YKIS 2016/06/13

Quantum Entanglement ofLocal Operators in Various CFTs
Masahiro Nozaki
Kadanoff Center for Theoretical Physics, The University of Chicago
1. arXiv:1401.0539 [hep-th]
2. arXiv:1405.5875 [hep-th]
3. arXiv:1405.5946 [hep-th]
4. arXiv:1507.04352[hep-th]
5. arXiv:1512.08132 [hep-th]
6. arxiv:16xx.xxxxx[hep-th]

Introduction
Recently, (Renyi) entanglement entropy((R)EE) has a center of wide interest in a broadarray of theoretical physics.
• It is useful to study the distinctive features ofvarious quantum state in condensed matterphysics.
• (Renyi) entanglement entropy is expected tobe an important quantity which may shedlight on the mechanism behind the AdS/CFT
correspondence .(Gravity ↔ Entanglement)

Introduction
Recently, (Renyi) entanglement entropy((R)EE) has a center of wide interest in a broadarray of theoretical physics.
• In the lattice gauge theory, it is expected thatentanglement entropy is a new orderparameter which helps us study QCD more.
• (R)EE is expected to be entropy in nonequilibrium system.

Introduction
Recently, (Renyi) entanglement entropy((R)EE) has a center of wide interest in a broadarray of theoretical physics.
• In the lattice gauge theory, it is expected thatentanglement entropy is a new orderparameter which helps us study QCD more.
• (R)EE is expected to be entropy in nonequilibrium system.

It is important to study the propertiesof (Renyi) entanglement entropy.

Motivation• Fundamental Property of Entanglement
Dynamics of Quantum Entanglement

Motivation• Fundamental Property of Entanglement
Dynamics of Quantum Entanglement• How amount of Quantum Informationis stored in Subsystem

Motivation• Fundamental Property of Entanglement
DynamicsofQuantum
Entanglement
Measuring(Renyi) Entanglement Entropy• How amount of Quantum Informationis stored in Subsystem

SetupWe study the property of (R)EE for
1. The size of subsystem is infinite.
A half of the total system:lz

SetupWe study the property of (R)EE for
2. A state is defined by acting witha local operator on the ground state:tx 1= - lBA
t=-tx1

Quantity
We would like to focus on the time evolution of the (R)EE.
We define: (R)EE forthe excess of the (R)EE:( Reduced Density Matrix for: (R)EE for the ground state)Quantity
Replica Method

τ
X1A

τ
X1

A

τ
X1

A

Quantity
Replica Method

τX12n-Point Function
τ
on
X1

AA

τ
X1

A

Quantity
Replica Method

τX12n-Point Function
τ
on
X1

A

2-Point Function
Aon

τ
X1

A

Quantity
Replica Method

τ
X1

A

This formulaτ holds for any local operators
A dimensions.
in general
QFT
inany
X1
τ
X1

A

Field Theory
1. Free massless scalar field theory
2. U(N) or SU(N) free massless scalar field theory
in Large N limit
3. Free massless fermionic field theory
4. Charged Renyi Entanglement Entropy (CREE)
5. Maxwell Theory in 4d

6. Holographic field theory

Free Field Theory(Massless Scalar)

Constant

At the late time,

ltlt

Holographic Field Theory

Free Field Theory(Massless Scalar)

Constant

At the late time,

ltlt

Holographic Field Theory

Free massless scalarS( 2)A

Example:t

Free massless fermion
Example:Red:
Green:Blue:
Orange: Log2t

Free massless scalarS( 2)A

Example:t
Depend on Theory and operators

Free massless fermion
Example:Red:
Green:Blue:
Orange: Log2

Depend on gamma matrix

Free massless scalarS( 2)A

Example:t
Depend on Theory and operators

How about Gauge Theory ?

Free massless fermion
Example:Red:
Green:Blue:
Orange: Log2

Depend on gamma matrixtQuasi-Particle
Example:for・Late time value:
Result by Replica Trick

We assume that late time value comes from entanglement
between quasi-particles

Quasi-Particle
We assume that late time value comes from entanglement
between quasi-particles.
We define an effective reduced density matrix:

ForDecomposition:

Quasi-Particle
We define an effective reduced density matrix:

ForDecomposition:=Left mover which corresponds to theparticles included in B
=Right mover which corresponds to theparticles included in A

Quasi-Particle
We define an effective reduced density matrix:

For

Quantization:

Quasi-Particle
We define an effective reduced density matrix:

For

Quantization:

ConsistentQuasi-Particle
We define an effective reduced density matrix:

For

How about Gauge Theory ?

