Entanglement Loss Along RG Flows
Entanglement and Quantum Phase Transitions
José Ignacio Latorre
Dept. ECM, Universitat de Barcelona
Newton Institute, Cambridge, August 2004
•Entanglement in Quantum Critical Phenomena
G. Vidal, J. I. Latorre, E. Rico, A. Kitaev. Phys. Rev. Lett. 90 (2003) 227902
•Ground State Entanglement in Quantum Spin Chains
J. I. Latorre, E. Rico, G. Vidal. Quant. Inf. & Comp. 4 (2004) 48
•Adiabatic Quantum Computation and Quantum Phase Transitions
J. I. Latorre, R. Orús, PRA, quant-ph/0308042
•Universality of Entanglement and Quantum Computation Complexity
R. Orús, J. I. Latorre, Phys. Rev. A69 (2004) 052308, quant-ph/0311017
•Fine-Grained Entanglement Loss along Renormalization Group Flows
J. I. Latorre, C.A. Lütken, E. Rico, G. Vidal. quant-ph/0404120
Entanglement loss along RG flows




Introduction
Scaling of entropy
Entanglement loss along RG flows
Preview of new results
HEP
Condensed Matter
• Black hole entropy
• Conformal field theory
• Spin networks
• Extensions of DMRG
Scaling of entropy
Quantum Information
• Entanglement theory
• Efficient simulation
Entanglement measures for many-qubit systems

Few-qubit systems





Formation, Distillation, Schmidt coefficients,…
N=3, tangle (for GHZ-ness) out of 5 invariants
Bell inequalities, correlators based measures
Entropy, negativity, concurrence,…
Many-qubit systems
vac | Oi O j | vac connected
e |i  j|/ 

| i  j |

Scaling of correlators

Concurrence does not scan the system
We need a measure that obeys scaling and does
not depend on the particular operator content of a theory

Reznik’s talk
Reduced density matrix entropy

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  U ik kV  kl

|  AB   pi |  i  A |  i  B
i 1
=min(dim HA, dim HB) is the Schmidt number
The Schmidt number relates to 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 log 2  A    pi log 2 pi  S B
i 1
• =1 corresponds to a product state
• Large  implies large superpositions
• e-bit
1
 A   B  Tr |   | I
2


1 1
1
1
S A  S B   log 2  log 2   1
2 2
2
2

Maximum Entropy for N-qubits
1
 N  N I 2N
2

1 
 1
S (  N )    N log 2 N  N
2 
i 1  2
2N
Strong subadditivity theorem
S ( A, B, C )  S ( B)  S ( A, B)  S ( B, C )
Smax=N
implies concavity on a chain of spins
SL+M
S L M  S LM
SL 
2
SL
SL-M

n→ -party entanglement
Ground state reduced density matrix entropy
SL measures the quantum correlations with the rest of the system
Goal: Analyze SL as a function of L for relevant theories
Note that ground state reduced density matrix entropy SL

Measures the entanglement corresponding to the block spins
correlations with the rest of the chain
Depends only the ground state, not on the operator content of the
theory
(Relates to the energy-momentum tensor!!)
Scans different scales in the system: Is sensitive to scaling!!

Has been discussed in other branches of theoretical physics





Black hole entropy
Field Theory entanglement, conformal field theory
No condensed matter computations
Scaling of entropy for spin chains

XY model
1  x x 1  y y
z
 
 l  l 1 
 l  l 1   l 
2

l 1  2
N
H XY

Quantum Ising model in a transverse magnetic field
N

H QI    lx lx1   lz

l 1

Heisenbeg model
1 x x

    l  l 1   ly ly1   lz lz1    lz 

l 1  2
N
H XXZ
XY plane
massive fermion
massive scalar
Quantum phase transitions occur at T=0.
Espectrum of the XY model
1  x x
1  y y
z 
 
 l  l 1 
 l  l 1   l 
 2

2
l1
N
H XY


l

,  m  2i    lm l

Jordan-Wigner transformation to spinless fermions
Lieb, Schultz, Mattis (1961)
bi   zj i
j i
b , b  

