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Quantum error-correction in black holes Beni Yoshida (Perimeter Institute) @ Taipei (Sep 2016) [1] Holographic quantum error-correcting codes, [2] Chaos in quantum channel, [3] Complexity by design, arXiv:1609:xxxxx String Daniel Harlow (Harvard MIT) Fernando Pastawski (Caltech Berlin) John Preskill (Caltech) String CMT Xiao-liang Qi (Stanford) QI QI Daniel Roberts (MIT IAS) Simons collaborations A black hole is a quantum errorcorrecting code Anti-de Sitter/Conformal field theory (AdS/CFT) correspondence • [Conjecture] Equivalence of string (gravity) theory in bulk with CFT on boundary (Maldacena) Hyperbolic space (negatively curved) Boundary D-dimensional conformal field theory (without gravity) Bulk (D+1)-dimensional theory with gravity on AdS space Anti-de Sitter/Conformal field theory (AdS/CFT) correspondence • [Conjecture] Equivalence of string (gravity) theory in bulk with CFT on boundary (Maldacena) • [Holography] Bulk degrees of freedom are encoded in boundary, like a hologram. “holography” Hyperbolic space (negatively curved) 1-dimensional strongly interacting “qubits” 2-dimensional “qubits” on AdS Quantum entanglement in AdS/CFT • [Ryu-Takayanagi formula] QM GR 1 S(A) = min(area( 4GN A A )) Minimize over spatial bulk surfaces homologous to A A A1 A2 (For “large N“) Space-time as a tensor network • [Swingle’s conjecture] AdS/CFT correspondence can be expressed as a MERA ? minimal surface 1 saturation ? S(A) 1 min(area( 4GN A A )) (1) MERA = Multi-scale Entanglement Renormalization Ansatz (Vidal) 2⇡kB RE Concrete and simple toy tensor network model (joint with Harlow, Pastawski and Preskill) Lattice gauge theory Dictionary : Correspondence of Yoshida operators Beni (Dated: April 12, 2015) boundary system Abelian and non-Abelian bulklattice system gauge theory, based A note on discrete, 1 1 O (1) O1 O2 1 O1 2 O1 O2 1 2 1 S(A) min(area( A )) 4GN A 1 S(A) min(area( O1 A )) O2 | O2 1 p 1 (2) i+| L 1 2 2 (1) (1) i + ··· (2) n and non-Abelian lattice gauge theory, based on Kogut’s 1979 review. Lattice gauge theory an and non-Abelian lattice gauge theory, based on Kogut’s 1979 review. Lattice gauge theory Bulk operator vs boundary operator Beni Yoshida Beni Yoshida wedge reconstruction] • [Entanglement (Dated: April 12, 2015) (Dated: April 12, A bulk operator can2015) be represented by some integral of local boundary operators A note on discrete, Abelian and non-Abelian lattice gauge theory, based on Kog supported on A if is contained inside the entanglement wedge of A. nd non-Abelian lattice gauge theory, based on Kogut’s 1979 review. A OA 1 p pL L A1 A2 OA 1 A 2 | i+| | i+| n Y p t Remarks (1 + e L ) j=1 j 1 1 (1) (1) (2) (2) (1) i + ··· · OA A ⇠ OAi + · · ⇠ 1 2 ⇠ OA p ⇠ OL A1 A2 (2) • Entanglement wedge may go beyond black hole horizons (i.e. no firewall). • “Proven” by using a generalized RT formula (Jefferisnet al, Dong et al, Bao et al) n Y Y jt more than 0̃i is known |1̃i for • No explicit|recipe j t one intervals(1 + e ) |0̃i |1̃i(1 + e ) j=1 | j=1 i + | i + ··· Bulk locality puzzle • The reconstruction recipe leads to a paradox All the bulk operators must correspond to identity operators on the boundary ? If so, the AdS/CFT seems very boring … Quantum error-correction in AdS/CFT ? • The AdS/CFT correspondence can be