YITP-19-80 ; IPMU19-0114

Entanglement Wedges from Information Metric in Conformal Field Theories

Yuki Suzukia, Tadashi Takayanagib,c and Koji Umemotoba Faculty of Science, Kyoto University,

Kitashirakawa Oiwakecho,Sakyo-ku, Kyoto 606-8502, JapanbCenter for Gravitational Physics,

Yukawa Institute for Theoretical Physics, Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku,

Kyoto 606-8502, Japan andcKavli Institute for the Physics and Mathematics of the Universe (WPI),

University of Tokyo, Kashiwa, Chiba 277-8582, Japan(Dated: August 28, 2019)

We present a new method of deriving the geometry of entanglement wedges in holography directlyfrom conformal field theories (CFTs). We analyze an information metric called the Bures metric ofreduced density matrices for locally excited states. This measures distinguishability of states withdifferent points excited. For a subsystem given by an interval, we precisely reproduce the expectedentanglement wedge for two dimensional holographic CFTs from the Bures metric, which turns outto be proportional to the AdS metric on a time slice. On the other hand, for free scalar CFTs, we donot find any sharp structures like entanglement wedges. When a subsystem consists of disconnectedtwo intervals we manage to reproduce the expected entanglement wedge from holographic CFTswith correct phase transitions, up to a very small error, from a quantity alternative to the Buresmetric.

1. Introduction

An important and fundamental question in the anti-de Sitter space/conformal field theory (AdS/CFT) cor-respondence [1] is “Which region in AdS corresponds toa given subregion A in a CFT ?”. The answer to thisquestion has been argued to be the entanglement wedgeMA [2–4], i.e. the region surrounded by the subsystem Aand the extremal surface ΓA whose area gives the holo-graphic entanglement entropy [5, 6]. Here the reduceddensity matrix ρA on the subregion A in a CFT gets dualto the reduced density matrix ρbulkMA

on the entanglementwedge MA in the dual AdS.

Normally this bulk-boundary subregion duality is ex-plained by combining several ideas: the gravity dual ofbulk field operator (called HKLL map [7]), the quantumcorrections to holographic entanglement entropy [8, 9]and the conjectured connection between AdS/CFT andquantum error correcting codes [10, 11]. However, sincethis explanation highly employs the dual AdS geometryand its dynamics from the beginning, it is not clear howthe entanglement wedge geometry emerges from a CFTitself. The main aim of this article is to derive the geom-etry of entanglement wedge purely from CFT computa-tions. We will focus on two dimensional (2d) CFTs fortechnical reasons. The AdS/CFT argues that a specialclass of CFTs, called holographic CFTs, can have classi-cal gravity duals which are well approximated by generalrelativity. A holographic CFT is characterized by a largecentral charge c and very strong interactions, which leadto a large spectrum gap [12, 13]. Therefore we expectthat the entanglement wedge geometry is available onlywhen we consider holographic CFTs. Our new frame-

work will explain how entanglement wedges emerge fromholographic CFTs.

For this purpose we consider a locally excited state ina 2d CFT, created by acting a primary operator O(w, w̄)on the vacuum. We focus on the 2d CFT which liveson an Euclidean complex plane R2, whose coordinate isdenoted by (w, w̄) or equally (x, τ) such that w = x+ iτ .By choosing a subsystem A on the x-axis, we define thereduced density matrix on A, tracing out its complementB:

ρA(w, w̄) = TrB[O(w, w̄)|0〉〈0|O†(w̄, w)

], (1)

first introduced in [14] to study its entanglement entropy.We assume that the (chiral and anti chiral) conformal

dimension h of O satisfies 1� h� c. In this case, we canneglect its back reaction in the gravity dual and can ap-proximate the two point function 〈O(w1, w̄1)O†(w2, w̄2)〉by the geodesic length in the gravity dual between thetwo points (w1, w̄1) and (w2, w̄2) on the boundary η → 0of the Poincare AdS3

ds2 = η−2(dη2 + dwdw̄) = η−2(dη2 + dx2 + dτ2), (2)

where we set the AdS radius one. Therefore, by project-ing on the bulk time slice τ = 0, the state ρA(w, w̄) isexpected to be dual to a bulk excitation at a bulk point Pdefined by the intersection between the time slice τ = 0and the geodesic, as depicted in Fig.1.