i
j
ij
i, j
n
1 n 



H   bi 1bi  bi bi 1  bi 1bi  bi bi 1     bi bi
2 i 1
i 1
Fourier plus Bogoliubov transformation
1 n1 ijk 2 / n
ck 
e
bj

n j 0
n/2
H

k   n / 2 1
2k 

2
2 2k 
k k
   cos
   sin
n 
n

For γ=0, Ek=λ-cos(2πk/n)
 k | 0  0
  | 0  0
k
 k  uk ck  iv k ck
2
Coordinate space correlators can be reconstructed
Some intuition
The XY chain reduces to a gaussian hamiltonian
• We have the exact form of the vacuum
• We can compute exact correlators
The partial trace of N-L does not imply interaction
• Each k mode becomes a mixed state
| 0     (1)   ( 2)     ( L )  
L
  L  ~ (1)  ~ ( 2)   ~ ( L )

(k )
~ ( k )
1 0  0 0
, 

 
 0 0  0 1
 1  i

 2
 0


0 

1  i 

2 
1  k
 1  k 

L   
ck ck 
ck ck 
2
2

k 1 
L
1  k 1  k
1  k 
 1  k
S L   
log 2

log 2

2
2
2
2 
k 1 
L
Universality of scaling of entanglement entropy

At the quantum phase transition point
c
S L L
 log 2 L

3
Quantum Ising
XY
XX
Heisenberg
c=1/2
c=1/2
c=1
c=1
free fermion
free fermion
free boson
free boson
Universality
Logarithmic scaling of entropy controled by the central charge

Conformal Field Theory

A theory is defined through the Operator Product Expansion
Oi ( x)O j ( y ) 
 ij
| x y|
hi  h j

Cij
| x y|
k
hi  h j  hk
Ok ( y )  
Scaling dimensions=anomalous dimensions
Structure constants

In d=1+1, the conformal group is infinite dimensional:
the structure of “descendants” is fixed
the theory is defined by Cijk and hi
c
1
T ( z )T ( w) 

T ( w)  
4
2
| z w| | z w|
Stress tensor
Central charge

Away from criticality
Saturation of entanglement
Quantum Ising
S L  N / 2
c
 log 2 | 1   |
6

Connection with previous results





Srednicki ’93 (entanglement entropy)
Fiola, Preskill, Strominger, Trivedi ’94 (fine-grained entropy)
Callan, Holzey, Larsen, Wilczeck ’94 (geometric entropy)
Poor performance of DMRG at criticality
Area law for entanglement entropy
B
A
Schmidt decomposition
SA= SB → Area Law
Entropy comes from the entanglement of modes at each side of the boundary
Entanglement depends on the connectivity!
i
A
Area law
Entanglement bonds
Area law in d>1+1 does not depend on the mass
Valence bond representation of ground state
Plenio’s talk
Verstraete’s talk

Entanglement in higher dimensions, “Area Law”, for free theories
L
S  c1  
 
d 1
n
d 1
d
c1 is an anomaly!!!!

 sm2
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 scheme dependent is d>1+1?
Yes
L
S  c1  
 
d 1
No
c1=1/6 bosons
c1=1/12 fermionic component
Entanglement along quantum computation

Spin chains are slightly entangled → Vidal’s theorem

Schmidt decomposition

If
A

B
poly(n) << en
max(AB)
Then
The register can be classically represented in an efficient way!
All one- and two-qubit gates actions are also efficiently simulated!!
Quantum speed-up needs large entanglement !!!
The idea for an efficient representation of states is to store and
manipulate information on entanglement, not on the coefficients!!
d
d
i1 1
in 1
|    ... ci1 ... in | i1...in 
ci1 ... in 


