Download Quantum error-correction in black holes

Quantum error-correction in
black holes
Beni Yoshida (Perimeter Institute)
@ Taipei (Sep 2016)
[1] Holographic quantum error-correcting codes,
[2] Chaos in quantum channel,
[3] Complexity by design, arXiv:1609:xxxxx
String
Daniel Harlow (Harvard
MIT)
Fernando Pastawski (Caltech
Berlin) John Preskill (Caltech)
String
CMT
Xiao-liang Qi (Stanford)
QI
QI
Daniel Roberts (MIT
IAS)
Simons collaborations
A black hole is a quantum errorcorrecting code
Anti-de Sitter/Conformal field theory (AdS/CFT)
correspondence
• [Conjecture] Equivalence of string (gravity) theory in bulk with CFT on
boundary (Maldacena)
Hyperbolic space
(negatively curved)
Boundary D-dimensional conformal field
theory (without gravity)
Bulk (D+1)-dimensional theory
with gravity on AdS space
Anti-de Sitter/Conformal field theory (AdS/CFT)
correspondence
• [Conjecture] Equivalence of string (gravity) theory in bulk with CFT on
boundary (Maldacena)
• [Holography] Bulk degrees of freedom are encoded in boundary, like a hologram.
“holography”
Hyperbolic space
(negatively curved)
1-dimensional strongly
interacting “qubits”
2-dimensional “qubits” on AdS
Quantum entanglement in AdS/CFT
• [Ryu-Takayanagi formula]
QM
GR
1
S(A) =
min(area(
4GN A
A ))
Minimize over spatial bulk
surfaces homologous to A
A
A1
A2
(For “large N“)
Space-time as a tensor network
• [Swingle’s conjecture]
AdS/CFT correspondence can be expressed as a MERA ?
minimal surface
1
saturation ?
S(A) 
1
min(area(
4GN A
A ))
(1)
MERA = Multi-scale Entanglement Renormalization Ansatz (Vidal)
2⇡kB RE
Concrete and simple toy tensor network model
(joint with Harlow, Pastawski and Preskill)
Lattice gauge theory
Dictionary : Correspondence
of Yoshida
operators
Beni
(Dated: April 12, 2015)
boundary
system Abelian and non-Abelian
bulklattice
system gauge theory, based
A note
on discrete,
1
1
O
(1)
O1
O2
1
O1
2
O1 O2
1
2
1
S(A) 
min(area( A ))
4GN A
1
S(A) 
min(area(
O1
A ))
O2
|
O2
1
p
1
(2)
i+|
L
1
2
2
(1)
(1)
i + ···
(2)
n and non-Abelian lattice gauge theory, based on Kogut’s 1979 review.
Lattice
gauge
theory
an and non-Abelian lattice gauge theory,
based on
Kogut’s
1979 review.
Lattice gauge theory
Bulk operator vs boundary operator
Beni Yoshida
Beni Yoshida
wedge reconstruction]
• [Entanglement
(Dated: April 12, 2015)
(Dated:
April 12,
A bulk operator
can2015)
be represented by some integral of local boundary operators
A note
on discrete, Abelian and non-Abelian lattice gauge theory, based on Kog
supported on A if is contained inside the entanglement wedge of A.
nd non-Abelian lattice gauge theory, based on Kogut’s 1979 review.
A
OA
1
p
pL
L
A1
A2
OA 1 A 2
|
i+|
|
i+|
n
Y
p t
Remarks (1 + e L )
j=1
j
1
1
(1)
(1)
(2)
(2)
(1)
i + ···
· OA A
⇠ OAi + · · ⇠
1
2
⇠ OA
p
⇠ OL
A1 A2
(2)
• Entanglement wedge may go beyond black hole horizons (i.e. no firewall).
• “Proven” by using a generalized RT formula (Jefferisnet al, Dong et al, Bao et al)
n
Y
Y
jt
more than
0̃i is known
|1̃i for
• No explicit|recipe
j t one intervals(1 + e
)
|0̃i
|1̃i(1 + e )
j=1
| j=1 i + |
i + ···
Bulk locality puzzle
• The reconstruction recipe leads to a paradox
All the bulk operators must correspond to identity operators on the boundary ?
If so, the AdS/CFT seems very boring …
Quantum error-correction in AdS/CFT ?
• The AdS/CFT correspondence can be viewed as a quantum error-correcting code.
[Almheiri-Dong-Harlow].
