Projection Measurement and Holography - Osaka Universityseminar/pdf_2017... · Seminar@Osaka, April...

Seminar@Osaka, April 10, 2017

Projection Measurement and Holography

Tokiro Numasawa

Osaka University

Based on JHEP08(2016)077 arXiv:1604.01772 [hep-th]

Collaboration with N.Shiba(Harvard) T.Takayanagi and K.Watanabe(YITP)

Particle Physics Theory

(1)Introduction

Recently. quantum information theory gives useful tools to understand Holography or AdS/CFT correspondence.

・Entanglement Entropy・Tensor Networks ・Quantum Error Corrections ・Complexity・Information metric

・・・

(1)

Entanglement Entropy and AdS/CFT・EntanglementLet’s consider two spins.

�i(1) �j(2)

HA HB

| i = |"i |"i

h |�i(1)�j(2) | i

(2) | 0i = 1p2(|"i |"i+ |#i |#i)

h 0|�i(1)�j(2) | 0i

(no entanglement)

(entangled)

・Entanglement Entropynon local correlation (consider i=j= z )

⇢A = TrB [| i h |]Reduced Density Matrix:Entanglement Entropy:

ex) | i = |"i |"i SA = 0

(2) | 0i = 1p2(|"i |"i+ |#i |#i) SA = log 2

SA = �TrA[⇢A log ⇢A]

= h |�i(1) | i · h |�j(2) | i

�h 0|�i(1) | 0i · h 0|�j(2) | 0i 6= 0

⇒ Ground states of QFTs have entanglement !

Ex) EE for an interval in CFT2

・Entanglement in QFTs

h0|�(x)�(y) |0i 6= 0

SA =

c

3

log

l

✏

Al

c :central charge:UV cutoff✏

In AdS3/CFT2

A

We still do not fully understand how operational aspects inquantum information theory work in gravity through holography.

In quantum information, we can consider

・Local operations(LO)

X

i

AiA†i = 1

X

j

BjB†j = 1

with trace preserving condition⇢AB !X

i,j

(Ai ⌦Bj)⇢AB(A†i ⌦B†

j )

include projection measurements and unitary transformation

・Classical Communication (CC)⇒These operations are combined and called LOCC

A B

A B

Ai Bj

LOCC and entanglement entropy

One of interesting things is that LOCC give an operational meaning to entanglement entropy.

A B

A B

LOCC

SA

EPR pairsSA

gives maximal number of EPR pairs by LOCC

(1) Introduction(2) Local Projection measurement in a CFT(3) Partial Entangling and Swapping in two CFTs(4)Holographic duals of Local projections(5)Conculusion

(2) Local Projection measurement in a CFT

As we saw, Projection Measurements are important operations in QI.

In QFTs, we can consider local projection measurements where the state at each point in a region P is projected:

P =⇣ Y

x2P

| x

i h x

|⌘⌦⇣ Y

x2P

c

Ix

⌘

| i | i | i

|�i

After the projection, there is no real space entanglement in the region P.

PC P

Let’s consider Conformal Field Theories (CFT).

In CFTs, one of such states with no real space entanglement is boundary state (Cardy state).|Bi [Miyaji-Ryu-Wen-Takayanagi, 2014]

ex) Ising CFT (c=1/2 minimal model)

|0i : spin up |"i |"i · · · |"i

|#i |#i · · · |#i

|!i |!i · · · |!i

| 12i : spin down

| 116

i :free

= critical point of H =X

i

�z

i

�z

i+1 + �x

i

Therefore, we obtain the following path integral representation of the states after local projection measurements :

e�pHP |0i =| i =

| i

Density matrix :| i h | = h |

h |O(x1)O(x2) | i

Correlation functions :

= O(x1)O(x2)

X

p+it

p-it

ζ

ξ w y

x

0

-2π

0 log(ρ)

Identify

1

SE22e

Identify

2q

1U

U

X(⇣) = 2ip⇣K(⇣/

p⇢) +K(⇣

p⇢)� 1

2

⌘� t

K(⇣) =⇣

⇣ � 1+

1X

k=1

⇣ ⇢2k

⇣ � ⇢2k� ⇢2k

⇣�1 � ⇢2k

⌘

Conformal mapsWe obtain a geometry with two holes

→We can map this geometry to the annulus (or cylinder)

0.2 0.4 0.6 0.8

10

20

30

40

50

q

p

⇢

q

p⇡ 2⇡

| log ⇢|

EE in Free Dirac Fermion CFT

!10 !5 5 10

!0.15

!0.10

!0.05

�q q0

P A

x

�SA(x)

x� 0.5x+ 0.5

⇢ = 0.6

p = 0.5

q = 5.3

Reduction of EE due to the projection

⌘ S

A

(x)� S

ground

A

(x)

EE in Free Dirac Fermion CFT

�q q0P

A

⇢ = 0.6

p = 0.5

q = 5.3

5 10 15

!0.15

!0.10

!0.05

0.05

0.10

0.15

�0.5 0.5

t

�SA(t) ⌘ SA

(t)� Sground

A

(t)

Linear growth of Global quenches

Entangled pairs go away from A(result from relativistic propagation)

Local Mearuement (without cutoff)

⇠

O1(⇠1)

O2(⇠2)O1(w1)

O2(w2)

w

⇠ =

rq + w

q � w

Without cutoff p , we obtain only one boundary ( for projection) :

The calculation becomes easy.