Now we are interested in how we can distinguish thetwo states: ρA(w, w̄) and ρA(w′, w̄′) when w 6= w′, cre-ated by the same operators. They are dual to bulkstates with two different points excited. On the time

arX

iv:1

908.

0993

9v1

[he

p-th

] 2

6 A

ug 2

019

2

A

ΓA

O(w1)

O(w2)

O(w1)

O(w2)

BoundaryBulk AdS

P Pτ=0 time slice

FIG. 1. A sketch of entanglement wedge MA for an intervalA in AdS3/CFT2 and holographic computations of two pointfunctions dual to geodesics. The blue (or green) geodesic does(or does not) intersect with MA.

slice τ = 0, the location of bulk excitations are given by(η, x) = (τ, x) and (η, x) = (τ ′, x′). The entanglementwedge reconstruction argues we cannot distinguish thetwo excited bulk states when both excitations are out-side of MA, while we can distinguish them if at least oneof them is inside of MA.

A useful measure of distinguishability between twodensity matrices ρ and ρ′ is the Bures distance DB , de-fined by (refer to e.g. [15])

DB(ρ, ρ′)2 = 2(1− Tr[√√

ρρ′√ρ]), (3)

Moreover we can define the information metric when thedensity matrix is parameterized by continuous valuablesλi, denoted by ρ(λ):

DB(ρ(λ+ dλ), ρ(λ))2 ' Gijdλidλj ≡ dD2B , (4)

where dλi are infinitesimally small. This metric Gij iscalled the Bures metric, which measures the distinguish-ablility between nearby states.

The quantum version of Cramer-Rao theorem [16] tellsus that when we try to estimate the value of λi fromphysical measurements, the errors of the estimated valueis bounded by the inverse of the Bures metric as follows

〈〈δλiδλj〉〉 ≥ (G−1)ij . (5)

This shows as the Bures metric gets larger, the errors dueto quantum fluctuations get smaller.

As an exercise, consider the case where A coversthe total system, where ρA(w, w̄) becomes a pure state|φ(w)〉〈φ(w)|. The Bures distance DB is simplified as

DB(|φ〉, |φ′〉)2 = 2(1− |〈φ(w)|φ(w′)〉|),|〈φ(w)|φ(w′)〉| = |w − w̄|2h|w′ − w̄′|2h|w − w̄′|−4h. (6)

This leads to the Bures metric

dD2B =

h

τ2(dτ2 + dx2). (7)

In this way, the information metric is proportional to theactual metric in the gravity dual (2) on the time sliceτ = 0. This coincidence is very natural because the dis-tinguishability should increase as the bulk points are ge-ometrically separated and was already noted essentially

in [17]. However, this result is universal for any 2d CFTs.Soon later we will see this property largely changes forreduced density matrices, where results crucially dependon CFTs. We will be able to find the entanglement wedgestructure only for holographic CFTs.

Before we go on, let us mention that for technical con-veniences, we often calculate (introduced in [18])

I(ρ, ρ′) =Tr[ρρ′]√

Tr[ρ2]Tr[ρ′2], (8)

instead of Tr[√√

ρρ′√ρ] to estimate distinguishability. If

ρ = ρ′ we find I(ρ, ρ′) = 1, while we have 0 < I(ρ, ρ′) < 1when ρ 6= ρ′.

2. Single Interval CaseWe choose the subsystem A to be an interval 0 ≤ x ≤ L

at τ = 0. The surface ΓA in the bulk AdS is given bythe semi circle (x− L/2)2 + η2 = L2/4. Therefore if theentanglement reconstruction is correct, the informationmetric should vanish if the intersection P is outside ofthe entanglement wedge given by

|w − L/2| = L/2. (9)

In this example, the entanglement wedge is equivalent tothe causal wedge [19].

Let us start with the calculation of the quantity I(ρ, ρ′)defined by (8), for ρ = ρA(w, w̄) and ρ′ = ρA(w′, w̄′).Since this calculation is essentially that of Tr[ρρ′], weperform the conformal transformation:

z2 =w

w − L, (10)

which maps two flat space path-integrals which produceρ(w, w̄) and ρ(w′, w̄′) into a single plane (z−plane). Referto [20, 21] for similar calculations in the context of en-tanglement entropy of such states. The insertion pointsof the four primary operators on the z−plane are givenby (remember w = x+ iτ)

z1 =

√−x− iτL− x− iτ

(≡z), z2 =

√−x+ iτ

L− x+ iτ(≡ z̄),

z′3 =−√−x′ − iτ ′L− x′ − iτ ′

(≡−z′), z′4 =−√−x′ + iτ ′

L− x′ + iτ ′(≡−z̄′).

