Dynamics for holographic codes - QuTech...Definition.Thompson’s group ! is the group of piecewise...

Dynamicsforholographiccodes

TobiasOsborneDeniz Stiegemann

1706.08823arXiv:

JHEP06(2015)149

Published for SISSA by Springer

Received: April 22, 2015

Accepted: May 26, 2015

Published: June 23, 2015

Holographic quantum error-correcting codes: toy

models for the bulk/boundary correspondence

Fernando Pastawski,a,1 Beni Yoshida,a,1 Daniel Harlowb and John Preskilla

aInstitute for Quantum Information & Matter and Walter Burke Institute for Theoretical Physics,California Institute of Technology,1200 E. California Blvd., Pasadena CA 91125, U.S.A.

bPrinceton Center for Theoretical Science, Princeton University,400 Jadwin Hall, Princeton NJ 08540, U.S.A.

E-mail: [email protected], [email protected],

[email protected], [email protected]

Abstract: We propose a family of exactly solvable toy models for the AdS/CFT corre-

spondence based on a novel construction of quantum error-correcting codes with a tensor

network structure. Our building block is a special type of tensor with maximal entangle-

ment along any bipartition, which gives rise to an isometry from the bulk Hilbert space

to the boundary Hilbert space. The entire tensor network is an encoder for a quantum

error-correcting code, where the bulk and boundary degrees of freedom may be identified as

logical and physical degrees of freedom respectively. These models capture key features of

entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula

and the negativity of tripartite information are obeyed exactly in many cases. That bulk

logical operators can be represented on multiple boundary regions mimics the Rindler-

wedge reconstruction of boundary operators from bulk operators, realizing explicitly the

quantum error-correcting features of AdS/CFT recently proposed in [1].

Keywords: AdS-CFT Correspondence, Lattice Integrable Models

ArXiv ePrint: 1503.06237

1These authors contributed equally to this work.

Open Access, c⃝ The Authors.Article funded by SCOAP3.

doi:10.1007/JHEP06(2015)149

ThePoincarédisk

interior oftheunitdisk

distances aredifferent!


distancesaredifferent!

|c| = 1


geodesics

arediametersandcircles that

meettheboundaryatrightangles

tessellationintotriangles

allvertices ofthetriangleslieontheboundary

tessellationintotriangles

0 ⇠= 1

1/4

1/2

3/4

standard dyadictessellation

Definition. Thompson’s group 𝑇 is the group of piecewise linear homeomorphisms of the circle 𝑆# (understood as the interval 0, 1with endpoints identified) that• map dyadic rational numbers to dyadic rational numbers,• are differentiable except at finitely many dyadic rational numberssuch that• on intervals of differentiability the derivatives are powers of 2.

a

2n

(a, n 2 N)

Definition. Thompson’s group 𝑇 is the group of piecewise linear homeomorphisms of the circle 𝑆# (understood as the interval 0, 1with endpoints identified) that• map dyadic rational numbers to dyadic rational numbers,• are differentiable except at finitely many dyadic rational numberssuch that• on intervals of differentiability the derivatives are powers of 2.

a

2n

(a, n 2 N)

for example1

2,

3

4,

7

8

0 ⇠= 1

1/4

1/2

3/4

)',!((!1-$ /+&&+99%/$"*&:

;%

Cutoffs

�

4!"!#$ >"941+

�

4!"!#$ >"941+

*&"0$5 )+0$"*

�

4!"!#$ >"941+

*&"0$5 )+0$"*

#"4*5+5'#8*-&($'+*2%0$+"22$*$/+98'1%*8'.$&'$(!*(/3%/'

�0 � �

CdCd

Cd

CdCd

Cd

Cd

�

H� =

|�|O

n=1

Cd

CdCd

Cd

CdCd

Cd

Cd

holographicstatesareelementsoftheseHilbertspaces

Perfect Tensor

V : Cd ⌦ Cd ! Cd

=

=

=

=+)2+%)$%*/'4*5+)')"/%/$"*&

forexample,

?"9"0)%=3$

?"9"0)%=3$<

� �0 �00

H� H�0 H�00

H� H�0 H�00

H� H�0 H�00

2$*+C0)%$*$*0

$5)*#-3+&"$ 3"9"0)%=3$

Dynamics

Dynamics ?

Ordinaryquantummechanics

Thompson’sgroup

T

Definition. Thompson’s group ! is the group of piecewise linear homeomorphisms of the circle "# (understood as the interval $% &with endpoints identified) that! map dyadic rational numbers to dyadic rational numbers,! are differentiable except at finitely many dyadic rational numberssuch that! on intervals of differentiability the slopes are powers of '.

EF'GF'H%**"*B'GF'EF'I9"85B'GF'JF';%))8B'!"#$%&'(#%$)*+%#,-*%"*./(01$&*20%34-%"-5-*6$%'4-B'K*&F'L%/3F'MN'OPQQRSB'*"F'TUMB'NPVUNVR


0

1

1


1

8

1

4


1

2

3

4


1

2= 2�1

1 = 20

2 = 21

W+*+)%/")&'"2'!

EF'GF'H%**"*B'GF'EF'I9"85B'GF'JF';%))8B'!"#$%&'(#%$)*+%#,-*%"*./(01$&*20%34-%"-5-*6$%'4-B'K*&F'L%/3F'MN'OPQQRSB'*"F'TUMB'NPVUNVR

How does Thompson’s group T act

on tessellations,

cutoffs, and

holographic states?

Tactsontheboundary ofthedisk.

Tactsontheboundary ofthedisk.

(Allvertices ofthetessellationslieontheboundary.)

1/4

1/2

3/4

1/4

1/2

3/4

0 ⇠= 10 ⇠= 1

0 1/2

1

1/2

1

JF';+**+)B'LF'Y1#+)/B';F'Z"

|⌦�i 7! |fi ⌘ ⇡(f)|⌦�i

Theorem (Jones). The action

⇡(f)|gi ⌘ |fgi

defines a unitary representation of Thompson’s group T on the Hilbert space spanned by all states of the form

|fi ⌘ ⇡(f)|⌦�i.

V.F.R.Jones.arXiv:1607.08769

Wehavefoundagroupthat

matchesourchoiceoftessellations;✓


matchesourchoiceoftessellations;

hasaunitaryrepresentationonourchoiceofHilbertspaces;

✓

✓


canbeunderstoodasadiscreteversionofdiff+(S1).

matchesourchoiceoftessellations;

hasaunitaryrepresentationonourchoiceofHilbertspaces;

✓

✓

✓

FutureWork

FutureWork

fieldoperatorsforThompson’sgroup

FutureWork


MERA insteadoftrees

FutureWork


MERA insteadoftrees

other geometries&groups

In September 2018 I’ll be looking for postdoc positions.

ThedynamicsfortheseholographicstatesisgivenbyThompson’sgroupT.

Dynamics for holographic codes - QuTech...Definition.Thompson’s group ! is the group of piecewise...

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