Quantization:

ConsistentField Theory
1. Free massless scalar field theory
2. U(N) or SU(N) free massless scalar field theory
in Large N limit
3. Free massless fermionic field theory
4. Charged Renyi Entanglement Entropy (CREE)

5. Maxwell Theory in 4d
6. Holographic field theory

This work will appear on arXiv soon!
Collaborate with Naoki Watamura.

Our Claim
Subtleties : How to divide Hilbert space

Related to the D.O.F around the entangling surface

In terms of REE, the terms
which depends on UV cutoff
are related to the D.O.Faround entangling surface
For, these terms are subtracted .

∂AAB
on a certain time slice

This work will appear on arXiv soon!
Collaborate with Naoki Watamura.

Our Claim
Subtleties : How to divide Hilbert space

Related to the D.O.F around the entangling surface

Subtleties are negligible !!
In terms of REE, the terms
which depends on UV cutoff
are related to the D.O.F
around entangling surface
For, these terms are subtracted .

∂AABon a certain time slice

Results.1
We studyfor.

Time evolution ofshows that it is invariant
under the transformation:

Results.1
Time evolution ofshows that it is invariant
under the transformation:

S

Example:

( 2)
Aort

Results.1
Time evolution ofshows that it is invariant
for
for
under the transformation:

S

Example:

( 2)
Aort

Results.1
Time evolution ofshows that it is invariant
under the transformation:

forgrows faster thanfor

S

Example:

( 2)
Aort

Results.1
Time evolution ofshows that it is invariant
for
for
under the transformation:

They approach tosame number (log 2).
S

Example:

( 2)
Aort

Results.1
Time evolution ofshows that it is invariant
for
for
under the transformation:

This can be interpreted in termsof scalar quasi-particles
S

Example:

( 2)
Aort

Results. 2 Quasi-particle Interpretation
The late time values offor operators suchas
, which are notconstructed of both
andcan beinterpreted in terms of scalar quasi-particles.

Results. 2 Quasi-particle Interpretation
The late time values offor operators suchas
, which are notconstructed of both
andcan beinterpreted in terms of scalar quasi-particles.

The late time values offor operators suchas
, which are constructed of both
andcan not be interpreted in termsof scalar quasi-particles.

Results. 2 Quasi-particle Interpretation
The late time values offor operators suchas
, which are notconstructedof bothand quasi-particles
can beWe needelectromagnetic
interpreted in terms of scalar quasi-particles.

The late time values offor operators suchas
, which are constructed of bothandcan not be interpreted in termsof scalar quasi-particles.

Results. 2 Quasi-particle Interpretation
Example: Late time value offor

Results. 2 Quasi-particle Interpretation=Left mover which corresponds to thequasi-particles included in B
=Right mover which corresponds to thequasi-particles included in A

QuantizationC is real number.
same as scalar quasi-particles.

Quantization

C is real number.

Quantization

Feature of gauge field

C is real number.

Quantization
Electromagnetic fields can have the effect on thelate-time structure of entanglement differently
Unique for gauge theory
from scalar fields.
C is real number.

Summary• We check that
duality.
•respects electric-magnetic

for the operators constructed of bothelectric and magnetic fields can have the effecton the late-time structure of quantum
entanglement differently from scalar operator.

Summary• We check that
duality.
•respects electric-magnetic

for the operators constructed of bothelectric and magnetic fields can have the effecton the late-time structure of quantum
entanglement differently from scalar operator.

Future directions
・Weak Interacting F.T.
・Non-relativistic case・Maxwell Theory in general d・Non-local Operator

Propagator-Probability
Correspondence
Density matrix:
Smearing ParameterIn the path-integral formalism:We take ε → 0 .

Only three diagrams contribute at the leading
order :

We take ε → 0 .

Only three diagrams contribute at the leading
order :

Quasi-particles• Decomposition:• Quantization:• Reduced Density matrix:

Compare=Compare=Compare=Late time algebra:

Compare=In the same manner,
we derive the late-time algebra for gauge theory .

Late time algebra: