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)

18/07/31 Strings and Fields 2018@ YITP, Kyoto Entanglement Wedge Cross Section from Conformal Field Theory -Towards Entanglement of Purification for CFTs- Kotaro Tamaoka (Osaka U.) Based on [1803.10539] with Hayato Hirai, Tsuyoshi Yokoya (Osaka U.) PTEP 2018, no.6, 063B03 (2018)

[Umemoto-Takayanagi ’17], [Nguyen-Devakul-Halbasch-Zaletel-Swingle ’17] Question Can we get EWCS directly from CFT? Entanglement wedge cross section (EWCS) A bulk object, away from the boundary

B A Our result Yes! we can get the EWCS from CFT 2 4-point functions with twist number-1 channel − ∂ � → − ∂ � � � ∂ nG ( z, ¯ z ) ∂ nG σ n ( z, ¯ z ) = E W ( A, B ) − � � � � n → 1 n → 1 x 1 x 2 h O ( x 1 ) O ( x 2 ) ¯ O ( x 3 ) ¯ O ( x 4 ) i 1 = O G ( z, ¯ z ) | x 12 | 2 ∆ O | x 34 | 2 ∆ ¯ the leading contribution x 3 x 4 OO ∼ σ n for the holographic CFT

Outline 1. EWCS/EoP & “Replica method” 2. EWCS from Holographic CFT 2 3. Summary and Discussion

[Calabrese, Cardy ’04] EE from twist op.s correlation function (Replica method) S in = � ∂ � σ n � ∂ n h σ n ¯ σ n i � � n → 1 in out CFT n / Z n ！ Scaling dim. of twist op. → from Conformal WT id. ¯ σ n ✓ ◆ n − 1 ∆ n = c 12 n

̶ ̶ EWCS(EoP) as HEE for a new boundary x 1 x 2 AB A’ B’ A B A’ B’ x 3 x 4 AB EWCS: minimal surface of a new boundary!

c.f. Miyaji-Takayanagi ‘15 ̶ ̶ [Pastawski, Yoshida, Harlow, Preskill '15] EWCS(EoP) as HEE for a new boundary (H)EE → “Replica method” x 1 x 2 AB S AA 0 | min. φ n A’ B’ = � ∂ � ∂ n h Ψ opt. | φ n ¯ � φ n | Ψ opt. i � A B � n → 1 A’ B’ ~ 2pt function in the bulk! x 3 x 4 ¯ φ n AB Verification: EoP computation using holographic code model

[Ryu, Takayanagi ’06] AB A’ B’ B’ A B B A ̶ ̶ A’ AB ̶ For pure states, EoP → EE σ n φ n ¯ φ n ¯ σ n Take the size of → 0: the original RT formula AB

r → ∞ “Bulk twist op.” → twist op. in CFT Boundary condition φ n − → σ n m 2 φ n R 2 AdS = ∆ n ( ∆ n − 2) ! Within framework of the holographic code model, φ n is just the HKLL map of σ n ✓ ◆ n − 1 ∆ n = c 12 n

Outline 1. EWCS/EoP & “Replica method” → The twist op. exchange in the bulk is important ! 2. EWCS from Holographic CFT 2 → Can be obtained from twist op. conformal blocks 3. Summary and Discussion

A B CFT 2 4pt functions with twist number-1 channel 1 h O ( x 1 ) O ( x 2 ) ¯ O ( x 3 ) ¯ O ( x 4 ) i = O G ( z, ¯ z ) | x 12 | 2 ∆ O | x 34 | 2 ∆ ¯ Twist number (n+1)/2, for example x 1 x 2 Also, assume the holographic CFT; CB for twist op. σ n will dominate ! x 4 x 3 − ∂ � → − ∂ � � � ∂ nG ( z, ¯ z ) ∂ nG σ n ( z, ¯ z ) = E W − � � � � n → 1 n → 1

Conformal Block from AdS geodesics [Hijano-Kraus-Perlmutter-Snively ’15] O 1 O 2 A Conformal Block = X ( λ ) Z Z d λ 0 G ∆ , 0 bb ( X ( λ ) , Y ( λ 0 )) d λ φ ( ↔ O ) ∼ e − ∆ σ min Y ( λ 0 ) 1 ⌧ ∆ ' mR AdS O 3 O 4 At the semiclassical limit, only σ min contributes! (Now we are treating so-called “light operators” (1<< Δ i , Δ <<c))

B A CFT 2 4pt functions with twist number-1 channel captures the EWCS @ large-c − ∂ � → − ∂ � � � ∂ nG ( z, ¯ z ) ∂ nG σ n ( z, ¯ z ) = E W ( A, B ) − � � � � n → 1 n → 1 = c 1 σ min = E W 6 σ min = 4 G N x 1 x 2 12 ( n − 1 n ) σ min G σ n ∼ e − c [Hijano-Kraus-Perlmutter-Snively ’15] σ min. 1 h O ( x 1 ) O ( x 2 ) ¯ O ( x 3 ) ¯ O ( x 4 ) i = O G ( z, ¯ z ) x 4 x 3 | x 12 | 2 ∆ O | x 34 | 2 ∆ ¯

Extension to the BTZ blackholes BTZ t=0 σ min. σ min. ・ Start from the heavy-light correlators h O H |O L ( x 1 ) O L ( x 2 ) ¯ O L ( x 3 ) ¯ O L ( x 4 ) |O H i ・ Reduce to the heavy-light Virasoro CBs

B A Summary Can obtain the EWCS from CFT 2 4-point functions with twist number-1 channel − ∂ � → − ∂ � � � ∂ nG ( z, ¯ z ) ∂ nG σ n ( z, ¯ z ) = E W ( A, B ) − � � � � n → 1 n → 1 x 1 x 2 ・ EWCS for static BTZ blackhole ・ In the holographic code model, actually related to the EoP x 3 x 4 [Hirai, KT, Yokoya ‘18]

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 Future directions Path integral representation of 4pt functions

Back up

[Terhal-Horodecki-Leung-DiVincenzo '02] Entanglement of Purification a correlation measure for two subregions ˆ H A H B ρ AB H C | ψ i ABC H A H B ρ AB = Tr C | ψ ih ψ | s.t. H A 0 H B 0 E P ( A : B ) = min S ( ρ AA 0 ) H A H B | ψ i ABA 0 B 0