Refer to Fig.2 for this conformal mapping. It is importantto note that the boundaries of the wedges (9) on thew−planes are mapped into the diagonal lines z = ±iz̄.

The quantity Tr[ρρ′] is expressed as a correlation func-tion on the z−plane:

Tr[ρρ′]=

[∣∣∣∣ dz1

dw1

∣∣∣∣ ∣∣∣∣ dz2

dw2

∣∣∣∣ ∣∣∣∣ dz′3dw′3

∣∣∣∣ ∣∣∣∣ dz′4dw′4

∣∣∣∣]2h

·H(z1, z2, z′3, z′4)·Z(2)

(Z(1))2,

H(z1, z2,z′3,z′4)≡ 〈O†(z1,z̄1)O(z2,z̄2)O†(z′3,z̄

′3)O(z′4,z̄

′4)〉

〈O†(w1,w̄1)O(w2,w̄2)〉〈O†(w′3,w̄′3)O(w′4,w̄′4)〉

,

where 〈· · ·〉 denotes the normalized correlation functionsuch that 〈1〉 = 1 and we also write the vacuum partition

3

ρ=

ρ'= A

A

w1

w4

w3

w2w1

w2

w3

w4

z2

z1

z3

z4

z2z3

z4 z1

FIG. 2. A sketch of conformal transformation for the calcula-tion of Tr[ρρ′]. Green Points (or bule points) are outside (orinside) of the wedge (9).

function on a n-sheeted complex plane by Z(n). Finallywe obtain

I(ρ, ρ′) =F (z1, z2, z

′3, z′4)√

F (z1, z2, z3, z4)F (z′1, z′2, z′3, z′4), (11)

F (z1, z2,z′3,z′4)≡〈O†(z1,z̄1)O(z2,z̄2)O†(z′3,z̄

′3)O(z′4,z̄

′4)〉.

In holographic CFTs, we can approximate the correla-tion functions by regarding the operators are generalizedfree fields [22] so that we simply take the Wick contrac-tions of two point functions (we set z = z1 and z′ = −z3):

F (z1, z2, z′3, z′4) ' |z−z̄|−4h·|z′−z̄′|−4h+|z+z̄′|−8h. (12)

The value of I(ρ, ρ′) as a function of w′ = x′ + iτ ′ isplotted in the left two graphs in Fig.3. The upper leftgraph is the case where w is inside the wedge (9) and wehave I = 1 iff w = w′, while 0 < I < 1 iff w 6= w′, asexpected. This shows that we can correctly distinguishthe states. On the other hand, if w is outside the wedge(see the lower left graph), we find I ' 1 (i.e. indistin-guishable) if w′ is also outside, while we have I ' 0 ifw′ is inside. We can see that the border is precisely theCFT counterpart of the entanglement wedge (9). Thisborder gets very sharp when h � 1 as we are assumingto justify the geodesic approximated. These behaviorsperfectly agree with the distinguishability of bulk statesin the AdS/CFT.

When we calculate the information metric we assumew ' w′ (or equally z ' z′). In this case the first termin (12) dominates when |z − z̄| ≤ |z + z̄| and this con-dition precisely matches with that for the outside wedgecondition. Indeed, if we only keep this first term, we im-mediately find I(ρ, ρ′) = 1. On the other hand, whenit is inside, the second term is dominant and the resultis identical to the case where A is the total space (i.e.ρA = |φ(w)〉〈φ(w)| is pure).

It is instructive to calculate I(ρ, ρ′) for non-holographicCFTs. As an example, we consider a 2d free masslessscalar CFT (the scalar field is denoted by ϕ) and choosethe primary operator to be O(w, w̄) = eiαϕ(w,w̄), which

FIG. 3. The profiles of I(ρ, ρ′) as a function of x′ (horizontalaxis) and τ ′ (depth axis) for the choice A = [0, 2] (i.e.L = 2).The left two ones are for a 2d holographic CFT while theright ones for a 2d free scalar CFT. In the upper two graphswe chose h = 1/2 and (x, τ) = (1, 0.1) and in the lower two,we chose h = 10 and (x, τ) = (−1, 0.1).

has the dimension h = α2/2. Then we explicitly obtain

I(ρ, ρ′) =

(|z + z′|2|z + z̄||z′ + z̄′|

4|z||z′||z + z̄′|2

)4h

. (13)

The result is plotted as right two graphs in Fig.3. Clearlyin this free CFT, we cannot find any sharp structure ofentanglement wedge as opposed to holographic CFTs,though they have qualitative similarities (refer to thelower right picture).

