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(nl )
S ( l ) Trl 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
n1
1 ... n1
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
A23 A34 ....An1
2 ... n1
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 j1
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
A23 A34 ....An1
2 ... n1
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 a1
2m a 1
A
[1]i1
1 2
[ 2]i2
[ n ]in
n 1
A23 ....A
i ( x1 )i ( x2 )i ( xn )
1
2
n
2 ... n1
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 ) i1i2in 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 n1
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 55
i
i j a i
a
j i
j
2
55 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
Aq 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 )
logl 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
xy
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 22
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!!!