn, hence:

HMG〈tr(ρ2#)〉 = 〈uiAu∗jAujBu∗iB〉 = (D.11)

=1

nK(nK + 1)(1iAjA1jBiB + 1iAiB1jAjB) =

K + n

nK + 1.

Once we put together C.11 and C.2, initial expression can indeed be recovered.

65

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