3. Bures MetricNow let us calculate the genuine Bures metric when A

is a single interval. We evaluate Tr[√√

ρρ′√ρ] from

An,m = Tr[(ρmρ′ρm)n]. (14)

via the analytical continuation n = m = 1/2. We applythe conformal transformation (we set k = (2m+ 1)n)

zk =w

w − L, (15)

so that the path-integrals for 2mn ρs and n ρ′s aremapped into that on a single plane. This leads to

An,m=〈O†(w1)O(w2)· · ·O†(w2k−1)O(w2k)〉·Z(k)∏k

i=1〈O†(w2i−1)O(w2i)〉·(Z(1))k.(16)

Refer to [23, 24] for analogous computations of relativeentropy. After the conformal mapping (15), we find

An,m =

2k∏i=1

∣∣k−1(zi)1−k∣∣2h · k∏

j=1

|(z2j−1)k − (z2j)k|4h

×〈O†(z1)O(z2) · · ·O†(z2k−1)O(z2k)〉 · Z(k)

(Z(1))k. (17)

Note that we have

z1 =

(−x− iτL− x− iτ

)1/k

, z2(= z̄1) =

(−x+ iτ

L− x+ iτ

)1/k

,

z2s+1 = e2πik sz1, z2s+2 = e

2πik sz2, (s = 1, 2, · · ·, k − 1).

4

Let us evaluate An,m in holographic CFTs, using thegeneralized free field approximation. We take w ' w′ tocalculate the Bures metric. When w and w′ are outsideof the entanglement wedge (9), or equally |z2j−1− z2j | <|z2j−2− z2j−1|, the 2k point function is approximated as

〈O†(z1)O(z2) · · ·O†(z2k−1)O(z2k)〉

'k∏j=1

〈O†(z2j−1)O(z2j)〉 'k∏j=1

|z2j−1 − z2j |−4h, (18)

and this leads to the trivial result An,m = 1, leading thevanishing Bures metric dD2

B = 0. This agrees with theAdS/CFT expectation that ρA cannot distinguish twodifferent bulk excitations outside of entanglement wedge.

On the other hand, when w and w′ are inside ofthe entanglement wedge (9), or equally |z2j−1 − z2j | >|z2j−2 − z2j−1|, we can approximate as

〈O†(z1)O(z2) · · ·O†(z2k−1)O(z2k)〉

'k∏j=1

〈O†(z2j−2)O(z2j−1)〉 'k∏j=1

|z2j−2 − z2j−1|−4h.

In the limit n→ 1/2 and m→ 1/2, this leads to

A1/2,1/2 = |w − w̄|2h|w′ − w̄′|2h|w′ − w̄|−4h,

dD2B =

h

τ2(dx2 + dτ2). (19)

This Bures metric for ρA(w, w̄) coincides with that forthe pure state (7) and reproduces the bulk AdS metricon the time slice τ = 0.

Similarly, in a 2d holographic CFT with a circle com-pactification x ∼ x+ 2π, we obtain the Bures metric

dD2B =

h

sinh2 τ(dτ2 + dx2), (20)

if w is inside the wedge. In a 2d holographic CFT atfinite temperature T , the Bures metric is computed as

dD2B = h

(2πT )2

sin2 (2πTτ)(dτ2 + dx2). (21)

if w is inside the wedge. These metrics agree with thoseon the time slice τ = 0 of global AdS3 and BTZ blackhole, by projecting the point (x, τ) at the AdS bound-ary into the time slice along each geodesic. In summary,our CFT calculations for these setups show that in holo-graphic CFTs, we can distinguish two excitations if theyare inside the entanglement wedge.

It is intstructive to calculate the Bures metric in the2d massless free scalar CFT for the primary O = eiαϕ.

ρ= ρ'=A1

w1

w2

w1

w2

z2

z1

z3

z4

z2z3

z4 z1

A2 A1 A2

π

-π

0

Re[z]-J 0 J

Im[z]

w3w3

w4w4

FIG. 4. A sketch of conformal transformation for Tr[ρAρ′A]

in the double interval case. We assumed the phase (i), wherethe entanglement wedge is connected, as depicted by the col-ored region. The lower picture describes the geometry afterthe transformation and is given by a torus by identifying theedges. Blue (or Green) points are outside (or inside) of MA.