Lattice gauge theory
Lattice gauge theory Lattice gauge theory
Beni Yoshida
(Dated: April 12, 2015)
Beni Yoshida
Beni Yoshida
(Dated:
April 12, 2015)
(Dated: April
12, 2015)
A note
on discrete,
Abelian
and
gauge
based1979
on Kogut’s
A note on
discrete,
Abelian
and
non-Abelian
lattice gauge
theory,
basedtheory,
on Kogut’s
review.
elian and non-Abelian
lattice
gauge
theory,
based
onnon-Abelian
Kogut’s
1979lattice
review.
OAB
OBC
OCA
OAB
OBC
OABOCAOBC
OCA
They are different operators, but act in the same manner in a low energy subspace.
cf. Quantum secret-sharing code
Let’s construct a toy model
A simple toy model
1 bulk qubit
in total, just 6 qubits
5 boundary qubits
A bulk operator must have representations on any region with three qubits.
logical qubit
physical qubits
Entanglement wedge reconstruction !
tal SPT phases and their gauged models
may
be an interesting
A note
on discrete,
Abelian andfuture
non-Abelian lattice gauge theory, b
ruct an SPT Hamiltonian protected by fractal-like symmetry operators.
h fractal SPT phases and their gauged models may be an interesting future
Lattice gaugecode
theory
Five-qubit
S1 = X ⌦ Z ⌦ Beni
Z ⌦ Yoshida
X ⌦I
a|0i + b|1i (73)
(Dated:
April
2015)
system
of14,
five
qubits.(73)
• EncodeS1a=single
X ⌦ Zlogical
⌦ Z ⌦qubit
X ⌦ Iinto a
= I⌦and
⌦I X
⌦ Z ⌦ lattice
Z ⌦X
(74)
S1 A=note
X on
⌦ discrete,
Z ⌦ ZSAbelian
⌦2 X
(73)
non-Abelian
gauge theory, based
on Kogut’s 1979 review.
X⌦⌦ZZ⌦⌦XZ⌦⌦IX
SS12==XI ⌦ Z
(73)(74)
input
S
=
XZ⌦ I ⌦ X ⌦ Z ⌦ Z
3ZZ
S2 S=
ZX⌦
⌦⌦
X I⌦
⌦⌦Z
⌦⌦
XX
S23=I=I⌦
X⌦X
⌦
Z
⌦
S⌦
=⌦⌦
ZZZ⌦ X ⌦ I ⌦ X ⌦ Z
⌦⌦
X
S34=X
=XZ⌦⌦⌦IIX
⌦4ZZ
X
S3 S=
⌦
XI⌦
a|0i + b|1i
4 = Z ⌦X ⌦I ⌦X ⌦Z
S4 S=
Z ⌦X ⌦I ⌦X ⌦Z
(74)(75)
(74)
(75)(76)
(75)
encode
(76)
• codeword space
output
a|0̃i + b|1̃i
a|0̃i + b|1̃i
C = {| i : Sj | i = | i 8j}
(76)
(76)
OAB
Acknowledgment
OBC
OCA
(77)
C
=
{|
i
:
S
|
i
=
|
i
8j}
j
thank Aleksander Kubica, John Preskill and Burak Şahinoğlu for helpful
code
has
3
• Five-qubit
omments.
ICalso
Burak
Şahinoğlu
this problem to
= {|thank
i:S
= | distance
i 8j}for suggesting
j | icode
Acknowledgment
(75)
(77)
(77)
set of states with distance 1
O
O
O
AB
BCfor Theoretical
CA
ork was completed during the visit to the Kavli Institute
thank Aleksander
Kubica,byJohn
Preskill
Burak
for helpful
owledge
funding provided
the Institute
for
Information
and
| 0and
i Quantum
| 1 iŞahinoğlu
(78)
omments.
I also Center
thank Burak
Şahinoğlu
forGordon
suggesting
Physics
Frontiers
with support
of the
and this
Bettyproblem
Moore to
|
0i
|
p
(78)
The code can correct single-qubit
L
1i
ts No.
and PHY-1125565).
am supported
David
work
wasPHY-0803371
completed during
the visit to theI Kavli
Instituteby
forthe
Theoretical
errors
!
distance 3
tdoctoral
fellowship.
This research
supported
in part byInformation
the Nationaland
owledge funding
provided
by the was
Institute
for Quantum
| i = ↵|
0i
+ |
1i
n under Frontiers
Grant No.Center
NSF PHY11-25915.
Physics
with support of the Gordon and Betty Moore
(79)
Five-qubit code is a quantum gravity
Perfectness of five qubit code
• Let’s view the five-qubit code as a six-leg tensor.
output
input
T
T
• Any leg can be used as an input of quantum codes.
