Transcript Document

Area law and Quantum Information
José Ignacio Latorre
Universitat de Barcelona
Cosmocaixa, July 2006
Bekenstein-Hawking black hole entropy
S BH
Ah

4G
Entanglement entropy
|  AB
A
B
 A  TrB |  AB  |
S A  Tr  A log2  A 
Goal of the talk
Entropy sets the limit for the simulation of QM
Area law in QFT
PEPS in QI
Some basics
Schmidt decomposition
H  H A  HB
A
|  AB 
B
dim H A dim H B
 A |u 
ij
i 1
j 1
i
A
| vj B
Aij  Uik kV  kl

|  AB   pi | i  A |  i  B
i 1
=min(dim HA, dim HB) is the Schmidt number
The Schmidt number measures entanglement

|  AB   pi | i  A |  i  B
i 1
Let’s compute the von Neumann entropy of the reduced density matrix

 A  TrB |  AB   |   pi | i  i |
i 1

S A  Tr  A log2  A    pi log2 pi  S B
i 1
=1 corresponds to a product state
Large  implies large number of superposed states
A
B
Srednicki ’93:
S A  S B  Area
Maximally entangled states (EPR states)
1
| A |B  | A |B 
|  
2
1
| A |B  | A |B 
|  
2


Each party is maximally surprised when ignoring the other one
1
 A   B  Tr |   | I
2


1 1
1
1
S A  S B   log2  log2   1
2 2
2
2
Ebits are needed for e.g. teleportation
(Hence, proliferation of protocoles of distillation)
1 ebit
Maximum Entropy for N-qubits
1
 N  N I 2N
2
1 
 1
S (  N )    N log2 N  N
2 
i 1  2
2N
Strong subadditivity
S ( A, B, C )  S ( B)  S ( A, B)  S ( B, C )
implies concavity
S L  M  2S L  S L  M d 2 S L

0
2
2
dL
Quantum computation
preparation
evolution
measurement
U
entanglement
simulation
quantum computer
How accurately can we simulate entanglement?
Exponential growth of Hilbert space
n
d
d
i1 1
in 1
|   ... ci1 ...in | i1...in 
Classical representation requires dn complex coefficients
A random state carries maximum entropy
l  Tr(nl )  
S ( l )  Trl log l   l log d
Efficient description for slightly entangled states
H  H A  HB
Back to Schmidt decomposition
A
|  AB 
B
dim H A dim H B

i1 1
 ci1i2 | i1 A | i2  B
ci1i2  U i1k pk Vki2
i2 1


|  AB   pk |  k  A |  k  B
k 1

ci1i2   
k 1
[1]i1
k
 = min(dim HA, dim HB)
Schmidt number
k 
A product state corresponds to
[ 2 ]i2
k
 1
d
i1 1
in 1
|   ... ci1 ...in | i1...in 
Vidal: Iterate this process
ci1...in 
d


 
      ....
[1]i1 [1]
1
1
[ 2]i2 [ 2]
1 2
[3]i3
2
2 3
[ n ]in
n1
1 ... n1
A product state iff
i  1
# parameters nd 2  d n
Slight entanglement iff poly(n)<< dn
• Representation is efficient
• Single qubit gates involve only local update
• Two-qubit gates reduces to local updating
efficient simulation
Small entanglement can be simulated efficiently
quantum computer more efficient than classical computer
if
large entanglement
Matrix Product States
d
d
i1 1
in 1
|   ... ci1 ...in | i1...in 
i
α
A[11]i12 A[22]i23 A[33]i34 A[44]i45 A[55]i56 A[66]i67 A[77]i78
ci1...in 
A

 
[1]i1
1 2
[ 2]i2
[3]i3
[ n ]in
A23 A34 ....An1
2 ... n1
canonical form
A
i
[i ]
[i ]
A
I
[ i ] [ i 1] [ i ]
[i ]
A

A



i
Approximate physical states with a finite  MPS
Graphic representation of a MPS
  1,, 

[ j ]i j
A j j1
i  1,, d
Efficient computation of scalar products
operations
d
2
nd 3
Intelligent way to represent entanglement!!
ci1...in 
A

 
[1]i1
1 2
[ 2]i2
[3]i3
A23 A34 ....An1
2 ... n1
Efficient
Efficient
Efficient
Efficient
[ n ]in
representation
preparation
processing
readout
Ex: retain 2,3,7,8 instead of 6,14,16,21,24,56
Matrix Product States for continuous variables
Iblisdir, Orús, JIL
Harmonic chains
1 n 2
2