For α = 1, we find the following analytical result:

A 12 ,

12

=(√z +√z′)(√z̄ +√z̄′)(√z +√z̄)(√z′ +

√z̄′)

4√|z||z′|(

√z +√z̄′)(√z̄ +√z′)

,

dD2B = − L2(dw)2

16w2(L− w)2− L2(dw̄)2

16w̄2(L− w̄)2

+L2(dw)(dw̄)

2|w||w − L|(√

w̄(w − L) +√w(w̄ − L)

)2 . (22)

Note that we cannot find any sharp structure ofentanglement wedge as opposed to the holographicCFT. However, in the limit τ → 0, we find the metricds2 ' h

τ2 (dτ2 + dx2) for 0 ≤ x ≤ L.

4. Double Interval CaseFinally, we take the subsystem A to be a union of two

disconnected intervals A1 and A2, which are parameter-ized as A1 = [0, s] and A2 = [l+ s, l+ 2s], without losinggenerality. We conformally map the w-plane with twoslits along A1 and A2 into a z−cylinder via (see e.g. [25])

z = f(w) =J(κ2)

2− J(κ2)

2K(κ2)sn−1(w̃, κ2), (23)

where we introduced

w̃ =2

l

(w − s− l

2

), J(κ2) = 2π

K(κ2)

K(1− κ2),

K(κ2) =

∫ 1

0

dx√(1− x2)(1− κ2x2)

, κ =l

l + 2s.

The function sn−1(w̃, κ2) is the Jacobi elliptic function:

sn−1(w̃, κ2) =

∫ w̃

0

dx√(1− x2)(1− κ2x2)

. (24)

It is useful to note the relation sn−1(w̃, 0) = arcsin(w̃).We can calculate I(ρ, ρ′) using the map (23) both for

ρA(w, w̄) and ρA(w′, w̄′) and the formula (11). The two

5

w−planes are mapped into a torus, described by thez−plane with the identification Re[z] ∼Re[z] + 2J andIm[z] ∼Im[z] + 2π, as depicted in Fig.4.

In holographic CFTs, we need to distinguish twophases depending on the moduli of the torus:

(i) Connected phase : J < π or equally κ < 3− 2√

2,

(ii) Disconnected phase : J > π or equally κ > 3− 2√

2.

We can confirm the phase (i) (or (ii)) coincides with thecase in the gravity dual where the entanglement wedgegets connected (or disconnected), and the circle Re[z] (orIm[z]) shrinks to zero size in the bulk, respectively. Thisis the standard Hawking-Page transition [26] and agreeswith the large c CFT analysis [12]. The holographic twopoint functions on the torus in each phase behaves like

〈O†(z,z̄)O(z′,z̄′)〉(i)'Maxn1∈Z

∣∣∣∣sin(π(z+2πin1−z′)2J

)∣∣∣∣−4h

,

〈O†(z,z̄)O(z′,z̄′)〉(ii)'Maxn2∈Z

∣∣∣∣sinh

((z+2Jn2−z′)

2

)∣∣∣∣−4h

.

Let us estimate 4−point functions F in (11) by thegeneralized free field prescription, where we again assumew ' w′. There are two contributions: the trivial Wickcontraction and the non-trivial one as in (12). The trivialone leads to I(ρ, ρ′) = 1, which tells us that we cannotdistinguish the two nearby states. Therefore we againfind that the entanglement wedge corresponds to the re-gion where non-trivial contractions get dominant.

The non-trivial Wick contraction is dominant when

Minn1∈Z

∣∣∣sin( π2J

(z2−z1−2n1πi))∣∣∣ ≥ ∣∣∣sin( π

2J(z3−z2)

)∣∣∣ ,in the connected case, and when∣∣∣∣sinh

(z2 − z1

2

)∣∣∣∣ ≥ Minn2∈Z

∣∣∣∣sinh

(z2 − z3 − 2n2J

2

)∣∣∣∣ ,in the disconnected case. We plotted these regions interms of the coordinate w̃ of (23) in Fig. 5.

