Area law and Quantum Information

José Ignacio LatorreUniversitat de Barcelona

Cosmocaixa, July 2006

Bekenstein-Hawking black hole entropy

G

AS h

BH 4

Entanglement entropy

A

B

AB|

AAA TrS 2log

|| ABBA Tr

Entropy sets the limit for the simulation of QM

Goal of the talk

Area law in QFT PEPS in QI

Schmidt decomposition

BiAii

iAB p

|||1

BjA

B

ij

A

vuA i

H

j

H

iAB

|||dim

1

dim

1klkikij

VUA

A B

=min(dim HA, dim HB) is the Schmidt number

BA HHH

Some basics

The Schmidt number measures entanglement

BiAii

iAB p

|||1

Let’s compute the von Neumann entropy of the reduced density matrix

Bi

iiAAA SppTrS

1

22 loglog

1

||||i

iiiABBA pTr

=1 corresponds to a product stateLarge implies large number of superposed states

A

B Srednicki ’93: AreaSS BA

Maximally entangled states (EPR states)

BABA ||||2

1| BABA ||||

2

1|

Each party is maximally surprised when ignoring the other one

ITrBA 2

1||

12

1log

2

1

2

1log

2

122

BA SS 1 ebit

Ebits are needed for e.g. teleportation

(Hence, proliferation of protocoles of distillation)

Maximum Entropy for N-qubits

Strong subadditivity

implies concavity

NINN 22

1 NS

N

iNNN

2

12 2

1log

2

1)(

),(),()(),,( CBSBASBSCBAS

02

22

2

dL

SdSSS LMLLML

Uentanglement

preparation evolution measurement

quantum computer

simulation

Quantum computation

How accurately can we simulate entanglement?

Exponential growth of Hilbert space

d

i

d

inii

n

niic

1 11...

1

1...|...|

Classical representation requires dn complex coefficients

n

A random state carries maximum entropy

)( lnl Tr

dlTrS lll loglog)(

Efficient description for slightly entangled states

BkAkk

kAB p

|||1

BA

H

i

H

iAB iic

B

ii

A

21

dim

1

dim

1

|||2

21

1

2121 kikkiii VpUc

A B

= min(dim HA, dim HB) Schmidt number

BA HHH Back to Schmidt decomposition

1

]2[]1[ 21

21k

ikk

ikiic

A product state corresponds to 1

d

i

d

inii

n

niic

1 11...

1

1...|...|

n

n

n

n

iniiiiic

][

...

]3[]2[]2[]1[]1[... 1

11

3

322

2

211

1

11....

Slight entanglement iff poly(n)<< dn

• Representation is efficient• Single qubit gates involve only local update• Two-qubit gates reduces to local updating

Vidal: Iterate this process

A product state iff 1i

ndndparameters 2#

efficient simulation

Small entanglement can be simulated efficiently

quantum computer more efficient than classical computerif

large entanglement

Matrix Product States

d

i

d

inii

n

niic

1 11...

1

1...|...|

1

21

]1[ iA 2

32

]2[ iA 3

43

]3[ iA 4

54

]4[ iA 5

65

]5[ iA 6

76

]6[ iA 7

87

]7[ iA

n

n

n

n

iniiiii AAAAc ][

...

]3[]2[]1[1... 1

12

3

43

2

32

1

21....

i

α

Approximate physical states with a finite MPS

IAA i

i

i ][][ ][][]1[][ iii

i

i AA canonical form

Graphic representation of a MPS ,,1

di ,,1j

jj

ijA ][

1

Efficient computation of scalar products

operations2d

3nd

n

n

n

n

iniiiii AAAAc ][

...

]3[]2[]1[1... 1

12

3

43

2

32

1

21....

Intelligent way to represent entanglement!!

Ex: retain 2,3,7,8 instead of 6,14,16,21,24,56

Efficient representationEfficient preparationEfficient processingEfficient readout

Matrix Product States for continuous variables

211

2

2

1

aa

n

aa xxp

mH

)()()(.... 21][1

...