[Rajabpour 15]

※)One of the merit to include cutoff is that we can follow the time evolution by analytic continuation

It is also useful to consider the projection measurement without cutoff.

P 2 = P

PA

ex1) Single interval A ending on P

w

l

2q

SA =

c

6

log

2l(l + 2q)

q✏+ �b

�n(w1)

⇒

To calculate Renyi EE, we need the 1-pt func of twist op

1-pt on UHP can be determined by conformal symmetry.

�b: boundary entropy

[Rajabpour 15]

�n

ex1) General single interval A

PA2q

w

�n(w1)

To calculate Renyi EE, we need the 2-pt func.

�n(w2)

l1 l2

For free Dirac fermion,

SA =

1

6

log

2l1(l1 + 2q)

q✏+

1

6

log

2l2(l2 + 2q)

q✏+2�b +

1

3

log

ql1+2q

l1�

ql2+2q

l2ql1+2q

l1+

ql2+2q

l2

Generically, it is difficult to determine the full expression.If we take the limits, we obtain

PA2q

l1 l2

l1 ⌧ qP

A2q

l1 l2

SA =

(c6 log

2l1(l1+2q)q✏ +

c6 log

2l2(l2+2q)q✏ + 2� (l1 ⌧ q),

c3 log

l2�l1✏ (l1 � q)

l1 � q(region A is very close to P) ( A is very far from P and ignore

the existence of P )

b

(3) Partial Entangling and Swapping in two CFTs

In QI, EPR pair (maximal entanglement) is also an important object.

(3-1)Partial Entangling

PC P

Thus, we consider to create EPR pairs on an interval P between the identical QFT

This can also be seen as the projection measurement on P in the theory QFT1×QFT2.

P =Y

x2P

⇣X

n

x

|nx

i1 |nx

i2⌘⇣X

m

x

hmx

|1 hmx

|2⌘⌦

Y

x2P

c

(I1x

⌦ I2x

)

CFT1 CFT2

P

CFT1 CFT2

P’

Each sheet is mapped to annulus

P1 P2

P1 P2

Since each plane with two cuts is conformally mapped into a cylinder as we have seen, our doubled geometry is conformal to a torus:

We define the period of this torus as ⌧ = ⌧1 + i⌧2

Entanglement entropy between the two CFTs created by this operation can be found as follows assuming ,⌧2 � 1

Sent =⇡c

3⌧2

(3-2)Partial Swapping Two CFTs

We cut out P1 and P2 from CFT1 and CFT2. After we exchange them, we glue them again.

CFT1 CFT2

P

CFT1 CFT2

P’

w

-q-ip q-ip

-q+ip q+ipCy

Cx=

・Usual Elliptic curve・The period is different from the previous one

⌧

P1 P2

Entanglement Entropy as a function of q/p

w

-q-ip q-ip

-q+ip q+ipCy

Cxy

2 = (x� ip� q)(x� ip+ q)(x+ ip� q)(x+ ip+ q)

Equation of elliptic curve:

Period: ⌧2 =

RC

y

dx

yRC

x

dx

y

When p/q << 1, we find

Sent ⇡2c

3

log

⇣qp

⌘= 2· Sinterval

Sinterval =c

3

log

⇣ l

✏

⌘※ ) Entanglement entropy of single interval is given by

[Holzhey-Larsen-Wilczek 1994]

swappingA1A2

CFT1 CFT2

A1 A2

⇒A1 A2

CFT1 CFT2

A1 A2

Physical interpretation

SA1 =

c

3

log

q

pSA2 =

c

3

log

q

p

Sent = SA1 + SA2

entangled entangled

entangled

entangled

key: bulk extension of ①conformal map②boundary

First, we consider the holographic dual of projection measurement w/o cutoff

⇠

O1(⇠1)

O2(⇠2)O1(w1)

O2(w2)

w

⇠ =

rq + w

q � w

(4)Holographic duals of Local projections

① Conformal Mapping in AdS3/CFT2

AdS metric in Poincare patch:

ds2 = R2AdS

d⌘2 + 2d⇠d⇠

⌘2

⇠ = f(w)� 2z2(f 0)2f 00

8f 0f 0 + z2f 00f 00

⇠ = f(w)� 2z2(f 0)2f 00

8f 0f 0 + z2f 00f 00

⌘ =8z(f 0f 0)