In both cases, the regions are very close to the trueentanglement wedges, respecting the expected connectedor disconnected geometry (note our holographic relationin Fig.1). The deviation is always within a few percent,depicted in Fig.5. This small deviation arises as the cor-rect distinguishability should be measured by the Buresmetric. Our analysis using I(ρ, ρ′) only gives an approx-imation, much like the Renyi entropy compared with thevon-Neumann entropy. As sketched in Fig.6, this wedgefrom I(ρ, ρ′) obeys the following rules: (a) the wedge forA1∪A2 is larger than the union of the wedges for A1 andA2 and (b) the wedge for A is the complement to the onefor Ac.

Thus it would be ideal if we can calculate the genuineBures distance DB(ρ, ρ′) in the double interval case.This is very complicated as the trace Tr[(ρmρ′ρm)n] cor-responds to a partition function on a genus n(2m+1)−1

w∼iw∼10i

-10 -5 0 5 10

-10

-5

0

5

10

w∼i

-0.10 -0.05 0.00 0.05 0.10

0.96

0.98

1.00

1.02

1.04 w∼10i

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

9.80

9.85

9.90

9.95

10.00

10.05

10.10

w∼3+2i

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

w∼3+2i

2.90 2.95 3.00 3.05 3.10

1.90

1.95

2.00

2.05

2.10

FIG. 5. The plots of the locations of the operator insertionon w̃−plane where the non-trivial Wick contraction is favored(blue colored regions in the left pictures). The upper picturesare for κ = 0.1 where MA is connected, while the lower onesare for κ = 0.2 where MA is disconnected. In the uppermiddle and right picture, blue curves are the borders betweenthe non-trivial and trivial contraction, while orange curvesdescribe the borders of the entanglement wedge. The same istrue for the lower right picture.

Entanglement Wedge in AdS

Wedge from I(ρ,ρ’)

A1 A2 Entanglement Wedge in AdS

Wedge from I(ρ,ρ’)

A1 A2

FIG. 6. Sketches which exaggerate small deviations betweenthe wedges from I(ρ, ρ′) and the actual entanglement wedges.

Riemann surface. However, since we will finally taken = m → 1/2 limit (genus 0 limit), it might not besurprising to obtain the expected metric (7) whichcoincides with the case where A is the total space.

5. DiscussionsIn this article, we presented a general mechanism how

entanglement wedges emerge from holographic CFTsand gave several successful examples. One importantfurture problem is to repeat the same procedure by usingthe genuine localized operator in the bulk [7] or the state[17], which may give us more refined results. Anotherinteresting direction will be to extend this constructionto the higher dimensional AdS/CFT. Moreover, it maybe useful to consider other distance measures such astrace distances [27]. It is also intriguing to explore therelationship between our approach and the path-integraloptimization [28]. It might also be fruitful to considerconnections between our results and the recent propos-als for entanglement wedge cross sections [29–34]. Wewould like to come back to these problems soon later [35].

Acknowledgements We thank Pawel Caputa,Veronika Hubeny, Henry Maxfield, Mukund Rangamani,Hiroyasu Tajima and Kotaro Tamaoka for useful con-versations. TT is supported by the Simons Foundationthrough the “It from Qubit” collaboration. TT is sup-

6

ported by JSPS Grant-in-Aid for Scientific Research (A)No.16H02182 and by JSPS Grant-in-Aid for Challeng-ing Research (Exploratory) 18K18766. TT is also sup-ported by World Premier International Research Cen-ter Initiative (WPI Initiative) from the Japan Ministryof Education, Culture, Sports, Science and Technology(MEXT). KU is supported by Grant-in-Aid for JSPS Fel-lows No.18J22888. We are grateful to the long term work-shop Quantum Information and String Theory (YITP-T-19-03) held at Yukawa Institute for Theoretical Physics,Kyoto University and participants for useful discussions.TT thanks very much the workshop “Quantum Informa-tion in Quantum Gravity V,” held in UC Davis, wherethis work was presented.

[1] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998)231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].

[2] B. Czech, J. L. Karczmarek, F. Nogueira and M. VanRaamsdonk, “The Gravity Dual of a Density Matrix,”Class. Quant. Grav. 29 (2012) 155009 doi:10.1088/0264-9381/29/15/155009 [arXiv:1204.1330 [hep-th]].

[3] A. C. Wall, “Maximin Surfaces, and the Strong Sub-additivity of the Covariant Holographic EntanglementEntropy,” Class. Quant. Grav. 31 (2014) no.22, 225007doi:10.1088/0264-9381/31/22/225007 [arXiv:1211.3494[hep-th]].

