Entanglement and irreversibility on the light-cone

Gonzalo TorrobaCentro Atómico Bariloche

Argentina

It from Qubit Workshop. Bariloche, 2017

QFT has emerged as the framework for quantum many body systems, in high energy and CM physics

Fundamental problem: given a microscopic QFT, find its long distance dynamics

➡ Systematic approach: the renormalization group (RG)

• Physics at a scale described by EFT. • Integrate out d.o.f. with • Produces a flow in the space of couplings

⇤E > ⇤

EdgIdE

= �I(g)

Think about “space of QFTs”, RG flows & fixed points

K. G. Wilson and J. Kogut, The renormalization group and the expansion 169

on the critical surface between ~ and ~ the velocity cannot change direction withoutgoing through zero.The above discussion takes literally fig. 12.8, in which the space S is two-dimensional and the

critical surface is one-dimensional. The general definition of n-unstable fixed points for arbitraryn in an infinite-dimensional space S will be given later.

Fig. 12.8. Topology of the renormalization groupin a region having three fixed points PA.,.,, ~and PC,,,, on the critical surface. PA=, and ~

are once-unstable. ~B= is twice-unstable.

There are unique renormalized trajectories GA and Gc leaving the fixed points PA,, and Pc,,.,,analogous to the unique trajectory G in fig. 12.6 or the gully G in the classical analogue. Inaddition there is an infinite set of renormalized trajectories leaving the point RB.,’. The point PB,is analogous to the top of a hill in the classical analogue while PA,,., and Pc~are analogous tosaddle points. The appropriate contour map is given in fig. 12.9. The curves GB and Gj~in bothfigs. 12.8 and 12.9 are examples of trajectories leaving PB,,,,.

Fig. 12.9. Contour map of the topology of fig. 12.8.

When there are several fixed points an additional problem arises. Suppose one has a canonicalsurface (C in fig. 12.8). The question is which fixed point is connected via the renormalizationgroup to the canonical surface. In fig. 12.8 the renormalization group trajectories leaving thecanonical surface go to either P0 or PA,~.In the case of an interaction on the critical surface thetrajectory goes to PA~~’.So, the critical behavior is governed by PA’~~,not PB,. or Pa,., andrenormalization starting from the canonical surface is governed by the single renormalizedtrajectory GA.Associated with each fixed point ~JC”in S there is a domain D(~IC*)of interactions in S. The

domain D(~C*)is defined as the set of all initial interactions W0 E S such that the trajectory ~JC~

THE RENORMALIZATION GROUPAND THE ~EXPANSION

Kenneth G.WILSON

Institute for Advanced Study, Princeton, N.J. 08540, USAand Laboratory of Nuclear Studies, Cornell University, Ithaca, N Y. 14850, USA

andJ.KOGUT

Institute forAdvanced Study, Princeton, NJ. 08540, USA

INORTH~HOLLANDPUBLISHING COMPANY — AMSTERDAM

Crucial property: irreversibility of the RGIntuition: Loss of information about UV d.o.f.

E.g. C-thm

✓g-theorem

✓c-theorem

✓F-theorem

✓a-theorem

Use quite different tools: unitarity, dilaton, entanglement …

➡ Underlying principle for irreversibility of the RG?

➡ Useful way of comparing QFTs, and defining distance in theory space?

➡ What happens in D>4?

Results on RG evolution as dissipative process:

✓holographic C-thms

[Affleck, Ludwig; Friedan, Konechny …]

[Zamolodchikov; Cappelli, Friedan …]

[Myers, Sinha; Jafferis et al; Casini, Huerta]

[Cardy; Komargodski, Schwimmer …]

0 + 1

1 + 1

2 + 1

3 + 1

any D [Freedman et al; Myers, Sinha, …]

Review recent progress on understanding the RG using quantum information theory.