32

8f 0f 0 + z2f 00f 00

dual coordinate transformation of Conformal map

ds2 = R2AdS

⇣dz2

z2+ L(w)dw2 + L(w)2dw2 +

⇣ 2

z2+

z2

2L(w)L(w)

⌘dwdw

⌘

Tww ⌘ L(w) =3(f 00)2 � 2f 0f 000

4(f 0)2 : Schwarzian deriv (× (-1/2))

[Roberts 12]

② AdS3/BCFT2

AdS boundary with boundary P

Bulk extension of boundary

In addition to the AdS boundary P , there appears another boundary Q in the bulk.

[Takayanagi 11][Fujita-Takayanagi-Tonni 11]

Computation of HEE in AdS/BCFT [Takayanagi 11][Hartman-Maldacena 13]

There are two types of geodesics:①Connected Geodesics②Disconnected Geodesics

The smaller one gives the holographic EE.

A�A

�AA

Holographic Projection measurement (w/o cutoff)

w ξ

P

Im[w]=0 ξ=f(w)

-q q f(P)

i

Extend to the bulk

AA A

SA =

Area(�A)

4GN=

c

6

log

2l1(l1 + 2q)

q✏

Holographic calculation leads to

Matches with CFT calculation !

・ Single interval A ending on P

PAl

2q

l

�AA

General interval A

0.00 0.02 0.04 0.06 0.08 0.102.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Phase-1 Phase-2

P P

A A

Q Q

l1q

�SAdifferent l2 � l1

q

sharp transition only in large N limit

PA2q

l1 l2

A

�A

SA =

c

6

log

2l1(l1 + 2q)

q✏+

c

6

log

2l2(l2 + 2q)

q✏+

c

3

log

ql1+2q

l1�

ql2+2q

l2

2(

l1+2ql1

l2+2ql2

)

14

Phase-1

Phase-2

SA =

c

6

log

2l1(l1 + 2q)

q✏+

c

6

log

2l2(l2 + 2q)

q✏

(4)Holographic dual of Projection measurements (w/ cutoff) and Partial Entangling

Partial Entangling of Two CFTs

・Geometry is torus⇒dual: BTZ black hole (= Solid torus)

Entanglement Entropy between CFT1 and CFT2 is given by BH entropy.

Local projection measurement (with cutoff)・Geometry is cylinder ⇒The bulk dual is given by half of the solid torus

[cf:Hartman-Maldacena 13]

Connected Geodesic

BH Horizon

Disconnected Geodesics

We can evaluate the Holographic EE from the geodesics in the BTZ black hole geometry.

ds

2 =R

2AdS

z

2

hdz

2 +⇣1� �

2z

2

2

⌘2dx

2 +⇣1 +

�

2z

2

2

⌘2dy

2i

Holographic EE for a single interval ⇢ = 0.6p = 0.5q = 5.3

!10 !5 5 10

!1.0

!0.5

0.5

1.0

1.5

2.0

�SA Connected geodesics

Disconnected geodesics(ending on the bounadry Q)

x

P A�q 0 q

x� 0.5 x+ 0.5

Entanglement is generated

Entanglement is reduced

※) Choose the smaller one

2 4 6 8 10 12

!2

!1

0

1

2

3

Disconnected geodesics(ending on the bounadry Q)

Connected geodesics

Time evolutoin of Holographic EE for a single intervalP

A

�q 0 q

+0.5�0.5

5 10 15

!0.15

!0.10

!0.05

0.05

0.10

0.15

cf) Free fermion

�SA(t)

・sharp transition because of large N・qualitatively agree with the CFT results

(5)Conclusions and Discussion

・We considered Projection measurements, partial entangling and entangling swapping in CFTs.

・Bulk duals are given by cutting or connecting the geometry

・We also followed the time evolution after a projection measurements both in CFT side and gravity side, and showed matches.

Future problems

・Entanglement swapping in one CFT

⇒crosscap appears

・Partial Entangling through interfaces.CFT1 CFT2

P

CFT1 CFT2

P’

・Can we consider the quantum computation in CFT/holography?

Detail of AdS/BCFT

IE = � 1

16⇡GN

Z

N

pg(R� 2⇤� Lmatter)�

1

8⇡GN

Z

Q

ph(K � LQ

matter)

・Action

Gibbons-Hawking term

Lagrangian of matter on boundary Q

Boundary E.O.M leads to Neumann boundary condition:

Kab �Khab = 8⇡GNTQab

Conformal invariance⇒ Tab = �Thab

T is the tension of Q and related to boundary entropy

For simplicity we set T=0

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