[4] M. Headrick, V. E. Hubeny, A. Lawrence andM. Rangamani, “Causality and holographic en-tanglement entropy,” JHEP 1412 (2014) 162doi:10.1007/JHEP12(2014)162 [arXiv:1408.6300 [hep-th]].

[5] S. Ryu and T. Takayanagi, “Holographic derivation ofentanglement entropy from AdS/CFT,” Phys. Rev. Lett.96 (2006) 181602; “Aspects of holographic entanglemententropy,” JHEP 0608 (2006) 045.

[6] V. E. Hubeny, M. Rangamani and T. Takayanagi, “ACovariant holographic entanglement entropy proposal,”JHEP 0707 (2007) 062 [arXiv:0705.0016 [hep-th]].

[7] A. Hamilton, D. N. Kabat, G. Lifschytz andD. A. Lowe, “Holographic representation of localbulk operators,” Phys. Rev. D 74 (2006) 066009doi:10.1103/PhysRevD.74.066009 [hep-th/0606141].

[8] T. Faulkner, A. Lewkowycz and J. Maldacena, “Quan-tum corrections to holographic entanglement entropy,”JHEP 1311 (2013) 074 doi:10.1007/JHEP11(2013)074[arXiv:1307.2892 [hep-th]].

[9] D. L. Jafferis, A. Lewkowycz, J. Maldacena andS. J. Suh, “Relative entropy equals bulk relative entropy,”JHEP 1606 (2016) 004 doi:10.1007/JHEP06(2016)004[arXiv:1512.06431 [hep-th]].

[10] A. Almheiri, X. Dong and D. Harlow, “Bulk Lo-cality and Quantum Error Correction in AdS/CFT,”JHEP 1504 (2015) 163 doi:10.1007/JHEP04(2015)163[arXiv:1411.7041 [hep-th]]; D. Harlow, “The Ryu-Takayanagi Formula from Quantum Error Correction,”arXiv:1607.03901 [hep-th].

[11] X. Dong, D. Harlow and A. C. Wall, “Reconstruction

of Bulk Operators within the Entanglement Wedge inGauge-Gravity Duality,” Phys. Rev. Lett. 117 (2016)no.2, 021601 doi:10.1103/PhysRevLett.117.021601[arXiv:1601.05416 [hep-th]].

[12] M. Headrick, “Entanglement Renyi entropies in holo-graphic theories,” Phys. Rev. D 82 (2010) 126010doi:10.1103/PhysRevD.82.126010 [arXiv:1006.0047 [hep-th]].

[13] T. Hartman, C. A. Keller and B. Stoica, “Universal Spec-trum of 2d Conformal Field Theory in the Large c Limit,”JHEP 1409 (2014) 118 doi:10.1007/JHEP09(2014)118[arXiv:1405.5137 [hep-th]].

[14] M. Nozaki, T. Numasawa and T. Takayanagi, “Quan-tum Entanglement of Local Operators in ConformalField Theories,” Phys. Rev. Lett. 112 (2014) 111602doi:10.1103/PhysRevLett.112.111602 [arXiv:1401.0539[hep-th]].

[15] M. Hayashi, “Quantum Information Theory”, GraduateTexts in Physics, Springer.

[16] C. W. Helstrom, Minimum mean-square error estimationin quantum statistics, ” Phys. Lett. 25A, 101-102 (1976).

[17] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi andK. Watanabe, “Continuous Multiscale EntanglementRenormalization Ansatz as Holographic Surface-State Correspondence,” Phys. Rev. Lett. 115 (2015)no.17, 171602 doi:10.1103/PhysRevLett.115.171602[arXiv:1506.01353 [hep-th]].

[18] J. Cardy, “Thermalization and Revivals after a QuantumQuench in Conformal Field Theory,” Phys. Rev. Lett.112 (2014) 220401 doi:10.1103/PhysRevLett.112.220401[arXiv:1403.3040 [cond-mat.stat-mech]].

[19] V. E. Hubeny and M. Rangamani, “Causal Holo-graphic Information,” JHEP 1206 (2012) 114doi:10.1007/JHEP06(2012)114 [arXiv:1204.1698 [hep-th]].

[20] M. Nozaki, T. Numasawa and T. Takayanagi, “Quan-tum Entanglement of Local Operators in ConformalField Theories,” Phys. Rev. Lett. 112 (2014) 111602doi:10.1103/PhysRevLett.112.111602 [arXiv:1401.0539[hep-th]]; M. Nozaki, “Notes on Quantum Entan-glement of Local Operators,” JHEP 1410 (2014)147 doi:10.1007/JHEP10(2014)147 [arXiv:1405.5875[hep-th]].