]2[]1[1 21

12

2

32

1

2 niiiinii xxxAAA

n

n

n

n

Harmonic chains

MPS handles entanglement Product basis

di ,,1

Truncate tr dtr

2,,1n

d

Iblisdir, Orús, JIL

][][ AHA

i

iiHH 1,Nearest neighbour interaction

][AH

][A

0][][

][][][

AA

AHA

A i

Minimize by sweeps(periodic DMRG,Cirac-Verstraete)

Choose Hermite polynomials for local basis )()exp()( 2 xhaxx ii

optimize over a

Results for n=100 harmonic coupled oscillators(lattice regularization of a quantum field theory)

dtr=3 tr=3

dtr=4 tr=4

dtr=5 tr=5

dtr=6 tr=6

Newton-raphson on a

Quantum rotor(limit Bose-Hubbard)

Eigenvalue distribution for half of the infinite system

i i i

ii

UJH

2

2

1 2)cos(

Simulation of Laughlin wave function

i

iz

jijin ezzzz2

2

1

1 )(),,(

2

2

1

)(za

i ezz

Local basis: a=0,..,n-1

Analytic expression for the reduced entropy nnk

nknS

log),(

Dimension of the Hilbert space nn

i

iz

jijin ezzzz2

2

1

1 )(),,(

nn in

iiiiijijin zzzzzzz 2121

211 )(),,(

52121 nn iiiiii Tr

Iabba 2, 1105

n

nn

a n

22dim

Exact MPS representation of Laughlin wave function

Clifford algebra

nS

k

n222

Optimal solution!

(all matrices equal but the last)

5521212121 nnnn jjjiiijjjiii TrTr

551

naaTr

aji

jia

i

ji

m=2

i

izm

jijin ezzzz2

2

1

1 )(),,(

5555

22dimn

maoptimal

Spin-off?

Problem: exponential growth of a direct product Hilbert space

Computational basis

MPS

Neural network

i1 i2in

niic ...1

MPS

Product states

H

NN

Non-critical1D systems

?

11,...,

)()1(

1

4

1...,...

...|....

...||

1

1

1

21

,1

1

ii

iic

nini

nii

iiimage

n

n

n

n

n

i1=1 i1=2

i1=3 i1=4

| i1 i2=1 i2=2

i2=3 i2=4

| i2 i1 105| 2,1

Spin-off 1: Image compression

pixel addresslevel of grey

RG addressing

QPEG

• Read image by blocks

• Fourier transform

• RG address and fill

• Set compression level:

• Find optimal

• gzip (lossless, entropic compression) of

• (define discretize Γ’s to improve gzip)• diagonal organize the frequencies and use 1d RG• work with diferences to a prefixed table

niic ...1

Lowfrequencies

highfrequencies

}{ )(a

}{ )(a

= 1

PSNR=17 = 4

PSNR=25 = 8

PSNR=31

Max = 81

Spin-off 2: Differential equations

0),,(],[ 1 nxxfxO

)()()(),,( 1][]1[

1 1

1nii

inin xxAATrxxf

n

n

2

][min OfA

Good if slight correlations between variables

Limit of MPS

1D chains, at the quantum phase transition point : scaling

Lc

SLL 2log

3

Quantum Ising , XY c=1/2 XX , Heisenberg c=1

Universality

|1|log6 22/ c

S NLAway from criticality: saturation

MPS are a faithful representation for non-critical 1D systems but deteriorate at quantum phase transitions

Vidal, Rico, Kitaev, JILCallan, Wilczeck

622cS

L

Exact coarse graining of MPS

niii iiiAAA n 21

21

,

),min(

1

)()(

22

ll

d

l

pql

pqqp VUAAA

,' lllRGp VAA Optimal choice!

VCLRW

remains the same and locks the physical index!