T
T
A holographic quantum error-correcting code
• A tiling of the five qubit code
five qubit
code
output
input
AdS geometry
bulk
legs
boundary
legs
Entanglement wedge reconstruction
• 1 in & 3 out (operator pushing)
D
D
C
InputInput
Input
Input
A
A
AA
B
B
C
Holographic state
• The Ryu-Takayanagi formula holds exactly (tiling of perfect tensors)
Coarse-graining (RG transformation) = Distillation of EPR pairs along the geodesic
B
A
perfect tensor
EPR pairs
0
0
k+1
0
2n
B
A an en
A
to orthogonal. Such an isometry can
be viewed
as
EPR
(b)
ame, the boundary CFT operatorerror-correcting
O(x,(a)
t) can be written
as
code where k -spin input states are enco
Perfect
tensors
0
† Uwhere
B
states
k
input
as logical
A
UB = legs. A
= legs can
O(x, t) = U O (t = 0)U
(1.4)beVviewed
constructed
from
a perfect
tensor, will be called a perfect co
entanglement
in any
bipartition
• A pure state with maximal
operator O0 (t = 0) depends only on O(x,
the B
B is within
A
A t) where x
A
EPR(2n
pair legs)
state (x
(2n0 ,spins)
hus, the bulkPerfect
operator
r, t = 0) can be represented by
boundary
Perfect
tensor
(a)
the causal wedge.
ensor
B
UA
T
U=B =
T
A
(b)
T
VB
UB
distillation
duality
A
B
B
A
B
A
Figure 1. (a) Duality of unitary operators in a perfect state. (b) Distillation of EP
we introduce the notion of perfect tensors. Consider a system of 2n
the number of states per spin. A wavefunction | i is said to be a
We then introduce the notion of perfect tensors. A perfect state | i can be
s reduced density matrix ⇢A for allbyAa such
|A|  n is maximally
tensorthat
as follows:
⇢A / I A
for all |A|  n
v 1 X
v 1
X
v 1
X
2.
| i Figure
=
· · · (a) TPerfect
. . . in i (b) Perfect
i1 i2 ...in |i1 i2state.
(2.1)
i1 =0 i2 =0
in =0
For readers
from quantum
entity •matrix
on A.code
So, for any bipartition,
the entanglement
entropyinformation science, it may b
Five qubit
alue. Let | i be a perfect state, a
A quantum
be a subseterror-correcting
of spins with |A| 
n by a standard notation [[n
code
mplement of A. Since A is maximally
with B,
quantum
the entangled
total number
ofany
spins,
k represents
the number of logi
–5–
6 leg perfect tensor
T
on A has a dual quantum operation
acting onkB. Namely, letTUA ⌦ I
there are v orthogonal states in the codeword space), d re
nitary operator which acts exclusively on A. Then there always exists
and v represents the number of states in each spin. Acc
perfect code is represented by [[2n 1, 1, n]]v . Similarly, a
nsor with 2n legs. Let Mi1 ,...,ik be a tensor representation of an input state | i which
a k spin state. By contracting legs i1 , . . . , ik of Mi1 ,...,ik and Ti1 i2 ...i2n , one obtains a
w tensor Nik+1 ,...,i2n , which corresponds to the output state | i of the map:
Random tensors
X
Nik+1 ,...,i2n =
Ti1 i2 ...i2n Mi1 ,...,ik .
• Perfect tensors are very rare…
(2.5)
i1 ,...,ik
v ⌦k
v ⌦2n k !
perfect
tensors
are
common
• But almost
is contraction
defines
the map
: (C
) pretty
! (C
)
whose input is Mi1 ,...,ik and
tput is Nik+1 ,...,i2n . This map is an isometry since pairwise orthogonal states remain
Pick a Haar random state.
orthogonal. Such an isometry can be viewed as an encoding map of a quantum
or-correctingDue
code
where
-spin input
states
are encoded
2n k-spin
output
to the
Page’sk theorem,
the state
is almost
maximallyin
entangled
along any
cut.
tes where k input legs can be viewed as logical legs. A quantum code with k = 1,
construct
a holographic
pickcode
tensors
nstructed
a perfect
tensor, willcode/state,
be called a just
perfect
(Fig.randomly.
2).
• Tofrom
(a)
(b)
logical leg
Random
tensor
Figure
2. (a)
Perfect state. (b) Perfect code.
