Relative entropy in CFT
Tonomori Ugajin (KITP)
Works in collaboration with Gabor Sarosi ( Budapest-> Pennsylvania)[1603.03057]+ To appear

YKIS 2016 , Kyoto
1Motivation• Microstates of a black hole
• They are “almost” same -> The foundation of the statistical mechanicalinterpretation of BH entropy.
• The difference between them is also important for information loss .

・In addition to this, usally we employ a coarse grained view of the system,-> How can we quantify the difference betweenand?
2

Relative entropy cf[Ooguri san, van Raamsdonk, Harlow, Casini’s talk]It measures the distance between

andAsymmetric under the exchange
It can be written by the modular Hamiltonian3

Calculations of Relative entropy
A Calculation of relative entropy itself remains to be a difficult task.
Holographic computations: Available only for
RT formula + the vacuum modular Hamiltonian ex [Blanco Casini Hung Myers]
cf [ Ooguri san, van Raamsdonk’s talk]

CFT computations: 2d free scalar[ Lashkari]

4

Calculations of Relative entropy
A Calculation of relative entropy itself remains to be a difficult task.
Holographic computations: Available only for
RT formula + the vacuum modular Hamiltonian ex [Blanco Casini Hung Myers]..
cf [ Ooguri san, van Raamsdonk’s talk]

CFT computations: 2d Free scalar[ Lashkari]
Problem: Computein an arbitrary CFT.

for generic two excited states5Calculations of Relative entropy
A Calculation of relative entropy itself remains to be a difficult task.
Holographic computations: Available only for
RT formula + the vacuum modular Hamiltonian ex [Blanco Casini Hung Myers]..
cf [ Ooguri san, van Raamsdonk’s talk]

CFT computations: 2d Free scalar[ Lashkari]
Problem: Computein an arbitrary CFT.

for generic two excited states

Main claim: We found a general formula for it, in the small subsystem size limit.
6

Relative entropy in 2d CFT

7

Set up・A 2d CFTon a Cylinder

・ A subsystem

・Excited states0

at2πx
・Reduced density matrices

8Main result
In the small interval limit x->0, the leading behavior of relative entropy is given by

Set of lightest primary operators with9

Main result
In the small interval limit x->0, the leading behavior of relative entropy is given by

Set of lightest primary operators withAt the leading order, the relative entropy is captured by the differenceof the one point function of these excited states V and W.

10

Main result(2)When the lightest operator is the stress tensor,

11Sketch of the derivation• Step1: Introduce the replica trick[ Lashkari]• Step2 : Write each term by a correlation function on an n-sheeted plane:• Step 3: expand these correlation functions in the x->0 limit, using OPEs
[Alcaraz,Berganza,Sierra], [CarabreseCardy Tonni], [Headrick]+ a lot of papers
12

Example(1):Relative entropy in free boson theory• Lashkari computed the relative entropy between two chiral vertex
operators,
in free boson theory,・By using our formula, this leading behavior is explained by the exchange of the U(1) current
operator,with the conformal dimension
and the gamma function factor=1/3.

13

Ex(2):Relative entropy from the modular Hamiltonian• The relative entropy between an excited states V and the ground state has been
calculated in the somewhat different way by using the modular Hamiltonian, ex [Blanco Casini HungMyers]..

is the difference of entanglement entropy. When there are no primary operator
lighter than the stress tensor ,we can computein the small x limit from the vacuum
conformal block. ThenThis is a special case of our formula,14

Ex(3):Relative entropy between two generalized free fields
Generalized free fields =The CFT dual of bulk free scalar fields.

The correlation functions of the generalized free fields are calculated as if they are freefields (ie Wick contractions).
This property allows us to compute the relative entropy between two generalized free fields
without using the general formula,

We checked this result can be reproduced from the general formula.
15

Generalizations to higher dimensions

16

Relative entropy in d dimensional CFT• Higher dimensional generalizations of this calculation are straightforward.
• We start from a CFT on, and take the subsystem A : a cap like region.

• By using the Casini Huerta Myers conformal map,andare now computed bythe correlation functions onwith the periodicity
・ Technically, the most difficult part is the analytic continuationof the sum of two point functions
onThis has been done in [Agon, Faulkner] in last year.

A.
017

The result• When the lightest operator is the stress tensor, then,

CT: the coefficient of the two point function of the stress tensor.
energy density of the excited state V.

When
the result agrees with the holographic result : the RT
surface area calculation on the AdS black brane with. [Blanco, Casini Hung Myers]18

Relative entropy of two disjoint intervals

19

A trick[Jafferis Maldacena Lewkowitz Suh] [Dong Harlow Wall] [Leichnauer: private communication]

We start from the entanglement entropy of:

Slightly deform the density matrix :Since this is true for any deformation, we can read off the modular Hamiltonian from it

20

Relative entropy of two disjoint intervals

AB

We find that in the small subsystem limit,
the mutual information of the excited state is given byCf. [ Wolf Verstraete Hastings Cirac]21

Relative entropy of disjoint intervals

22

Relative entropy of disjoint intervals

23

Relative entropy of disjoint intervals

Asymmetric under the exchange
Although the relative entropy for a single interval starts from the symmetric term, the part involving both A
and B contains the asymmetric part from the beginning.
24Conclusions
Relative entropy in CFT.
Time dependent Process?
Bulk reconstruction? Radon transformation?
Another quantum information theoretic quantities?

25