Holographic Entanglement of Purification

Transcription

1 20 Years Later: The Many Faces of Princeton, Oct.31-Nov.3, 2017 Holograhic ntanglement of Purification Tadashi Takayanagi Yukawa Institute for Theoretical Physics (YITP, Kyoto Univ. ased on arxiv: with Koji Umemoto (YITP Closely related work: arxiv: by Nguyen-Devakul-Halbasch-Zaletel-Swingle

2 1 Introduction ds/cft [Maldacena 1997, Gubser-Klebanov-Polyakov, Witten 1998,..] ``Geometrization of Dynamics in CFTs One manifestation of this feature is Quantum entanglement of CFTs ulk Geomety dy CFTd ulk dsd+1 Γ S Minimal surface Holograhic ntanglement ntroy Trρ logρ rea( Γ 4G [Ryu-TT 2006, Hubeny-Rangamani-TT 2007] [Derivation: Casini-Huerta-Myers 2009, Lewkowycz-Maldacena 2013] N

3 mergent sacetime from ntanglement [Swingle 2009, Raamsdonk 2009,.] Quantum States lgebraically comlicated, but geometrically look nicer! Γ? Tensor Network Γ ds SMin #(Γ TN In this talk, we would like to resent a new conjecture which generalizes the holograhic entanglement entroy.

4 Contents 1 Introduction 2 ntanglement Wedges 3 ntanglement of Purification 4 Holograhic ntanglement of Purification 5 Derivation via Tensor Network 6 Conclusions

5 2 ntanglement Wedges Which bulk region is dual to a given region in CFT? ntanglement Wedge M (note: we took a time slice M region surrounded by and Γ (on time slice Γ M Minimal Surface ρ in CFT ρ M in ds gravity [Hamilton-Kabat-Lifschytz-Lowe 2006, Czech- Karczmarek-Nogueira-Raamsdonk 2012, Wall 2012, Headrick-Hubeny-Lawrence-Rangamani 2014, Jafferis-Lewkowycz-Maldacena-Suh 2015, Dong-Harlow-Wall 2016,... ]

6 More recise definition of W ntanglement Wedge Domain of deendence of M Γ Note: Our arguments below assume a static asymtotically ds sacetime, which has a canonical time slice. However, it is straightforward to extend our arguments to general non-static setus in a covariant way.

7 W for Disconnected Subregions I ( : S + S S 0 Γ Γ M M Γ M Σ MM M, Γ Γ Γ I ( : > 0 Σ Minimal Surface which divides M into side and side

8 ntanglement Wedge Cross Section We define a quantity called W cross section by W rea( Σ 4G N M Σ

9 Our Conjecture W P ntanglement of Purification [Terhal-Horodecki-Leung-Divincenzo quant-h/ ] ρ Note: When is a ure state, we simly have W P S S.

10 3 ntanglement of Purification (op Purification given density matrix for : We can always describe this state as a ure state by extending the Hilbert sace: H C such that ρ C H C H Note: there are infinite many ways to do this. D H C Tr [ D ]. ρ C CD λi i i λ i i i C C i i. D

11 Definition of op ρ Consider all urifications ~ ~ of in the extended Hilbert sace: H H H H H ~ H ~. Then, ntanglement of Purification (op is defined by P Min ( ~ S ~ ~ ll urifications of ρ ρ Tr ~[ ~ ]. ntanglement ntroy Note: ( ρ 0 and 0 ρ ρ ρ.

12 Proerties of op [agchi-pat, ] [1] In general,. [2] [3] If ρ is ure, then I( : / 2. min{ S ( C I( : / 2 +, S }. S I( : C / S 2.. [4] If is ure, then [5] ρ C + ( C ( C. C. [Polygamy] Comment: These all follows from strong subadditivity.

13 Oerational Interretation of op ρ If is a ure state (, ntanglement ntroy C ρ S Tr[ ρ logρ # of PR airs ( needed to create via LOCC # of PR airs ( D we can create from via LOCC. Distillation ρ Cost When is a mixed state, we find C in general. ntanglement measure # for mixed states should satisfy D # ] Local Oerations and Classical Communications D C.

14 The entanglement of urification can be identified with the ntanglement Cost (c for the LOq rocess. P # of PR airs needed to create via LOq. ρ LOq Local Oerations + small number of communications. (Fact: LOq is enough to a convertion PR airs a ure state However, note that op is not a entanglement measure, but a correlation measure between and. D SQ C F P.

15 4 Holograhic ntanglement of Purification To test our conjecture, let us confirm the roerties of. W W P Proerty [1] min{ S, S }. If is ure, then S S. ρ Γ Σ ΓΣ

16 Proerty [2] rea( Γ rea( Γ rea( Γ rea( Γ rea( Γ + rea( Γ rea( Σ + rea( Σ 2 rea( Σ rea( Γ + rea( Γ rea( Γ Proerty [3] I( : / 2 Monogamy Mutual Info. I( : C I( : + [Freedman-Headrick 2016] holds for holograhic. [Hayden-Headrick-Maloney 2011] ( C I( : / 2 +. I( : C I( : C / 2. Proerty [4] Γ + ( C C C

17 Proerty [5] W nesting roerty: Therefore, C ( C C D M. M C M D M Strong Suer-additivity (Secial to Hol. CFTs (C M ~ ~ M M ~ ~, M M ~ ~ φ rea( Σ ~ ~ rea( Σ + rea( Σ ~ ~ ~ ~ + ~ ~ (.

18 xamle 1: Pure ds3/cft2 z W ( a ( b 2 1 c 6 a a 1 2 log 1 ( b ( b 2 2 b a 1 1 Σ a1 a2 b1 b2 c + 1, I( : log z 3 ( + 2z + z( z. w I(:/2 z1 z

19 xamle 2: TZ/Thermal CFT W c 3 Min β log, πε log β sinh πε ( πl / β. Σ H x β log(1 0 + w 2 Transition I(:/2 x0 xsize of H Σ Σ [Nice Matching with Finite Tem. MPS: Nguyen-Devakul-Halbasch-Zaletel-Swingle 2017]

20 5 ``Derivation via Tensor Network Σ Original Conjecture [Swingle 2009] MR Tensor Network [Vidal 2005] time slice of ds Many Imrovements Vac [Perfect TN: Pastawski-Yoshida-Harlow-Preskill 15] [Random TN: Hayden-Nezami-Qi-Thomas-Walter-Yang 16] : Γ S#Γ Princile of ds/tn Surface/State Duality [Miyaji-TT, 2014] Σ : a convex surface (codim.2 in ds Toologically trivial Φ ( Σ H CFT ure state

21 op from TN Γ Γ C Σ ρ Comression (Isometry ρ TrC TrD VV Vac ot 1 ot V Vac ot Maximum Comression Minimal surface Γ Γ ~ Vac C Min S ~ ~ Γ # Σ ~. rea( Σ 4G N

22 6 Conclusions We conjectured that a gravity dual of ntanglement of Purification is given by the minimal cross section of ntanglement Wedge. Future roblems (1 Field theory calculations of op? [ Free field theory: work in rogress with hattacharyya-umemoto] [cf. Path-integral Otimization: Cauta-Kundu-Miyaji-Watanabe-TT 2017] (2 Proof of Hol. op formula? (3 Correct ``tensor networks descrition of ds/cft? (4 Oerational Interretations of ds/cft? (What are LOCC and LOq in ds/cft?

23 Thank you very much!