Holographic code
For readers from quantum information science, it may be convenient to characterize
⇠ OA
⇠ OA 1 A 2
(1)
Coding properties: erasure threshold
1
• Remove qubits with probability p = 2
n
Y
(1 + e
✏
jt
(2)
)
A central bulk leg is contained in the
entanglement wedge of A (if |A|>|B|)(3)
j=1
“e
H
”
Erasure threshold = 1/2
e
iHt
(4)
New quantum codes from quantum gravity ?
| i
(5)
So far, no black holes…
Information loss puzzle
1
• Is quantum information lost ?
| i
Hawking radiation
“e
1X
P ⌦ P = SWAP
d P
| ji = U|
ji
H
(1)
”
(2)
(3)
1
Information loss puzzle
• Or hidden into some non-local degrees of freedom ?
| i
Hawking radiation
1X
P ⌦ P = SWAP
d P
(2)
Locally it looks like “e
| ji = U|
ji
(1)
H , but globally it is not .
”
(3)
n
Y
jt
(1 error-correction
+e
)
Quantum
j=1
• Scrambling is very similar to how quantum error-correcting codes work.
1
• Local indistinguishability.
⇢R ' ⇢0R
| i
(1)
logical qubit
time evolution
e
1X
d
P
iHt
Pblack
⌦ hole
P = SWAP
| ji = U| ji
“e
Hawking
n
radiation Y
= error (1
+e
j=1
(2)
H
”
codewords
| i
(3)e
iHt
jt
)
(Dated: May 31, 2016)
(Dated: May(Dated:
31, 2016)
May 27, 2016)
California Institute of Technology, Pasadena, California 9
Random
thoughts on scrambling
Choi-Jamilkowski
isomorphism
(Dated: August 12, 2015)
I. EQUATIONS
ndom thoughts on scrambling
Beni can
qubits
be viewedofasobservations
a state
2non
qubits.
• Quantum channel on n This
I. onEQUATIONS
is Yoshida
a collection
scrambling.
I.
EQUATIONS
Beni Institute
Yoshida for Quantum Information and Matter,
unitary operator as a state
of Technology,
Pasadena, California 91125, USA
tute forCalifornia
Quantum Institute
Information
and Matter,
X
X
U=
Ui,j(Dated:
|iihj|
|U91125,
i =12, 2015)
Ui,j |ii ⌦ |ji
August
ute of Technology,
Pasadena,
California
USA
i,j
i,j
August of
12,observations
2015)
This(Dated:
is a collection
on scrambling.
⇢in = {pj , | j i}
in
of observations on scrambling.
⌦
in
(1)
U
out
†
†
†
†
A(0)B(t)C(0)D(t)C (0)B (0)A (0)D (t)
⇢in = {pj , | j i}
↵
U
⇢out = {pj , | j i}
out
(1)
(2)
(1)
⇢out = {pj , | j i}
(2)
(1)
h
i
Xp
H
H
†
†
†
†
{pj ,i |( jt)e
i} 2 Bj (0)e 2 ' hAi ( t)Bj (0)A|i(2)
Tr Ai ( ⇢t)B
( it)B
(3)
out =
j (0)A
j (0)i
=
p
|
i ⌦ | ji
j
j
U
S
'
|A|
·
log
d
A
RECURRENCE
TIME
OF
A
PLANAR
PERFECT
TENSOR
NETWORK
SA ' |A| · log d
(2)
j
⇢in = {pj , | j i}
input
output
quantum
channel TENSOR NETWORK
IME OF A PLANAR
PERFECT
nsider a planar tensor network as shown in Fig. 1. This mimics the growth of the
I.
RECURRENCE TIME OF A PLANAR PERFECT TE
ein-Rosen
for the
interior
ofAai ⌦
two-sided
black
hole under
along
| 1.i =This
I |TFDi
(4)
r network bridge
as shown
in Fig.
mimics
the
growth
of the time-evolution
[Choi-Jamilkowski,
Hayden-Preskill,Hartman-Maldacena]
Perimeter
Institute
for Theoretical Physics
(Dated: April
14, 2016)
(Dated:
April 14, 2016)
(Dated: April 12, 2016)
Scrambling in a black hole
equations
ations
• Thermofield double state (finite T)
e e HH
⇢in⇢in==
TrTre e HH
| i=
X
e
⇢out
Ej /2
e
e
=
Tr e
iEj t
j
H
(1)
H
(1)
| ji ⌦ | ji
(1)
• A black hole geometry for the TFD state is the two-sided hole (AdS/CFT prediction)
†
†
†
†
hA(0)B(t)A
(0)B
(t)i
hA(0)B(t)A (0)B (t)i
(2)
length=O(T)
{U, ⇢in }
⌦
(2)
Let’s construct a tensor network toy
CFT 2
CFT 1
⌦
↵
↵
A(0)B(t)C(0)D(t)
A(0)B(t)C(0)D(t) {p ,U }
j
U=
X
j
(2)
{pj ,Uj }
Ui,j |iihj|
model !
