Towards Entanglement of Purification for Conformal Field Theories - - PowerPoint PPT Presentation
18/07/31 Strings and Fields 2018@ YITP, Kyoto Towards Entanglement of Purification for Conformal Field Theories Kotaro Tamaoka (Osaka U.) Based on [1803.10539] with Hayato Hirai, Tsuyoshi Yokoya (Osaka U.) PTEP 2018, no.6, 063B03 (2018)
Towards Entanglement of Purification for Conformal Field Theories
Kotaro Tamaoka (Osaka U.)
Based on [1803.10539] with Hayato Hirai, Tsuyoshi Yokoya (Osaka U.) PTEP 2018, no.6, 063B03 (2018)
18/07/31 Strings and Fields 2018@ YITP, Kyoto
Kotaro Tamaoka (Osaka U.)
Based on [1803.10539] with Hayato Hirai, Tsuyoshi Yokoya (Osaka U.) PTEP 2018, no.6, 063B03 (2018)
18/07/31 Strings and Fields 2018@ YITP, Kyoto
Entanglement Wedge Cross Section from Conformal Field Theory
A bulk object, away from the boundary
[Umemoto-Takayanagi ’17], [Nguyen-Devakul-Halbasch-Zaletel-Swingle ’17]
Question Can we get EWCS directly from CFT?
Entanglement wedge cross section (EWCS)
Our result
Yes! we can get the EWCS from CFT2 4-point functions with twist number-1 channel
− ∂ ∂nG(z, ¯ z)
− → − ∂ ∂nGσn(z, ¯ z)
= EW (A, B)
x1 x3 x2 x4
hO(x1)O(x2) ¯ O(x3) ¯ O(x4)i
= 1 |x12|2∆O|x34|2∆ ¯
O G(z, ¯
z)
OO ∼ σn
the leading contribution for the holographic CFT
A B
Outline
EE from twist op.s correlation function (Replica method)
in
σn ¯ σn
∆n = c 12 ✓ n − 1 n ◆
！Scaling dim. of twist op. → from Conformal WT id.
[Calabrese, Cardy ’04]
Sin = ∂ ∂n hσn¯ σni
CFTn/Zn
x1 x3 x2 x4
A B A’ A’ B’ B’ AB
̶
AB
̶
EWCS(EoP) as HEE for a new boundary
EWCS: minimal surface of a new boundary!
x1 x3 x2 x4
A B A’ A’ B’ B’
SAA0|min.
= ∂ ∂n hΨopt.|φn ¯ φn|Ψopt.i
φn ¯ φn
(H)EE → “Replica method”
~ 2pt function in the bulk! Verification: EoP computation using holographic code model
[Pastawski, Yoshida, Harlow, Preskill '15]
AB
̶
AB
̶
EWCS(EoP) as HEE for a new boundary
c.f. Miyaji-Takayanagi ‘15
For pure states, EoP → EE
A’ A’ B’ B’ A B B A
φn ¯ φn σn ¯ σn Take the size of →0: the original RT formula
AB
̶
AB
̶
AB
̶
[Ryu, Takayanagi ’06]
“Bulk twist op.” → twist op. in CFT
Boundary condition
! Within framework of the holographic code model, φn is just the HKLL map of σn
m2
φnR2 AdS = ∆n(∆n − 2)
r → ∞
∆n = c 12 ✓ n − 1 n ◆
Outline
→ The twist op. exchange in the bulk is important ! → Can be obtained from twist op. conformal blocks
CFT2 4pt functions with twist number-1 channel
hO(x1)O(x2) ¯ O(x3) ¯ O(x4)i =
1 |x12|2∆O|x34|2∆ ¯
O G(z, ¯
z)
− ∂ ∂nG(z, ¯ z)
− → − ∂ ∂nGσn(z, ¯ z)
= EW
Twist number (n+1)/2, for example
A B
x1 x2 x3 x4
Also, assume the holographic CFT; CB for twist op. σn will dominate !
Z dλ Z dλ0G∆,0
bb (X(λ), Y (λ0))
∼ e−∆σmin
1 ⌧ ∆ ' mRAdS
[Hijano-Kraus-Perlmutter-Snively ’15]
At the semiclassical limit, only σmin contributes!
O1 O2
O3
O4
φ(↔ O)
(Now we are treating so-called “light operators” (1<<Δi ,Δ<<c))
X(λ) Y (λ0)
Conformal Block from AdS geodesics
A Conformal Block =
= c 6σmin = 1 4GN σmin = EW
Gσn ∼ e− c
12(n− 1 n)σmin
CFT2 4pt functions with twist number-1 channel captures the EWCS @ large-c
hO(x1)O(x2) ¯ O(x3) ¯ O(x4)i =
1 |x12|2∆O|x34|2∆ ¯
O G(z, ¯z)
− ∂ ∂nG(z, ¯ z)
− → − ∂ ∂nGσn(z, ¯ z)
= EW (A, B)
A B
x1 x2 x3 x4
σmin.
[Hijano-Kraus-Perlmutter-Snively ’15]
Extension to the BTZ blackholes
σmin.
σmin.
BTZ t=0
・Reduce to the heavy-light Virasoro CBs
hOH|OL(x1)OL(x2) ¯ OL(x3) ¯ OL(x4)|OHi
・Start from the heavy-light correlators
Summary
Can obtain the EWCS from CFT2 4-point functions with twist number-1 channel
− ∂ ∂nG(z, ¯ z)
− → − ∂ ∂nGσn(z, ¯ z)
= EW (A, B)
・In the holographic code model, actually related to the EoP
x1 x3 x2 x4
A B [Hirai, KT, Yokoya ‘18]
・EWCS for static BTZ blackhole
Future directions
Path integral representation of 4pt functions
Extension to the higher dimension Calculation in RCFT or free theories
HKLL map for defects in the bulk?
Thank you !
“EWCS” for non-holographic CFT Twist number ±(n+1)/2 operators, for example
Entanglement of Purification
ˆ ρAB
[Terhal-Horodecki-Leung-DiVincenzo '02]
|ψiABC
a correlation measure for two subregions
HA HB
HA HB HC HA HB HA0 HB0
EP (A : B) = min
|ψiABA0B0
S(ρAA0) ρAB = TrC |ψihψ|
s.t.