➡ Based on collaborations with Horacio Casini, Ignacio Salazar & Eduardo Testé, at Bariloche

Goal of the talk:

A. Relative entropy on the light-cone

B. Irreversibility & Markov property

Outline:

C. Work in progress & future directions

A. Relative entropy on the light-cone

“Ndof

”

• QIT offers an appropriate framework for this.To make progress, start from the simplest possible RG:

Boundary RG flows

• Need: measure for along the RG &way of comparing theories at different scales

2d CFT

conformalboundary

BCFT characterized by ;measures at boundaryN

dof

g

Sthermal =⇡c

3

T L+ log g

E.g. Kondo model

[Casini, Salazar, GT 2016]

“g-theorem”: decreases along boundary RG flowsg

BCFTUV

BCFTIR

RG due to relevant boundary deformation

log gUV > log gIR

[Affleck, Ludwig; Friedan, Konechny]

Understand using QIT? g measured by EE:

S(r) =c

6

log

r

✏+ log g

[Cardy]

S = SBCFT +

Zdx0 �O(x)

x1

x0

)r

) log

gIRgUV

= SIR(r)� SUV (r)

Monotonicity properties of ?�S

Suggests using the relative entropy!Define the reduced density matrices on interval r :

Same operator content.Evolve w/different action

Then

⇢: UV BCFT

: theory w/relevant deformation

Srel(⇢|�) = tr(⇢ log⇢

�)

✓measure of distance between states

= �hHi ��S✓Central in QIT and various physics appl's

tr(⇢H)� tr(�H)

H = � log �

CFT modular Hamiltonian

Irreversibility of the RG, measured by , could then follow from monotonicity and positivity of relative entropy. But …

�S

This simple idea does not quite work: generically �hHi � �S

S(⇢)� S(�) = log

g(r)

g(0)

� = trV̄ |0ih0|

Then try to minimize . For this, change Cauchy surface: x

0

x

1

�S is indep of ⌃

�H =

Z

⌃ds ⌘µ⇠⌫ �hTµ⌫i

normal to ⌃CKV that keeps

interval fixed

• static limit , x

0 = 0 H = 2⇡

Z r

0dx1 r2 � (x1)2

2rT00(x

1)

Since �hT00i ⇠ �(x1) ) �hHi ⇠ r relative entropy distinguishes too much

• null limit , x

� = �r

H = 2⇡

Z r

�rdx+ r2 � (x+)2

2rT++(x

+)

�hT++i ⇠ �(x+ + r) ) �hHi = 0 boundary sits in high temperature region

Srel(⇢r|�r) = � log

g(r)

g(0)&

dSrel(r)

dr� 0 ) g0(r) 0Then

but

�hHi

C & area theorems [Casini, Testé, GT,2016]

Apply this approach to RG flows in d-dimsCFTUV

CFTIR

reduced density matrices for sphere of radius r

�r = trV̄ |0ih0|⇢r

Compare both theories using and minimize by taking the null limit

Srel �hHi

S = SCFT +

Zd

dx�O(x)

dim � < d

�hTµ⌫i⌃ ⇠ �2 ✏d�2�(⌘µ⌘⌫ � gµ⌫d

)

) �hHi⌃ ⇠ �2 ✏d�2�

Z

⌃⌘µ⇠µ

properties of CFT stress tensor imply

• �hHinull ⇠ �2 ✏d+2�2�rd�2

• �hHix

0const

⇠ �2 ✏d�2�rd

area law! (high T)

For � <d+ 2

2

, �hHinull = 0 ) Srel(⇢r|�r) = S(�r)� S(⇢r)

Monotonicity of relative entropy then implies

• d = 2 : Srel(r) ⇡1

3

(cUV � cIR) log(mr) ) cUV > cIR

• d > 2 : Srel(r) ⇡ (µUV � µIR) rd�2 ) µUV > µIR

C-theorem

“area theorem”⇡ ��(1

4GN) rd�2

Lessons so far:✓ on light-cone provides useful statistical distance in QFT✓ Quantum-information meaning for✓ Irreversibility of RG in terms of increased distinguishability or information loss

Srel

�g , �c , �µ

B. Irreversibility & Markov property

For a CFT in d-dims, consider the EE on a sphere:

S(r) = µd�2rd�2

+ µd�4rd�4

+ . . .+

⇢(�1)

d/2�14A log(r/✏)

(�1)