(3)
(3)
[Maldacena]
(3)
a hyperbolic network, representing the two asymptotically AdS boundaries. This n
extends infinitely from the UV into the IR thermal scale at the black hole horizon.
the middle is flat representing the black hole interior. The entire network grows as
by adding more layers in the middle flat region.
We would like to further elaborate on this proposal of tensor network representa
the black hole interior. We will study networks of perfect tensors and demonstrate
dynamics
by finding ballistic
of local unitary
operators and the linear gro
unitary
operators,
tiling growth
the wormhole
geometry.
the tripartite information until the scrambling time. For the rest of discussion, w
the infinite temperature = 0 limit so we can ignore the hyperbolic part and focu
the planar tiling of tensor networks representing the interior.
CFT 1
Right CFT
CFT 2
AdS
ERB
AdS
igure 8: Tensor network representation of the Einstein-Rosen bridge. We will consider a
etwork of perfect tensors.
length=O(T)
Left CFT
time
• Consider a network of random
ynamics by finding ballistic growth of local unitary operators and the linear growth of
he tripartite information until the scrambling time. For the rest of discussion, we take
he infinite temperature = 0 limit so we can ignore the hyperbolic part and focus in on
he planer tiling of tensor networks representing the interior.
Toy model of the Einstein-Rosen bridge
Figure 10: Tensor network representation of the Einstein-Rosen bridge. A four-leg
random
unitary
operators
lives at each node. We will
consider
a network
of perfect tensors.
18
This model has the additional nice property of implementing the holographic quantum error co
proposal of [41].
[Hosur-Qi-Roberts-BY]
How do we probe the interior of a
black hole ?
s
Equations
Beni Yoshida
Perimeter Institute for Theoretical Physics
Out-of-time
ordered
correlation
functions
(Dated: May 27, 2016)
Beni Yoshida
stitute for Theoretical Physics
We 27,
should
measure some “hidden” correlations (Kitaev 2014)
Dated: •May
2016)
.
I.
OTO = hA(0)B(t)C(0)D(t)i
(1)
EQUATIONS
local operators
(0)eiHt
B(t) = e
iHt
B(0)eiHt
D(t) = e
iHt
D(0)eiHt
C
B
D(t) = e
iHt
D(0)eiHt
(1)
(1)
A
D
H : Z ! X X t!
: Z!X X!
=
EQUATIONS
Z
Z
(2)
(2)
0
d
X
j=1
p
| i=
|iAB i.
j |iC i•⌦
Previously
d
X
p
j=1
j |iC i
⌦ |iAB i.
(3)
(3) in 1960s, and recently by Shenker and Stanford
considered by Larkin and Ovchinikov
(Dated: April 12, 2016)
(Dated: May 27, 2016)
PDF for equations
OTO figures
Time
evolution
OTO
figures of operators
I.Beni
EQUATIONS
Yoshida
operators
• OTOCs detect the growth ofBeni
Yoshida
Perimeter Institute for Theoretical Physics
⌦
↵Physics
Perimeter Institute
for Theoretical
†
†
• Consider(Dated:
OTO
= 12,
A(0)B(t)A
April
2016) (0)B (t)
ions
(Dated: April 12, 2016)
[A, B] = 0
then
(1)
Non-commutativity between A(0) and B(t)
PDF for equations
{A, B} = 0
• Expand B(t):
OTO = 1
(commutator) (1)
then
OTO =
1
(2)
OTO = hA(0)B(t)C(0)D(t)i
X iHt X
iHt
B(t)
= e= Be↵iHt
=
↵iHt
j Pj
B(t)iHt= e iHt
Be
j Pj
iHt
(2)
iHt
B(t) = e
B(0)e
D(t) = e
j
D(0)e
j
(3)
(1) (1)
high-weight
| 0 iPauli operators
| 1i
scrambling
OTO ' 0
H 0: Z !
X X !OTO
Z =1
[A, B] =
then
[A, B] 6= 0
then
1
| 0 (t)i OTO
| 1 (t)i
(3)
(4) (2)
(2)
(3) (4)
[Roberts-Stanford-Susskind]
Key Questions
• How do we define scrambling ?
B(t) = e
iHt
BeiHt =
X
↵ j Pj
j
• Quantum information theoretic meaning of OTO ?
• Is the converse true ?
scrambling
?
OTO ' 0
I will relate OTOCs to entanglement entropies (joint with Hosur, Qi and Roberts)
[A, B] = 0
then
OTO = 1
[A, B] 6= 0
then
OTO  1
IBD = SB + SD
SBD
[Theorem] Average value of OTO
• Average of OTO over local operators A and D at T=infty
⌦
A(0)D(t)A† (0)D† (t)
average over A, D
↵
=
1 X⌦
4a+d
=2
A(0)D(t)A† (0)D† (t)
A,D
(2)
n a d SBD
A
↵
D
[Hosur-Qi-Roberts-BY]
IBD
=
S
+
S
B
D
(2)
(2)
(2)
IBD = SB + SD
S
(2) BD
SBD
(21
[Theorem] Average value of OTO
• Average of OTO over local operators A and D at T=infty
X
⌦ ⌦
↵
⌦
↵
X
1
↵
⌦
↵
1
† †
†
† †
†
†
A(0)D(t)A
= a+d
A(0)D(t)A
(0)D
(t)
A(0)D(t)A(0)D
(0)D†(t)
(t) =
A(0)D(t)A
(0)D
(t)
a+d
44 A,D
A,D
average over A, D
n a d
n a
=2
=2
A
(2)
SBD(2)
d S
(22
Pauli operators (unitary 1-design)
BD
(23
D
[Hosur-Qi-Roberts-BY]
IBD
=
S
+
S
B
D
(2)
(2)
(2)
IBD = SB + SD
S
(2) BD
SBD
(21
[Theorem] Average value of OTO
• Average of OTO over local operators A and D at T=infty
X
⌦ ⌦
↵
⌦
↵
X
1
↵
⌦
↵
1
† †
†
† †
†
†
A(0)D(t)A
= a+d
A(0)D(t)A
(0)D
(t)
A(0)D(t)A(0)D
(0)D†(t)
(t) =
A(0)D(t)A
(0)D
(t)
a+d
44 A,D
A,D
n a d
n a
=2
=2
average over A, D
input
(2)
SBD(2)
d S
(22
Pauli operators (unitary 1-design)
BD
(23
B
A
A
D
U
output
C
D
[Hosur-Qi-Roberts-BY]
IBD
=
S
+
S
B
D
(2)
(2)
(2)
IBD
(2) = SB
(2) + S(2)
D
IBD = SB + SD
S
(2) BD
S(2)
BD
SBD
[Theorem] Average value of OTO
(21
(21)
• Average of OTO over local operators A and D at T=infty
X
⌦ ⌦
↵
⌦
↵
X
1
↵
⌦
↵
1
† †
††
† † ↵ †
†
X ⌦ A(0)D(t)A
⌦A(0)D(t)A
↵
1
A(0)D(t)A
(0)D
(t)
=
A(0)D(t)A
(0)D
(t)
=
(0)D
(t) (t)
†
†
†
†(0)D
a+d
a+d
A(0)D(t)A (0)D (t) = 44a+d
A(0)D(t)A (0)D (t)
A,D
4
A,D
A,D
(2)
(2)
n a d S(2)
BD
n
a
d
S
=2n a d SBDBD
average over A, D
input
=2
=2
A
Pauli operators (unitary 1-design)
Renyi-2 entropy
B
A
(22
(22)
(23
(23)
D
U
output
C
D
[Hosur-Qi-Roberts-BY]
r Institute
Beni YoshidaIBD = SB + SD
(2)
(2)
(2)
for Theoretical Physics
S
(2) BD
IBD
S(2)
(2) = SB
(2) + S(2)
D
BD
IBD = SBPhysics
+ SD
SBD
meter
Institute
for
Theoretical
(Dated: April 12,
2016)
[Theorem]
Average value
(21
(21)
of OTO
(Dated: April 12, 2016)
• Average of OTO over local operators A and D at T=infty
X
OTO
figures
⌦ ⌦
↵
⌦
X
1
↵
⌦
1
†
†
↵
↵
† † ↵ †
†
†
†
X ⌦ A(0)D(t)A
⌦A(0)D(t)A(0)D
↵
1
A(0)D(t)A
(t)
=
A(0)D(t)A
(0)D
(t)
=
(0)D
(t) (t)
(22
†
†
†
†(0)D
a+d
a+d
A(0)D(t)A (0)D (t) = 44a+d
A(0)D(t)A (0)D (t)
(22)
A,D
4 Beni
A,D
A,D YoshidaPauli operators (unitary 1-design)
(2)
(2)
n a d S(2)
BD
n
a
d
S
average over A, D
=2
(23
n a d SBD
BD
Perimeter
Institute
for
Theoretical
Physics
=2
=2
(23)
X
(t) = e
iHt
BeiHt
A =
input
A† (0)D† (t)
↵
output
• If
B
Renyi-2 entropy
↵ j Pj
(Dated: April 12, 2016)
A
j
X⌦
PDF for
1U equations
=
4a+d
C
A(0)D(t)A† (0)D† (t)
DA,D
(2)
a d SBD is large
0 n then,
OTO '=2
(2)
(2)
↵
(2)
This implies the mutual information IBD = SB + SD
B] = 0 Xthen
(1)
D
(1)
(2)
(2)
(2)
SBD is small
B and D are not correlated, so the system is scrambling.
⌦
OTO
=1
E /2
†
†
A(0)D(t)A (0)D (t)
iE t
↵
=
1 X⌦
a+d
(3)
↵
†
†
[Hosur-Qi-Roberts-BY]
A(0)D(t)A (0)D (t)
r Institute
Beni YoshidaIBD = SB + SD
(2)
(2)
(2)
for Theoretical Physics
S
(2) BD
IBD
S(2)
(2) = SB
(2) + S(2)
D
BD
IBD = SBPhysics
+ SD
SBD
meter
Institute
for
Theoretical
(Dated: April 12,
2016)
[Theorem]
Average value
(21
(21)
of OTO
(Dated: April 12, 2016)
• Average of OTO over local operators A and D at T=infty
X
OTO
figures
⌦ ⌦
↵
⌦
X
1
↵
⌦
1
†
†
↵
↵
† † ↵ †
†
†
†
X ⌦ A(0)D(t)A
⌦A(0)D(t)A(0)D
↵
1
A(0)D(t)A
(t)
=
A(0)D(t)A
(0)D
(t)
=
(0)D
(t) (t)
(22
†
†
†
†(0)D
a+d
a+d
A(0)D(t)A (0)D (t) = 44a+d
A(0)D(t)A (0)D (t)
(22)
A,D
4 Beni
A,D
A,D YoshidaPauli operators (unitary 1-design)
(2)
(2)
n a d S(2)
BD
n
a
d
S
average over A, D
=2
(23
n a d SBD
BD
Perimeter
Institute
for
Theoretical
Physics
=2
=2
(23)
X
(t) = e
iHt
BeiHt
A =
input
A† (0)D† (t)
↵
output
• If
B
Renyi-2 entropy
↵ j Pj
(Dated: April 12, 2016)
A
j
X⌦
PDF for
1U equations
=
4a+d
C
A(0)D(t)A† (0)D† (t)
DA,D
(2)
a d SBD is large
0 n then,
OTO '=2
(2)
(2)
↵
(2)
This implies the mutual information IBD = SB + SD
(1)
D
(1)
(2)
(2)
(2)
SBD is small
B and D are not correlated, so the system is scrambling.
• For finite T, we will consider the so-called Thermofield double state.
B] = 0 Xthen
⌦
OTO
=1
E /2
†
†
A(0)D(t)A (0)D (t)
iE t
↵
=
1 X⌦
a+d
(3)
↵
†
†
[Hosur-Qi-Roberts-BY]
A(0)D(t)A (0)D (t)
39]), and then an explicit tensor network model was proposed in [15] (see also [40]).18
Before we begin, let us review the proposal of [12]. The tensor network representation
of the thermofield double state is shown in Fig. 10. At the left and right ends, we have
a hyperbolic network, representing the two asymptotically AdS boundaries. This network
extends infinitely from the UV into the IR thermal scale at the black hole horizon. Then,
the middle is flat representing the black hole interior. The entire network grows as t grows
This captures key properties of scrambling for “large-N” theories.
by adding more layers in the middle flat region.
We would like
to Ballistic
further elaborate
on this proposal
of tensor network
Eg)
propagations
of entanglement,
RTrepresentation
formula in of
a wormhole geometry…
the black hole interior. We will study networks of perfect tensors and demonstrate chaotic
dynamics Operator
by finding ballistic
local unitary
operators
theitlinear
size growth
growsof linearly,
OTO
will and
pick
up. growth of
the tripartite information until the scrambling time. For the rest of discussion, we take
the infinite
temperature
= 0 limit
so we can
ignore the hyperbolic
part scrambling
and focus in on time is log(n) [Cleve et al 2006]
For
a non-local
random
quantum
circuit, the
the planar tiling of tensor networks representing the interior.
Scrambling phenomena in a black hole
AdS
Figure 10: Tensor network representation of the Einstein-Rosen bridge. A four-leg tensor
ives at each node. We will consider a network of perfect tensors.
