Emergence of Holographic Spacetime From Quantum Entanglement

Transcription

1 International Workshop on Condensed Matter Physics and Kavli IPMU, Tokyo, May 5-9, 015 Emergence of Holographic Spacetime From Qantm Entanglement ``Srface/State Correspodence Tadashi Takayanagi Ykawa Institte for Theoretical Physics YITP, Kyoto University

2 Based on [1] arxiv: with Masamichi Miyaji YITP, Kyoto ``Srface/State Correspondence [] arxiv: to appear in JHEP with Masamichi Miyaji YITP, Kyoto, Shinsei Ry Illinois, Urbana Champaign, and Xeda Wen Illinois, Urbana Champaign. [3] paper in preparation with Masamichi Miyaji YITP, Kyoto, Tokiro Nmasawa YITP, Kyoto, Nobro Shiba YITP, Kyoto, and Kento Watanabe YITP, Kyoto.

3 Contents 1 Introdction ds/cft and Tensor Network [Review] 3 Bondary State as Gravity Dal of Point-like Space [] 4 Srface/State Correspondence [1] 5 SS-correspondence in ds/cft [3] 6 Conclsions

4 1 Introdction String Theory a nified theory of qantm gravity It is still difficlt to compte qantm corrections in cosmological spacetimes like big bang, de-sitter etc. However, a generalization of ds/cft or holography may be able to resolve this problem: ``Qantm Gravity = Qantm Many-body Systems on Md+ on Md+1 For this, we need to nderstand the basic mechanism of ds/cft. key concept is qantm entanglement.

5 Entanglement Entropy EE log The best measre of qantm entanglement S Tr[ ] EE measres 1 How mch a many-body wave fnction is entangled. ctive degrees of freedom. ~central charges 3 qantm order parameter. 4 n observable in nmerical experiments. 5 a geometry of qantm many-body system.

6 rea law of EE [Bombelli-Kol-Lee-Sorkin 86, Srednicki 93] EE in QFTs incldes UV divergences. In a d+1 dim. QFT d>1 with a UV relativistic fixed point, the leading term of EE at its grond state behaves like S rea ~ d 1 sbleading terms, where is a UV ctoff i.e. lattice spacing. [d=1: log div.] Intitively, this property is nderstood like: Most strongly entangled

7 Holographic Entanglement Entropy HEE [Ry-TT 06; derived by Casini-Herta-Myers 09, Lewkowycz-Maldacena 13] S Min is the minimal area srface codim.= sch that and rea 4G N ~ homologos. CFTd 1 B z We omit the time direction. dsd z UV ct off Note: In time-dependent spacetimes, we need to take extremal srfaces. [Hbeny-Rangamani-TT 07] ds R dz dt d i 1 z dx i

8 The HEE sggests the following novel interpretation: `` spacetime in gravity = Collections of bits of qantm entanglement B Planck length Manifestly realized in the recently fond connection between ds/cft and tensor networks! [Swingle 09, ]

9 Crrent Stats Qantm Many-body Systems Cond-mat, QFTs, CFTs,.. ds/cft Holography Qantm gravity String theory HEE, BH info. ``ds/qi EE, ES, Top EE, Tensor networks, etc. Qantm Information Stat.Mech G N k B

10 Or Hope? Qantm Many-body System Gravity String Theory QI Stat.Mech. Qantm Entanglement New Unified Framework?

11 In this talk we want to psh a bit more in this direction by proposing the Srface / State correspondence as a new generalization of holography. Σ Codim. two convex srface H M Σd Gravity = Tensor network

12 ds/cft and Tensor Network -1 Tensor Network [See e.g. Cirac-Verstraete 09review] Tensor network states = Efficient variational ansatz for the grond state wave fnctions in qantm many-body systems. [ tensor network diagram = wave fnction] n ansatz shold respect the correct qantm entanglement of grond state. ~Geometry of Tensor Network

13 Ex. Matrix Prodct State MPS [DMRG: White 9,, Rommer-Ostlnd 95,..]

14 MPS with finite does not have enogh EE to describe 1d qantm critical points d CFTs : S log log CFT S L ~. 1 3 n 1 n In general, S N int ~ N int log, min[# Intersections of ].

15 - MER MER Mltiscale Entanglement Renormalization nsatz [Vidal 05] n efficient variational ansatz for CFT grond states. To increase entanglement in a CFT, we add disentanglers. S Min[# links] log L agrees with reslts in d CFT!

16 conjected relation to ds/cft [Swingle 09] Min[# Links] Min[rea] 0 ds e 1 IR Eqivalent dz dt dt dx z dx, where z e? CFT d 1 ds d However, MER has lattice artifacts. Continm limit? Other modifications: e.g.[qi 13, Pastawski-Yoshida-Harlow-Preskill 15].

17 -3 cmer [Haegeman-Osborne-Verschelde-Verstraete 11 reformlation and ds/cft interpretation: Nozaki-Ry-TT 1] To remove lattice artifacts, take a continm limit of MER: State at scale Kˆ : P exp i IR IR ds Kˆ s Ω. IR state disentangler at length scale ~ e : nentangled state in real space S 0 for any. What is this state? Or next topic!

18 Relation to discrete MER - By adding dmmy states 0>, we keep the dimension of Hilbert space for any to be the same. We can formally describe the real space RG by a nitary transformation.

19 3 Bondary State as Gravity Dal of Point-like Space Q. general constrction of the IR states Ω> in CFTs? rgment 1 [Miyaji-Ry-Wen-TT 14] We can realize disentangled states IR states Ω> Trivial Point-like spaces by performing a infinitely massive deformation: H m m H CFT m d 1 dx O x, the grond state of H O d m.

20 Now we apply the idea of qantm qenches. For t<0, we assme the grond state of the massive Hamiltonian Hm. Then at t=0, we sddenly change the Hamiltonian into HCFT as in [Calabrese-Cardy 05]. In this setp, the state at t=0 is identified with the bondary state: m t 0 B. We may introdce the UV ct off like m e H / m B. t H CFT H m Empty theory

21 Bondary states in CFTs assme d CFT bondary state Ishibashi state : B> = state which gives a conformally invariant bondary condition: L L B 0. In terms of the Virasoro algebra: B where k k 1,k,... represent 1 k k k L L n ~ n. 1 maximally entangled state between left and right moving sectors! Bt, the real space entanglement is qite sppressed! k k L k R

22 rgment : Correlation fnctions of local operators When xi-xj>>δ, there is no correlations! Disentangled! Bondary X1 X Xm δ~1/m Bondary. 1 1 n i i n x O x O x O x O

23 rgment 3: Direct calclation of EE For the reglarized IR state we can compte the EE explicitly in free fermion CFTs: S c 3 log [Finite], e H B, [Ugajin-TT 10] 0. Ths we can set S 0 when. Note: Bondary states can still have non-zero finite topological entanglement.

24 4 Srface/State Correspondence [Miyaji-TT 15] 4-1 Basic Principle Consider Einstein gravity on a d+ dim. spacetime M. We arge the following correspondence: Σ : an d dim. convex space-like srface in M which is closed and homologically trivial Σd Gravity pre state H M Md+

25 More generally, the qantm state dal to a convex srface Σ is a mixed state ρσ if Σ is open or topologically non-trivial. Md+ Md+ Σ BH Σ On the other hand, the zero size limit of Σ corresponds to the trivial state Ω> with no real space entanglement.

26 This srface/state correspondence is partially motivated by the tensor network description of holography.

27 4- Entanglement Entropy We can natrally generalize HEE for or setp : H H H S B, rea 4G N Tr. B [ ], Md+ ΣB Σ Σ ɤ Σ=Σ ΣB

28 4-3 Effective Dimension By dividing the srface Σ into infinitesimally small pieces Σ= i, we easily find: S i i rea 4G N. We interpret this as the log of effective dim. for Σ log[dim i H eff ] This is becase is expected to be maximally entangled except the dmmy states. [cf. Differential entropy: Balasbramanian-Chowdhry-Czech-deBoer-Heller 13]

29 4-4 Inner Prodcts and Information Metric nother intriging physical qantity is an inner prodct ' between two srfaces. Here focs on the two srfaces separated infinitesimally. Consider an information distance between them ds R d g x, dx dx.

30 Remember the correspondence:. The information metric is given by Define the Hamiltonian H s.t. is its grond state. Then the standard pertrbation theory leads to. 0 1 k k E H k d d 1 B G d d

31 This consideration leads to the following form: G B 1 d dx g x G N dy d g y P x, y, g x, g y,. If the metric is x-independent, we have G B ~ 1 G Example 1: a flat spacetime N dx g x K. [Translational inv. d. ] Example : an ds spacetime [Nozaki-Ry-TT 1]: B Vd d G Ndeg e grees with cmer for CFT d d d B G 0. 1

32 Qantm Estimation Theory qantm version of Cramer-Rao bond arges 1. B G Mean sqare error In the case of ds/cft, this leads to z z GN rea 1 log[dim [Helstrom 76] In the large N limit, this error is highly sppressed. Locality of the blk in the large N limit? ~ Some ncertainty principle of srfaces in QG? H eff ] rea G rea. N N.

33 5 SS-correspondence in ds/cft [Miyaji-Nmasawa-Shiba-Watanabe-TT, in preparation] We will show that the formlation of cmer can be generalized so that it describes the SS-dality. Let s focs on a ds3/cft setp. It is sefl to start with the symmetry of global ds3 space: ds R cosh whose isometry dt d SL,R SL sinh d,,r is generated by

34 In particlar, we are interested in the SL,R sbgrop which preserves the time slice t=0 i.e. H of the ds3. They are generated by, n which annihilates the bondary states. l n ~ L n L n 0, 1, The SL,R action which maps ρ=0 to the point, is given by l1 l. 1, il0 g e e t=0

35 cmer for the grond state of CFT is formlated as: 0 0 P exp i Kˆ d B 0 If we act the SL,R transformation. g, 0 0 P exp i ˆ, K d B0 where Kˆ g, Kˆ, bondary Ishibashi state for the identity sector, g,, we find 1.

36 More generally, we can describe the diffeomorphism by taking into accont The dal state of a srface expected in SS-correspondence is given by the form: :,3,.., ~ n L L l n n n. ξ l g g g g K g K B d K i P n n n g g 0 0 with exp where, ˆ ˆ ˆ ˆ, ˆ exp ˆ exp 0 B ds s K i P g 0 B

37 How to describe the blk excitation? We arge the identification:, Blk local operator 0, Blk CFT This is becase the local operator insertion does not change the blk metric = entanglement. P exp i 0 Kˆ, s ds B. Ishibashi state for primary, B

38 We arge this state is evalated as, CFT i L This reprodces the correct scalar field wave fnction This satisfies the correct EOM: g ds3 ~ L 0 0 H, e e J. some UV ct off We can compte the information metric:,, t CFT 0. SL Ishibashi, R State ds, 1, 1 G dx dx d sinh d. c as in ds/cft by choosing ab a b, c -1.

39 6 Conclsions Qantm entanglement represents a geometry of qantm state in many-body systems. Ex.1 HEE EE probes the metric of hol. geometry Ex. ds/tnmer, Entanglement = geometry Ex.3 Bondary state trivial point-like space This gravity/entanglement dality looks more general than ds/cft and even than holography. We proposed the srface/state correspondence.

40 The SS-dality arges Top. trivial convex srface a pre state Top. non-trivial srface Zero size srface a mixed state Bondary states rea of srface = log[eff. Dimension] Extrinsic crvatre Ftre problems Derivation of Einstein eq. = Information metric pplication to spacetime withot time-like bondary e.g. de-sitter spaces. nalysis of compact directions e.g. S5 in ds5 S5.