viewed as a quantum error-correcting code. [Almheiri-Dong-Harlow]. Lattice gauge theory Lattice gauge theory Lattice gauge theory Beni Yoshida (Dated: April 12, 2015) Beni Yoshida Beni Yoshida (Dated: April 12, 2015) (Dated: April 12, 2015) A note on discrete, Abelian and gauge based1979 on Kogut’s A note on discrete, Abelian and non-Abelian lattice gauge theory, basedtheory, on Kogut’s review. elian and non-Abelian lattice gauge theory, based onnon-Abelian Kogut’s 1979lattice review. OAB OBC OCA OAB OBC OABOCAOBC OCA They are different operators, but act in the same manner in a low energy subspace. cf. Quantum secret-sharing code Let’s construct a toy model A simple toy model 1 bulk qubit in total, just 6 qubits 5 boundary qubits A bulk operator must have representations on any region with three qubits. logical qubit physical qubits Entanglement wedge reconstruction ! tal SPT phases and their gauged models may be an interesting A note on discrete, Abelian andfuture non-Abelian lattice gauge theory, b ruct an SPT Hamiltonian protected by fractal-like symmetry operators. h fractal SPT phases and their gauged models may be an interesting future Lattice gaugecode theory Five-qubit S1 = X ⌦ Z ⌦ Beni Z ⌦ Yoshida X ⌦I a|0i + b|1i (73) (Dated: April 2015) system of14, five qubits.(73) • EncodeS1a=single X ⌦ Zlogical ⌦ Z ⌦qubit X ⌦ Iinto a = I⌦and ⌦I X ⌦ Z ⌦ lattice Z ⌦X (74) S1 A=note X on ⌦ discrete, Z ⌦ ZSAbelian ⌦2 X (73) non-Abelian gauge theory, based on Kogut’s 1979 review. X⌦⌦ZZ⌦⌦XZ⌦⌦IX SS12==XI ⌦ Z (73)(74) input S = XZ⌦ I ⌦ X ⌦ Z ⌦ Z 3ZZ S2 S= ZX⌦ ⌦⌦ X I⌦ ⌦⌦Z ⌦⌦ XX S23=I=I⌦ X⌦X ⌦ Z ⌦ S⌦ =⌦⌦ ZZZ⌦ X ⌦ I ⌦ X ⌦ Z ⌦⌦ X S34=X =XZ⌦⌦⌦IIX ⌦4ZZ X S3 S= ⌦ XI⌦ a|0i + b|1i 4 = Z ⌦X ⌦I ⌦X ⌦Z S4 S= Z ⌦X ⌦I ⌦X ⌦Z (74)(75) (74) (75)(76) (75) encode (76) • codeword space output a|0̃i + b|1̃i a|0̃i + b|1̃i C = {| i : Sj | i = | i 8j} (76) (76) OAB Acknowledgment OBC OCA (77) C = {| i : S | i = | i 8j} j thank Aleksander Kubica, John Preskill and Burak Şahinoğlu for helpful code has 3 • Five-qubit omments. ICalso Burak Şahinoğlu this problem to = {|thank i:S = | distance i 8j}for suggesting j | icode Acknowledgment (75) (77) (77) set of states with distance 1 O O O AB BCfor Theoretical CA ork was completed during the visit to the Kavli Institute thank Aleksander Kubica,byJohn Preskill Burak for helpful owledge funding provided the Institute for Information and | 0and i Quantum | 1 iŞahinoğlu (78) omments. I also Center thank Burak Şahinoğlu forGordon suggesting Physics Frontiers with support of the and this Bettyproblem Moore to | 0i | p (78) The code can correct single-qubit L 1i ts No. and PHY-1125565). am supported David work wasPHY-0803371 completed during the visit to theI Kavli Instituteby forthe Theoretical errors ! distance 3 tdoctoral fellowship. This research supported in part byInformation the Nationaland owledge funding provided by the was Institute for Quantum | i = ↵| 0i + | 1i n under Frontiers Grant No.Center NSF PHY11-25915. Physics with support of the Gordon and Betty Moore (79) Five-qubit code is a quantum gravity Perfectness of five qubit code • Let’s view the five-qubit code as a six-leg tensor. output input T T • Any leg can be used as an input of quantum codes. T T A holographic quantum error-correcting code • A tiling of the five qubit code five qubit code output input AdS geometry bulk legs boundary legs Entanglement wedge reconstruction • 1 in & 3 out (operator pushing) D D C InputInput Input Input A A AA B B C Holographic state • The Ryu-Takayanagi formula holds exactly (tiling of perfect tensors) Coarse-graining (RG transformation) = Distillation of EPR pairs along the geodesic B A perfect tensor EPR pairs 0 0 k+1 0 2n B A an en A to orthogonal. Such an isometry can be viewed as EPR (b) ame, the boundary CFT operatorerror-correcting O(x,(a) t) can be written as code where k -spin input states are enco Perfect tensors 0 † Uwhere B states k input as logical A UB = legs. A = legs can O(x, t) = U O (t = 0)U (1.4)beVviewed constructed from a perfect tensor, will be called a perfect co entanglement in any bipartition • A pure state with maximal operator O0 (t = 0) depends only on O(x, the B B is within A A t) where x A EPR(2n pair legs) state (x (2n0 ,spins) hus, the bulkPerfect operator r, t = 0) can be represented by boundary Perfect tensor (a) the causal wedge. ensor B UA T U=B = T A (b) T VB UB distillation duality A B B A B A Figure 1. (a) Duality of unitary operators in a perfect state. (b) Distillation of EP we introduce the notion of perfect tensors. Consider a system of 2n the number of states per spin. A wavefunction | i is said to be a We then introduce the notion of perfect tensors. A perfect state | i can be s reduced density matrix ⇢A for allbyAa such |A| n is maximally tensorthat as follows: ⇢A / I A for all |A| n v 1 X v 1 X v 1 X 2. | i Figure = · · · (a) TPerfect . . . in i (b) Perfect i1 i2 ...in |i1 i2state. (2.1) i1 =0 i2 =0 in =0 For readers from quantum entity •matrix on A.code So, for any bipartition, the entanglement entropyinformation science, it may b Five qubit alue. Let | i be a perfect state, a A quantum be a subseterror-correcting of spins with |A| n by a standard notation [[n code mplement of A. Since A is maximally with B, quantum the entangled total number ofany spins, k represents the number of logi –5– 6 leg perfect tensor T on A has a dual quantum operation acting onkB. Namely, letTUA ⌦ I there are v orthogonal states in the codeword space), d re nitary operator which acts exclusively on A. Then there always exists and v represents the number of states in each spin. Acc perfect code is represented by [[2n 1, 1, n]]v . Similarly, a nsor with 2n legs. Let Mi1 ,...,ik be a tensor representation of an input state | i which a k spin state. By contracting legs i1 , . . . , ik of Mi1 ,...,ik and Ti1 i2 ...i2n , one obtains a w tensor Nik+1 ,...,i2n , which corresponds to the output state | i of the map: Random tensors X Nik+1 ,...,i2n = Ti1 i2 ...i2n Mi1 ,...,ik . • Perfect tensors are very rare… (2.5) i1 ,...,ik v ⌦k v ⌦2n k ! perfect tensors are common • But almost is contraction defines the map : (C ) pretty ! (C ) whose input is Mi1 ,...,ik and tput is Nik+1 ,...,i2n . This map is an isometry since pairwise orthogonal states remain Pick a Haar random state. orthogonal. Such an isometry can be viewed as an encoding map of a quantum or-correctingDue code where -spin input states are encoded 2n k-spin output to the Page’sk theorem, the state is almost maximallyin entangled along any cut. tes where k input legs can be viewed as logical legs. A quantum code with k = 1, construct a holographic pickcode tensors nstructed a perfect tensor, willcode/state, be called a just perfect (Fig.randomly. 