Area law and Quantum Information
description
Transcript of Area law and Quantum Information
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!!!