Entanglement Loss Along RG Flows Entanglement and Quantum Phase Transitions

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Entanglement Loss Along RG Flows Entanglement
and Quantum Phase Transitions

• José Ignacio Latorre
• Dept. ECM, Universitat de Barcelona
• Newton Institute, Cambridge, August 2004

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• 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

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Entanglement loss along RG flows

• Introduction
• Scaling of entropy
• Entanglement loss along RG flows
• Preview of new results

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• HEP
• Black hole entropy
• Conformal field theory

• Condensed Matter
• Spin networks
• Extensions of DMRG

Scaling of entropy

• Quantum Information
• Entanglement theory
• Efficient simulation

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Entanglement measures for many-qubit systems

• Few-qubit systems
• F ormation, Distillation, Schmidt coefficients,
• N3, tangle (for GHZ-ness) out of 5 invariants
• Bell inequalities, correlators based measures
• Entropy, negativity, concurrence,
• Many-qubit systems
• 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

Rezniks talk
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Reduced density matrix entropy

• Schmidt decomposition

A B
?min(dim HA, dim HB) is the Schmidt number
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The Schmidt number relates to entanglement
Lets compute the von Neumann entropy of the
reduced density matrix

• ?1 corresponds to a product state
• Large ? implies large superpositions
• e-bit

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• Maximum Entropy for N-qubits
• Strong subadditivity theorem
• implies concavity on a chain of spins

SmaxN
SLM
SL
SL-M
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• 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
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• 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

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Scaling of entropy for spin chains

• XY model
• Quantum Ising model in a transverse magnetic
  field
• Heisenbeg model

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XY plane
massive fermion
massive scalar
Quantum phase transitions occur at T0.
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Espectrum of the XY model
Jordan-Wigner transformation to spinless
fermions Lieb, Schultz, Mattis (1961)
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Fourier plus Bogoliubov transformation
For ?0, Ek?-cos(2pk/n)
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Coordinate space correlators can be reconstructed
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• 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

L
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Universality of scaling of entanglement entropy

• At the quantum phase transition point

Quantum Ising c1/2 free
fermion XY c1/2
free fermion XX c1
free boson Heisenberg c1
free boson
Universality
Logarithmic scaling of entropy controled by the
central charge
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• Conformal Field Theory
• A theory is defined through the Operator Product
  Expansion
• In d11, the conformal group is infinite
  dimensional
• the structure of descendants is fixed
• the theory is defined by Cijk and hi

Scaling dimensionsanomalous dimensions
Structure constants
Stress tensor
Central charge
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• Away from criticality

Saturation of entanglement
Quantum Ising
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• 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
Schmidt decomposition
A
SA SB ? Area Law
Entropy comes from the entanglement of modes at
each side of the boundary
Entanglement depends on the connectivity!
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Area law
Entanglement bonds
Area law in dgt11 does not depend on the mass
Valence bond representation of ground state
Plenios talk


Verstraetes talk
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• Entanglement in higher dimensions, Area Law,
  for free theories

c1 is an anomaly!!!!
Von Neumann entropy captures a most elementary
counting of degrees of freedom
Trace anomalies
Kabat Strassler

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• Is entropy scheme dependent is dgt11?

Yes
No
c11/6 bosons c11/12 fermionic
component
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Entanglement along quantum computation

• Spin chains are slightly entangled ? Vidals
  theorem
• Schmidt decomposition
• If
• Then
• The register can be classically represented in an
  efficient way!
• All one- and two-qubit gates actions are also
  efficiently simulated!!

?max(?AB)
?poly(n) ltlt en
A B
Quantum speed-up needs large entanglement !!!
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The idea for an efficient representation of
states is to store and manipulate information on
entanglement, not on the coefficients!!

• Low entanglement iff ai1,,? and ?ltlt en
• Representation is efficient
• Single qubit gates involve only local update
• Two-qubit gates involve only local update

Impressive performance when simulating d11
quantum systems!
Holy GrailExtension to higher dimensions
Cirac,Verstraete - Vidal
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• Entanglement in Shors algorithm (Orús)

?r
r small easy small entanglement
no need for QM r large hard large
entanglement QM exponential speed-up
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• Entanglement and 3-SAT
• 3-SAT
• 3-SAT is NP-complete
• K-SAT is hard for k gt 2.41
• 3-SAT with m clauses easy-hard-easy around
  m4.2n
• 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
instance
For every clause, one out of eight options is
rejected
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• Adiabatic quantum evolution (Farhi-Goldstone-Gutma
  nn)

t
s(T)1
H(s(t)) (1-s(t)) H0 s(t) Hp
s(0)0
Inicial hamiltonian
Problem hamiltonian
Adiabatic theorem
if
E
E1
gmin
E0
t
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• Adiabatic quantum evolution for exact cover

0gt
0gt
0gt
1gt
1gt
1gt
1gt
0gt
(0gt1gt)
(0gt1gt)
(0gt1gt)
(0gt1gt)
.
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Typical gap for an instance of Exact Cover
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Scaling is consistent with gap 1/n
If correct, all NP problems could be solved
efficiently! Be cautious
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• Scaling of entropy for Exact Cover

A quantum computer passes nearby a quantum phase
transition!
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n6-20 qubits 300 instances n/2 partition
S .2 n Entropy seems to
scale maximally!
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Scaling of entropy of entanglement summary
Non-critical spin chains S ct
Critical spin chains S log2 n
Spin networks in d-dimensions Area Law S nd-1/d
NP-complete problems S n
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• 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 dgt11
• Next?

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Entanglement loss along RG
RG flow loss of Quantum information

• RG flow loss of information

1. Global loss of entanglement along RG
2. Monotonic loss of entanglement along RG
3. Fine-grained loss of entanglement along RG

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• Global loss of entanglement along RG
• Monotonic loss of entanglement along RG

c-theorem
SLUV ? SLIR
-1
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• Majorization theory
• Entropy provides a modest sense of ordering among
  probability distributions
• Muirhead (1903), Hardy, Littlewood, Pólya,,
  Dalton
• Consider such that

p are probabilities, P permutations
d cumulants are ordered
D is a doubly stochastic matrix
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• Fine-grained loss of entanglement

L
?L
RG
t t
?Lt ?Lt
?1 ? ?1 ?1 ?2 ? ?1 ?2 ?1 ?2 ?3 ? ?1
?2 ?3 ..
Strict majorization !!!
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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
• Efficient computations in theories with c?1/2,1

• Exact eigenvalues, equal spacing
• Exact majorization along RG
• Detailed partition function

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Recent sets of results II

• R. Orús, E. Rico, J. Vidal, J.I.L.
• Lipkin model
• Full connectivity (simplex) ? symmetric states ?
  SLltLog L

?1

• Entropy scaling characterizes a
• phase diagram as in XY c1/2 !!!
• Underlying field theory? SUSY?
• Effective connectivity of d1

??1
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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