H
p

x

x
 a a a1
2m a 1
 
A

 
[1]i1
1 2
[ 2]i2
[ n ]in
n 1
A23 ....A
i ( x1 )i ( x2 )i ( xn )
1
2
n
2 ... n1
MPS handles entanglement
  1,,   d
Truncate
Product basis
n
2
i  1,, d  
tr
dtr
 [ A] H  [ A]
H  [A]
 [A]
Nearest neighbour interaction
H   H i ,i 1
i
Minimize by sweeps
(periodic DMRG,
Cirac-Verstraete)

A[i ]
 [ A] H  [ A]
0
 [ A]  [ A]
Choose Hermite polynomials for local basis
i ( x)  exp(ax2 )hi ( x)
optimize over a
Results for n=100 harmonic coupled oscillators
(lattice regularization of a quantum field theory)
tr=3
dtr=3
dtr=4
dtr=5
dtr=6
tr=6
Newton-raphson on a
tr=4
tr=5
Quantum rotor
(limit Bose-Hubbard)
U
H   J  cos(i  i 1 ) 
2
i
2
i  2
i
Eigenvalue distribution for half of the infinite system
Simulation of Laughlin wave function
 ( z1 , , z n )   i  j ( zi  z j )e

1
2
 zi
2
i
1 2
 z
a
2
i ( z )  z e
Local basis: a=0,..,n-1
Dimension of the Hilbert space
nn
Analytic expression for the reduced entropy
n
S (n, k )  log   n n
k 
Exact MPS representation of Laughlin wave function
 ( z1 , , z n )   i  j ( zi  z j )e

1
2
 zi
2
i
( z1,, zn )  i j ( zi  z j )   i1i2in z1i1 z2i2 znin


a
,
b
 2
i1i2 in
ab
I
 Tr
i1
Clifford algebra
  dim  a  2
 n
  2     2n
k 
2
S
   5 
in
i2
n
2
 
 5   0 1  n1
 nn
Optimal solution!
(all matrices equal but the last)
 ( z1 , , zn )   i  j ( zi  z j ) e
m

1
2

zi
2
i
m=2
 i i i  j j  j  Tr i  i  i  5 Tr j  j  j  5 
12
n
1 2
n
1
2
n

1
 Tr  a1  an  55
i 
   
i  j a  i
a
j i
  

j
2

 55   5   5
optimal dim  a  2
n
m 
2
n
Spin-off?
Problem: exponential growth of a direct product Hilbert space
i1
in
i2
Computational basis
ci1 ...in
Neural network
MPS
NN
H
Product
states
MPS
Non-critical
1D systems
?
Spin-off 1: Image compression
| i|2i2,1
i11  
105|
i2=1
i2=2
i1=1
i2=3
i2=4
i1=2
RG addressing
i1=3
i1=4
|  image  


4
c
| i ...i 
i1 ...in
n
1
i1, ...,in 1
level of grey
  .... 



(1) i1
( n ) in
1 2
n 1
1 ,..., n 1
| in ...i1 
pixel address
Low
frequencies
QPEG
• Read image by blocks
high
frequencies
• Fourier transform
• RG address and fill
ci1 ...in
• Set compression level: 
• Find optimal
{(a) }
• gzip (lossless, entropic compression) of
{(a) }
• (define discretize Γ’s to improve gzip)
• diagonal organize the frequencies and use 1d RG
• work with diferences to a prefixed table
Max  = 81
=1
PSNR=17
=4
PSNR=25
=8
PSNR=31
Spin-off 2: Differential equations
O[ , x ] f ( x1,, xn )  0
f ( x1,, xn )  Tr( A[1]i1  A[n]in )i1 ( x1 )in ( xn )
min [ A] Of
2
Good if slight correlations between variables
Limit of MPS
1D chains, at the quantum phase transition point : scaling
c
S L L
 log 2 L

3
Quantum Ising , XY c=1/2
XX , Heisenberg
S
2
Universality
Vidal, Rico, Kitaev, JIL
Callan, Wilczeck
c=1
c
6
 2 L
Away from criticality: saturation
S L  N / 2
c
 log 2 | 1   |
6
MPS are a faithful representation for non-critical 1D systems
but deteriorate at quantum phase transitions
Exact coarse graining of MPS
   Ai Ai  Ai i1i2 in
1
2
n
p
A
Aq  A((pq)) 
min(d 2 ,  2 )
pq
U
 l lVl ,
l 1
Local basis
A  A'  lVl ,
p