[21] S. He, T. Numasawa, T. Takayanagi and K. Watan-abe, “Quantum dimension as entanglement entropy intwo dimensional conformal field theories,” Phys. Rev. D90 (2014) no.4, 041701 doi:10.1103/PhysRevD.90.041701[arXiv:1403.0702 [hep-th]].

[22] S. El-Showk and K. Papadodimas, “Emergent Space-time and Holographic CFTs,” JHEP 1210 (2012) 106doi:10.1007/JHEP10(2012)106 [arXiv:1101.4163 [hep-th]].

[23] N. Lashkari, “Relative Entropies in ConformalField Theory,” Phys. Rev. Lett. 113 (2014) 051602doi:10.1103/PhysRevLett.113.051602 [arXiv:1404.3216[hep-th]]; “Modular Hamiltonian for Excited States inConformal Field Theory,” Phys. Rev. Lett. 117 (2016)no.4, 041601 doi:10.1103/PhysRevLett.117.041601[arXiv:1508.03506 [hep-th]].

[24] G. Srosi and T. Ugajin, “Relative entropy of excitedstates in two dimensional conformal field theories,”JHEP 1607 (2016) 114 doi:10.1007/JHEP07(2016)114[arXiv:1603.03057 [hep-th]].

[25] M. A. Rajabpour, “Post measurement bipartite entan-

7

glement entropy in conformal field theories,” Phys. Rev.B 92 (2015) 7, 075108 doi:10.1103/PhysRevB.92.075108[arXiv:1501.07831 [cond-mat.stat-mech]]; “Fate of thearea-law after partial measurement in quantum fieldtheories,” arXiv:1503.07771 [hep-th]; “Entanglement en-tropy after partial projective measurement in 1 + 1dimensional conformal field theories: exact results,”arXiv:1512.03940 [hep-th].

[26] S. W. Hawking and D. N. Page, “Thermodynamics ofBlack Holes in anti-De Sitter Space,” Commun. Math.Phys. 87 (1983) 577. doi:10.1007/BF01208266

[27] J. Zhang, P. Ruggiero and P. Calabrese, “Sub-system Trace Distance in Quantum Field The-ory,” Phys. Rev. Lett. 122 (2019) no.14, 141602doi:10.1103/PhysRevLett.122.141602 [arXiv:1901.10993[hep-th]]; “Subsystem trace distance in low-lying statesof conformal field theories,” arXiv:1907.04332 [hep-th].

[28] P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi andK. Watanabe, “Anti-de Sitter Space from Optimizationof Path Integrals in Conformal Field Theories,” Phys.Rev. Lett. 119, no. 7, 071602 (2017), [arXiv:1703.00456[hep-th]]; “Liouville Action as Path-Integral Complexity:From Continuous Tensor Networks to AdS/CFT,” JHEP1711 (2017) 097 [arXiv:1706.07056 [hep-th]].

[29] K. Umemoto and T. Takayanagi, “Entanglement of pu-rification through holographic duality,” Nature Phys.14 (2018) no.6, 573 doi:10.1038/s41567-018-0075-2[arXiv:1708.09393 [hep-th]];

[30] P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zale-tel and B. Swingle, “Entanglement of purification: fromspin chains to holography,” JHEP 1801 (2018) 098doi:10.1007/JHEP01(2018)098 [arXiv:1709.07424 [hep-th]].

[31] J. Kudler-Flam and S. Ryu, “Entanglement negativityand minimal entanglement wedge cross sections in holo-graphic theories,” arXiv:1808.00446 [hep-th].

[32] P. Caputa, M. Miyaji, T. Takayanagi and K. Umemoto,“Holographic Entanglement of Purification from Con-formal Field Theories,” Phys. Rev. Lett. 122 (2019)no.11, 111601 doi:10.1103/PhysRevLett.122.111601[arXiv:1812.05268 [hep-th]].

[33] K. Tamaoka, “Entanglement Wedge Cross Section fromthe Dual Density Matrix,” arXiv:1809.09109 [hep-th].

[34] S. Dutta and T. Faulkner, “A canonical purification forthe entanglement wedge cross-section,” arXiv:1905.00577[hep-th].

[35] Y. Suzuki, T. Takayanagi and K. Umemoto, work inprogress.