(46)
(45)
i1 ,...,i2n
25
8A s.t |A| = n.
Ti1 ,...,i2n |i1 , . . . , i2n i
=
SA = |A| · log v
ERB
r normalization. We call a tensor T perfect if it is a associated with a
i, called a perfect state, which is maximally entangled along any bipartition.
AdS
time
input op
Random unitary (maximally entangled)
| i=
Right CFT
X
Left CFT
by briefly reviewing a definition of perfect tensors. Consider a tensor T with
ond dimension v. A tensor can be represented as a pure state
•
stic propagation of entanglement
•
sor network representation of the Einstein-Rosen bridge. We will consider a
rfect tensors.
finding ballistic growth of local unitary operators and the linear growth of
information until the scrambling time. For the rest of discussion, we take
mperature = 0 limit so we can ignore the hyperbolic part and focus in on
ing of tensor networks representing the interior.
•
[Hosur-Qi-Roberts-BY]
OTO figures
Beni Yoshida
Scrambling in AdS black hole
Perimeter Institute for Theoretical Physics
(Dated: April 12, 2016)
RT surface
PDF for equations
A
TFD(0)
B
OTO figures
mutual information I(A, B)
is
'large
exp(
const · T )
Beni
Yoshida
Perimeter Institute for Theoretical Physics
(Dated: April 12, 2016)
PDF for equations
TFD(T)
A
(2)
(2)
(2)
IBD = SB + SD
B
(2)
SBD
OTO correlators will be small.
~T
⌦ information
↵ ' (almost)
⌦ · T)
1 X
mutual
zero
exp(
const
† I(A,
† B) is
A(0)D(t)A (0)D (t)
=
4a+d
=2
†
†
A(0)D(t)A (0)D (t)
A,D
(2)
n a d SBD
↵
(
What did we learn ?
Lesson 1
The AdS/CFT correspondence is a quantum error-correcting code.
Bulk quantum information is encoded
in boundary like a hologram.
very entangled tensor
(eg random tensor)
nds infinitely from the UV into the IR thermal scale at the black hole horizon. Then,
middle is flat representing the black hole interior. The entire network grows as t grows
dding more layers in the middle flat region.
We would like to further elaborate on this proposal of tensor network representation of
lack hole interior. We will study networks of perfect tensors and demonstrate chaotic
mics by finding ballistic growth of local unitary operators and the linear growth of
OTO correlator
is the
of space-time
ripartite information
until the scrambling
time.probe
For the rest
of discussion, we take
nfinite temperature = 0 limit so we can ignore the hyperbolic part and focus in on
planar tiling of tensor networks representing the interior.
(46)
8A s.t |A| = n.
(45)
Ti1 ,...,i2n |i1 , . . . , i2n i
with a proper normalization. We call a tensor T perfect if it is a associated with a
pure state | i, called a perfect state, which is maximally entangled along any bipartition.
Namely,
re 10: Tensor network representation of the Einstein-Rosen bridge. A four-leg tensor
at each node. We will consider a network of perfect tensors.
5.1
his model has the additional nice property of implementing the holographic quantum error correction
sal of [41].
| i=
X
i1 ,...,i2n
25
OTO correlator detects scrambling/
chaos
SA = |A| · log v
AdS
Let us begin by briefly reviewing a definition of perfect tensors. Consider a tensor T with
2n legs and bond dimension v. A tensor can be represented as a pure state
ERB
Ballistic propagation of entanglement
AdS
Right CFT
Figure 8: Tensor network representation of the Einstein-Rosen bridge. We will consider a
network of perfect tensors.
Left CFT
time
dynamics by finding ballistic growth of local unitary operators and the linear growth of
the tripartite information until the scrambling time. For the rest of discussion, we take
the infinite temperature = 0 limit so we can ignore the hyperbolic part and focus in on
the planer tiling of tensor networks representing the interior.
Lesson 2
(Dated: April 11, 2016)
Lesson 3
OTO correlators are the probes of space-time (seeing the interior of
a black hole)
OTO = hA(0)B(t)C(0)D(t)i
(1)
B
A
t
0
A
B
~T
1970s
Black hole
=
Black hole
=
2010s
[1] Holographic quantum error-correcting codes
[2] Chaos in quantum channel,
Thank you !
[3] Complexity by design, arXiv:1609:xxxxx
String
Daniel Harlow (Harvard
MIT)
Fernando Pastawski (Caltech
Berlin) John Preskill (Caltech)
String
CMT
Xiao-liang Qi (Stanford)
QI
QI
Daniel Roberts (MIT
IAS)
Simons collaborations