After L spins are sequentially blocked

2)(

LA Entropy is bounded

Exact description of non-critical systems

Local basis

Area law for bosonic field theory

Geometric entropyFine grained entropyEntanglement entropy

S

QFT

0 Sgeometry

)()()(2

1 222

2 xxxxdH d

Radial discretization

ml

mlHH,

,

2

12/)1(

,

2/)1(

1,1

2,, )()1(2

1

2

1 N

jD

jml

D

jmlD

jmlml jjjH

2

,2

2

)2(jmlj

Dll

Srednicki ‘93

N

jijiji

N

ii xKxpH

1,1

2

2

1

2

1

xKxNN

T

eKxx 2

14/14/

10 det),,(

iml

iml

imlimlimlS ,

,

,,, log

1)1log(

ml

mlSS,

,

+ lots of algebra

Area Law for arbitrary dimensional bosonic theory Riera, JIL

2R

S

Vacuum order: majorization of renduced density matrix

Eigenvalues of Majorization in L: area lawMajorization along RG flows

Majorization theory

Entropy provides a modest sense of ordering among probability distributions

Muirhead (1903), Hardy, Littlewood, Pólya,…, Dalton

Consider such that dRyx

,

d

i

d

iii yx

1 1

1

yx

yPpx jj

p are probabilities, P permutations

k

ii

k

ii yx

11

d cumulants are ordered

yDx

D is a doubly stochastic matrix

)(yHxHyx

L

Lt L

t’

t t’RG

Vacuum reordering

Area law and gravitational anomalies

d

dd

nL

cS11

1

c1 is an anomaly!!!!

GscFscRcs

c

s

eds GF

sd

sm

eff 2210

2/

0

2

Von Neumann entropy captures a most elementary counting of degrees of freedom

Trace anomalies Kabat – Strassler

Is entropy coefficient scheme dependent is d>1+1?

1

1

dL

cS

Yes

No

c1=1/6 bosons c1=1/12 fermionic component

A

B

SA= SB → Area Law

Contour (Area) law

S ~ n(d-1)/d

Can we represent an

Area law?

Locality symmetry

iA

iA

ijkllk

ji

AAA

AA

''

''

''

'

'

4

22

22

dd AA

Efficient singularvalue decompositionBUT ever growing

Area Law and RG of PEPS

ProjectedEntangledPair

PEPS can support area law!!

Can we handle quantum algorithms?

Adiabatic quantum evolution Farhi-Goldstone-Gutmann

H(s(t)) = (1-s(t)) H0 + s(t) Hp

Inicial hamiltonian Problem hamiltonian

s(0)=0 s(T)=1t

Adiabatic theorem:

if

E1

E0

E

t

gmin

3-SAT

– 3-SAT

• 3-SAT is NP-complete• K-SAT is hard for k > 2.41• 3-SAT with m clauses: easy-hard-easy around m=4.2

– Exact Cover

A clause is accepted if 001 or 010 or 100

Exact Cover is NP-complete

0 1 1 0 0 1 1 0

For every clause, one out of eight options is rejected

instance

Beyond area law scaling!

n=6-20 qubits

300 instances

n/2 partition

S ~ .1 n

Orús-JIL

entropy

s

n=80 m=68 =10 T=600 Max solved n=100 chi=16 T=5000

New class of classical algorithms:

Simulate quantum algorithms with MPS

Shor’s uses maximum entropy with equidistribution of eigenvalues

Adiabatic evolution solved a n=100 Exact Cover!1 solution among 1030

Non-critical spin chains S ~ ct

Critical spin chains S ~ log2 n

Spin chains in d-dimensions

(QFT)

S ~ n(d-1)/d

Violation of area law!! (some 2D fermionic models)

S ~ n1/2 log2 n

NP-complete problems S ~ .1 n

Shor Factorization S ~ r ~ n

Summary

Beyond area law? VIDAL: Entanglement RG

Multiscale Entanglement Renormalization group Ansatz

Simulability of quantum systems

QPT MERA?

PEPSfinite Physics ?

QMA?

Area law

MPS

Quantum Mechanics

Classical Physics

+ classification of QMA problems!!!