 
      ....
[1]i1 [1]
1
1 ...
1
[ 2 ]i2 [ 2 ]
1 2
2
[ 3]i3
2 3
[ n ]in
n1
n1
Low entanglement iff αi=1,…, and << en
• Representation is efficient
• Single qubit gates involve only local update
• Two-qubit gates involve only local update
Impressive performance when simulating d=1+1 quantum systems!
Holy Grail=Extension to higher dimensions
Cirac,Verstraete - Vidal

Entanglement in Shor’s algorithm (Orús)
r
r small = easy = small entanglement
r large = hard = large entanglement
no need for QM
QM exponential speed-up

Entanglement and 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=4.2n
Exact Cover
A clause is accepted if 001 or 010 or 100
Exact Cover is NP-complete


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

Adiabatic quantum evolution for exact cover
|0> |1> |0> |1> |1> |0> |0> |1>
(|0>+|1>)(|0>+|1>) (|0>+|1>)…. (|0>+|1>)
Typical gap for an instance of Exact Cover
Scaling is consistent with
gap ~ 1/n
If correct, all NP problems could be solved efficiently! Be cautious

Scaling of entropy for Exact Cover
A quantum computer passes nearby a quantum phase transition!
n=6-20 qubits
300 instances
n/2 partition
S ~ .2 n
Entropy seems to scale maximally!
Scaling of entropy of entanglement summary
Non-critical spin chains
S ~ ct
Critical spin chains
S ~ log2 n
Spin networks in d-dimensions S ~ nd-1/d
“Area Law”
NP-complete problems
S~n
What has Quantum Information achieved?

“Cleaned” our understanding of entropy
Rephrased limitations of DMRG
Focused on entanglement
Represent and manipulate states through their entanglement
Opened road to efficient simulations in d>1+1

Next?



Entanglement loss along RG
flow =ofloss
of information
RG flowRG
= loss
Quantum
information
1. Global loss of entanglement along RG
2. Monotonic loss of entanglement along RG
3. Fine-grained loss of entanglement along RG

Global loss of entanglement along RG
+

c-theorem
Monotonic loss of entanglement along RG
-1
SLUV

SLIR

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
k
i
i 1
i
1
p are probabilities, P permutations
k
 x  y
i 1
x y
i 1


x   p j Pj y
 
xy
d
i
i 1
i
d cumulants are ordered


x  Dy
D is a doubly stochastic matrix
 


x  y  H x   H (y)

Fine-grained loss of entanglement
L
L
t
 Lt
RG

t’
Lt’
1  ’1
1 + 2  ’1 + ’2
1 + 2 + 3  ’1 + ’2 + ’3
……..
Strict majorization !!!
Recent sets of results I
Lütken, R. Orús, E. Rico, G. Vidal, J.I.L.

Analytical majorization along Rg
Exact results for XX and QI chains based on
Calabrese-Cardy hep-th/0405152, Peschel cond-mat/0403048
e H
L 
Tre  H
L 1
Z L   (1  e
k 0

L 1
H    k dk dk
k 0
 k
)

k 
2
2 LogL
2k  1
• Exact eigenvalues, equal spacing
• Exact majorization along RG
• Detailed partition function
Efficient computations in theories with c1/2,1
Recent sets of results II
R. Orús, E. Rico, J. Vidal, J.I.L.
Lipkin model
1
H 
N
  
i j
i
x
j
x
  yi  yj   h  zi
i
Full connectivity (simplex) → symmetric states → SL<Log L
=1
3/ 2
SL 
LogL
3
1
1
S L  LogL
3
• Entropy scaling characterizes a
phase diagram as in XY + c=1/2 !!!
• Underlying field theory? SUSY?
• Effective connectivity of d=1
Conclusion: A fresh new view on RG



RG on Hamiltonians
Wilsonian Exact Renormalization Group
RG on correlators
Flow on parameters from the OPE
RG on states
Majorization controls RG flows?
Lütken, Rico, Vidal, JIL
Cirac, Verstraete, Orús, Rico, JIL
The vacuum by itself may reflect irreversibility through a loss of entanglement
RG irreversibility would relate to a loss of quantum information