RG
l
Optimal choice!
VCLRW
remains the same and locks the physical index!
After L spins are sequentially blocked
( L   )  2 Entropy is bounded
Exact description of non-critical systems
 
A
Area law for bosonic field theory
Geometric entropy
Fine grained entropy
Entanglement entropy

S
0
S
QFT
geometry
Srednicki ‘93
2



1 d  2
2 2
H   d x  ( x)    ( x)    ( x) 


2


Radial discretization
H   H l ,m
l ,m
1  2
1

H l ,m    l m, j   j  
2 j 1 
2

N
D 1
2
l m, j 
 l m, j 1

 ( D 1) / 2  
( D 1) / 2
( j)
 ( j  1)

 l (l  D  2)
2 2
l m, j
 


2
j


1 N 2 1 N
H   pi   xi Kij x j
2 i 1
2 i , j 1
 0 ( x1 ,, xN )    N / 4 det K 1/ 4 e
1
 xT  K  x
2
+ lots of algebra
Sl m,i
l m,i
  log(1  l m,i ) 
logl m,i
1  l m,i
S   Sl ,m
l ,m
Area Law for arbitrary dimensional bosonic theory

Riera, JIL
S
R2
Vacuum order: majorization of renduced density matrix
Eigenvalues of 
Majorization in L: area law
Majorization along RG flows
Majorization theory
Entropy provides a modest sense of ordering among probability distributions
Muirhead (1903), Hardy, Littlewood, Pólya,…, Dalton
Consider
d
 
x, y  R d
such that


x   p j Pj y
 
xy
k
d
x y
i 1
i
i
i 1
1
p are probabilities, P permutations
k
 x  y
i
i 1
i 1


x  Dy
i
d cumulants are ordered
D is a doubly stochastic matrix
 


x  y  H x   H ( y )
Vacuum reordering
t
L
 Lt
RG

t’
Lt’
Area law and gravitational anomalies
 L
S  c1  
 
d 1
n
d 1
d
c1 is an anomaly!!!!

 sm 2
e
eff   ds d / 2
s
s0
 c0


c
R

c
Fs

c
Gs




1
2F
2G
 s

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?
Yes
 L
S  c1  
 
d 1
No
c1=1/6 bosons
c1=1/12 fermionic component
Can we represent an
B
Area law?
A
SA= SB → Area Law
S ~ n(d-1)/d
A
i

 

Contour (Area) law
Locality symmetry
Efficient singular
value decomposition
BUT ever growing
Area Law and RG of PEPS
Projected
Entangled
Pair
A
i

 

i
A
A

 

A
'
 

k
l
A

' 

A

 

d

ijkl
 '
 '  '
 '

 '
'
A
PEPS can support area law!!
j
A
2

2 22

d4
Can we handle quantum algorithms?
Adiabatic quantum evolution
Farhi-Goldstone-Gutmann
H(s(t)) = (1-s(t)) H0 + s(t) Hp
s(0)=0
Inicial hamiltonian
t
Problem hamiltonian
Adiabatic theorem:
if
E
E1
gmin
t
E0
s(T)=1
3-SAT
– 3-SAT
0
1
1
0
0
1
1
0
instance
For every clause, one out of eight options is rejected
• 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
Beyond area law scaling!
n=6-20 qubits
300 instances
entropy
n/2 partition
s
S ~ .1 n
Orús-JIL
n=80
m=68 =10
T=600
Max solved n=100 chi=16 T=5000
Adiabatic evolution solved a n=100 Exact Cover!
1 solution among 1030
New class of classical algorithms:
Simulate quantum algorithms with MPS
Shor’s uses maximum entropy with equidistribution of eigenvalues
Summary
Non-critical spin chains
S ~ ct
Critical spin chains
S ~ log2 n
Spin chains in d-dimensions S ~ n(d-1)/d
(QFT)
Violation of area law!!
S ~ n1/2 log2 n
(some 2D fermionic models)
NP-complete problems
S ~ .1 n
Shor Factorization
S~r~n
Beyond area law? VIDAL:
Entanglement RG
Multiscale Entanglement Renormalization group Ansatz
?
Simulability of quantum systems
QMA
MPS
QPT
PEPS
finite 
MERA?
Area law
Physics ?
Quantum Mechanics
Classical Physics
+ classification of QMA problems!!!