[(d�1)/2]F

Irreversibility of RG: look for inequalities in

d even

d odd�A , �F

Requires positivity of higher derivatives of

[Casini, Testé, GT, 2017]

�S = S(⇢)� S(�)

Suggests looking at multiple regions. We now argue that for a CFT and regions with surface on light-cone, the SSA is saturated

S�(A) + S�(B)� S�(A \B)� S�(A [B) = 0

) �S(A) +�S(B)��S(A \B)��S(A [B) � 0

Saturation of strong-subadditivity

Focus first on null plane and then map to light-cone

t

x?

x1

null plane

A

B

A ^B

x

+

Quick argument: entropy for region w/boundary

y

S� x

+ = �(y)should be invariant under boosts .x

+ ! �x

+

Taking

x

+ = �(y)curve

assume Lorentzinv. cutoff

� ! 0 ) S� indep of � ) S�A + S�B = S�A[�B + S�A\�B

Conformal map to light-cone:

AB

A ^B

r� = r � t = �(⌦)

• Only dependence on the curve can come from the cutoff

• This has to be local and extensive• Terms classified using Lorentz inv

) S� =

Zdd�2⌦ f(�(⌦), ✏)

=

Zd2⌦

⇢↵1

�2

✏2+ ↵2

✓log

�

✏� 1

2

(

r�

�)

2

◆+ . . .

�e.g. d=4

Hence ray by ray the terms in the SSA inequality cancel out and we get a saturation of SSA.

Markov property of the vacuum

We now discuss a complementary perspective

SSA saturation , log ⇢A[B = log ⇢A + log ⇢B � log ⇢A\B

✓This is called a quantum Markov state✓Tracing out a subsystem becomes a reversible process✓Roughly, no entanglement over different null lines

• Markov property also follows from result for modular Hamiltonian

x?

x1

null plane

H� = 2⇡

Zdd�2y

Zdx+(x+ � �(y))T++

Rindler result, null line by line

Follows from OPE of twist operatorsand from algebraic QFT methods

See also [Lashkari; Faulker et al]

RG irreversibility in d dims

Using the geometric setup of [Casini, Huerta, 2012], consider SSA for for boosted spheres on light-cone�S

N ! 1• boosted spheres

• intersections and unions give “wiggly”spheres.

• Problem: unlike d=3, for general dlim

N!1Swiggly 6= Ssphere

Crucial role of Markov property: limN!1

�Swiggly = �Ssphere

Differences between wiggly and regular spheres are UV as N ! 1Hence in SSA for �Swiggly ! �Ssphere�S ,

in SSA combination

in SSA formula

Repeated application of SSA:

�S(prR) � 1

N

NX

k=1

�Sk ⇡Z R

rd` �(`)�S(`)

union of intersectionof k spheres

density of spheresof radius `

As R ! r , r�S00(r)� (d� 3)�S0(r) 0

Unifies c, F and a theorems, and predicts new inequalities in higher d

S(r) = µd�2rd�2

+ µd�4rd�4

+ . . .+

⇢(�1)

d/2�14A log(r/✏)

(�1)

[(d�1)/2]F

• d = 2 :

• d = 3 :

• d = 4 :

• general d :

Recall

(r�S0(r))0 0 ) �c(r) = c(r)� cUV = r�S0(r)

�S00(r) 0 ) �F (r) = r�S0(r)��S(r)decrease

monotonically

rS00⇢ (r)� S0

⇢(r) 8AUV

r) AIR AUV entropic a-thm

d

dr(�S0

rd�3) 0 , lim

r!1�µd�4 � 0 renormalization of

gravitational terms

C. Future directions

•Work in progress: It seems previous arguments for independence on curve also work for all Renyi entropies:

Sn(A) + Sn(B)� Sn(A \B)� Sn(A [B) = 0

�(y)

Much stronger than Markov. CFT vacuum behaves as product state over null cone. It is possible to characterize all entropies on light-cone. Consequences?

• C-theorems in higher dimensions? Monotonic fc. in d=4?

• Applications of simple entanglement structure on light-cone. QNEC, generalized second law, monotones, …