2). • Tofrom (a) (b) logical leg Random tensor Figure 2. (a) Perfect state. (b) Perfect code. Holographic code For readers from quantum information science, it may be convenient to characterize ⇠ OA ⇠ OA 1 A 2 (1) Coding properties: erasure threshold 1 • Remove qubits with probability p = 2 n Y (1 + e ✏ jt (2) ) A central bulk leg is contained in the entanglement wedge of A (if |A|>|B|)(3) j=1 “e H ” Erasure threshold = 1/2 e iHt (4) New quantum codes from quantum gravity ? | i (5) So far, no black holes… Information loss puzzle 1 • Is quantum information lost ? | i Hawking radiation “e 1X P ⌦ P = SWAP d P | ji = U| ji H (1) ” (2) (3) 1 Information loss puzzle • Or hidden into some non-local degrees of freedom ? | i Hawking radiation 1X P ⌦ P = SWAP d P (2) Locally it looks like “e | ji = U| ji (1) H , but globally it is not . ” (3) n Y jt (1 error-correction +e ) Quantum j=1 • Scrambling is very similar to how quantum error-correcting codes work. 1 • Local indistinguishability. ⇢R ' ⇢0R | i (1) logical qubit time evolution e 1X d P iHt Pblack ⌦ hole P = SWAP | ji = U| ji “e Hawking n radiation Y = error (1 +e j=1 (2) H ” codewords | i (3)e iHt jt ) (Dated: May 31, 2016) (Dated: May(Dated: 31, 2016) May 27, 2016) California Institute of Technology, Pasadena, California 9 Random thoughts on scrambling Choi-Jamilkowski isomorphism (Dated: August 12, 2015) I. EQUATIONS ndom thoughts on scrambling Beni can qubits be viewedofasobservations a state 2non qubits. • Quantum channel on n This I. onEQUATIONS is Yoshida a collection scrambling. I. EQUATIONS Beni Institute Yoshida for Quantum Information and Matter, unitary operator as a state of Technology, Pasadena, California 91125, USA tute forCalifornia Quantum Institute Information and Matter, X X U= Ui,j(Dated: |iihj| |U91125, i =12, 2015) Ui,j |ii ⌦ |ji August ute of Technology, Pasadena, California USA i,j i,j August of 12,observations 2015) This(Dated: is a collection on scrambling. ⇢in = {pj , | j i} in of observations on scrambling. ⌦ in (1) U out † † † † A(0)B(t)C(0)D(t)C (0)B (0)A (0)D (t) ⇢in = {pj , | j i} ↵ U ⇢out = {pj , | j i} out (1) (2) (1) ⇢out = {pj , | j i} (2) (1) h i Xp H H † † † † {pj ,i |( jt)e i} 2 Bj (0)e 2 ' hAi ( t)Bj (0)A|i(2) Tr Ai ( ⇢t)B ( it)B (3) out = j (0)A j (0)i = p | i ⌦ | ji j j U S ' |A| · log d A RECURRENCE TIME OF A PLANAR PERFECT TENSOR NETWORK SA ' |A| · log d (2) j ⇢in = {pj , | j i} input output quantum channel TENSOR NETWORK IME OF A PLANAR PERFECT nsider a planar tensor network as shown in Fig. 1. This mimics the growth of the I. RECURRENCE TIME OF A PLANAR PERFECT TE ein-Rosen for the interior ofAai ⌦ two-sided black hole under along | 1.i =This I |TFDi (4) r network bridge as shown in Fig. mimics the growth of the time-evolution [Choi-Jamilkowski, Hayden-Preskill,Hartman-Maldacena] Perimeter Institute for Theoretical Physics (Dated: April 14, 2016) (Dated: April 14, 2016) (Dated: April 12, 2016) Scrambling in a black hole equations ations • Thermofield double state (finite T) e e HH ⇢in⇢in== TrTre e HH | i= X e ⇢out Ej /2 e e = Tr e iEj t j H (1) H (1) | ji ⌦ | ji (1) • A black hole geometry for the TFD state is the two-sided hole (AdS/CFT prediction) † † † † hA(0)B(t)A (0)B (t)i hA(0)B(t)A (0)B (t)i (2) length=O(T) {U, ⇢in } ⌦ (2) Let’s construct a tensor network toy CFT 2 CFT 1 ⌦ ↵ ↵ A(0)B(t)C(0)D(t) A(0)B(t)C(0)D(t) {p ,U } j U= X j (2) {pj ,Uj } Ui,j |iihj| model ! (3) (3) [Maldacena] (3) a hyperbolic network, representing the two asymptotically AdS boundaries. This n extends infinitely from the UV into the IR thermal scale at the black hole horizon. the middle is flat representing the black hole interior. The entire network grows as by adding more layers in the middle flat region. We would like to further elaborate on this proposal of tensor network representa the black hole interior. We will study networks of perfect tensors and demonstrate dynamics by finding ballistic of local unitary operators and the linear gro unitary operators, tiling growth the wormhole geometry. the tripartite information until the scrambling time. For the rest of discussion, w the infinite temperature = 0 limit so we can ignore the hyperbolic part and focu the planar tiling of tensor networks representing the interior. CFT 1 Right CFT CFT 2 AdS ERB AdS igure 8: Tensor network representation of the Einstein-Rosen bridge. We will consider a etwork of perfect tensors. length=O(T) Left CFT time • Consider a network of random ynamics by finding ballistic growth of local unitary operators and the linear growth of he tripartite information until the scrambling time. For the rest of discussion, we take he infinite temperature = 0 limit so we can ignore the hyperbolic part and focus in on he planer tiling of tensor networks representing the interior. Toy model of the Einstein-Rosen bridge Figure 10: Tensor network representation of the Einstein-Rosen bridge. A four-leg random unitary operators lives at each node. We will consider a network of perfect tensors. 18 This model has the additional nice property of implementing the holographic quantum error co proposal of [41]. [Hosur-Qi-Roberts-BY] How do we probe the interior of a black hole ? s Equations Beni Yoshida Perimeter Institute for Theoretical Physics Out-of-time ordered correlation functions (Dated: May 27, 2016) Beni Yoshida stitute for Theoretical Physics We 27, should measure some “hidden” correlations (Kitaev 2014) Dated: •May 2016) . I. OTO = hA(0)B(t)C(0)D(t)i (1) EQUATIONS local operators (0)eiHt B(t) = e iHt B(0)eiHt D(t) = e iHt D(0)eiHt C B D(t) = e iHt D(0)eiHt (1) (1) A D H : Z ! X X t! : Z!X X! = EQUATIONS Z Z (2) (2) 0 d X j=1 p | i= |iAB i. j |iC i•⌦ Previously d X p j=1 j |iC i ⌦ |iAB i. (3) (3) in 1960s, and recently by Shenker and Stanford considered by Larkin and Ovchinikov (Dated: April 12, 2016) (Dated: May 27, 2016) PDF for equations OTO figures Time evolution OTO figures of operators I.Beni EQUATIONS Yoshida operators • OTOCs detect the growth ofBeni Yoshida Perimeter Institute for Theoretical Physics ⌦ ↵Physics Perimeter Institute for Theoretical † † • Consider(Dated: OTO = 12, A(0)B(t)A April 2016) (0)B (t) ions (Dated: April 12, 2016) [A, B] = 0 then (1) Non-commutativity between A(0) and B(t) PDF for equations {A, B} = 0 • Expand B(t): OTO = 1 (commutator) (1) then OTO = 1 (2) OTO = hA(0)B(t)C(0)D(t)i X iHt X iHt B(t) = e= Be↵iHt = ↵iHt j Pj B(t)iHt= e iHt Be j Pj iHt (2) iHt B(t) = e B(0)e D(t) = e j D(0)e j (3) (1) (1) high-weight | 0 iPauli operators | 1i scrambling OTO ' 0 H 0: Z ! X X !OTO Z =1 [A, B] = then [A, B] 6= 0 then 1 | 0 (t)i OTO | 1 (t)i (3) (4) (2) (2) (3) (4) [Roberts-Stanford-Susskind] Key Questions • How do we define scrambling ? B(t) = e iHt BeiHt = X ↵ j Pj j • Quantum information theoretic meaning of OTO ? • Is the converse true ? scrambling ? OTO ' 0 I will relate OTOCs to entanglement entropies (joint with Hosur, Qi and Roberts) [A, B] = 0 then OTO = 1 [A, B] 6= 0 then OTO 1 IBD = SB + SD SBD [Theorem] Average value of OTO • Average of OTO over local operators A and D at T=infty ⌦ A(0)D(t)A† (0)D† (t) average over A, D ↵ = 1 X⌦ 4a+d =2 A(0)D(t)A† (0)D† (t) A,D (2) n a d SBD A ↵ D [Hosur-Qi-Roberts-BY] IBD = S + S B D (2) (2) (2) IBD = SB + SD S (2) BD SBD (21 [Theorem] Average value of OTO • Average of OTO over local operators A and D at T=infty X ⌦ ⌦ ↵ ⌦ ↵ X 1 ↵ ⌦ ↵ 1 † † † † † † † A(0)D(t)A = a+d A(0)D(t)A (0)D (t) A(0)D(t)A(0)D (0)D†(t) (t) = A(0)D(t)A (0)D (t) a+d 44 A,D A,D average over A, D n a d n a =2 =2 A (2) SBD(2) d S (22 Pauli operators (unitary 1-design) BD (23 D [Hosur-Qi-Roberts-BY] IBD = S + S B D (2) (2) (2) IBD = SB + SD S (2) BD SBD (21 [Theorem] Average value of OTO • Average of OTO over local operators A and D at T=infty X ⌦ ⌦ ↵ ⌦ ↵ X 1 ↵ ⌦ ↵ 1 † † † † † † † A(0)D(t)A = a+d A(0)D(t)A (0)D (t) A(0)D(t)A(0)D (0)D†(t) (t) = A(0)D(t)A (0)D (t) a+d 44 A,D A,D n a d n a =2 =2 average over A, D input (2) SBD(2) d S (22 Pauli operators (unitary 1-design) BD (23 B A A D U output C D [Hosur-Qi-Roberts-BY] IBD = S + S B D (2) (2) (2) IBD (2) = SB (2) + S(2) D IBD = SB + SD S (2) BD S(2) BD SBD [Theorem] Average value of OTO (21 (21) • Average of OTO over local operators A and D at T=infty X ⌦ ⌦ ↵ ⌦ ↵ X 1 ↵ ⌦ ↵ 1 † † †† † † ↵ † † X ⌦ A(0)D(t)A ⌦A(0)D(t)A ↵ 1 A(0)D(t)A (0)D (t) = A(0)D(t)A (0)D (t) = (0)D (t) (t) † † † †(0)D a+d a+d A(0)D(t)A (0)D (t) = 44a+d A(0)D(t)A (0)D (t) A,D 4 A,D A,D (2) (2) n a d S(2) BD n a d S =2n a d SBDBD average over A, D input =2 =2 A Pauli operators (unitary 1-design) Renyi-2 entropy B A (22 (22) (23 (23) D U output C D [Hosur-Qi-Roberts-BY] r Institute Beni YoshidaIBD = SB + SD (2) (2) (2) for Theoretical Physics S (2) BD IBD S(2) (2) = SB (2) + S(2) D BD IBD = SBPhysics + SD SBD meter Institute for Theoretical (Dated: April 12, 2016) [Theorem] Average value (21 (21) of OTO (Dated: April 12, 2016) • Average of OTO over local operators A and D at T=infty X OTO figures ⌦ ⌦ ↵ ⌦ X 1 ↵ ⌦ 1 † † ↵ ↵ † † ↵ † † † † X ⌦ A(0)D(t)A ⌦A(0)D(t)A(0)D ↵ 1 A(0)D(t)A (t) = A(0)D(t)A (0)D (t) = (0)D (t) (t) (22 † † † †(0)D a+d a+d A(0)D(t)A (0)D (t) = 44a+d A(0)D(t)A (0)D (t) (22) A,D 4 Beni A,D A,D YoshidaPauli operators (unitary 1-design) (2) (2) n a d S(2) BD n a d S average over A, D =2 (23 n a d SBD BD Perimeter Institute for Theoretical Physics =2 =2 (23) X (t) = e iHt BeiHt A = input A† (0)D† (t) ↵ output • If B Renyi-2 entropy ↵ j Pj (Dated: April 12, 2016) A j X⌦ PDF for 1U equations = 4a+d C A(0)D(t)A† (0)D† (t) DA,D (2) a d SBD is large 0 n then, OTO '=2 (2) (2) ↵ (2) This implies the mutual information IBD = SB + SD B] = 0 Xthen (1) D (1) (2) (2) (2) SBD is small B and D are not correlated, so the system is scrambling. ⌦ OTO =1 E /2 † † A(0)D(t)A (0)D (t) iE t ↵ = 1 X⌦ a+d (3) ↵ † † [Hosur-Qi-Roberts-BY] A(0)D(t)A (0)D (t) r Institute Beni YoshidaIBD = SB + SD (2) (2) (2) for Theoretical Physics S (2) BD IBD S(2) (2) = SB (2) + S(2) D BD IBD = SBPhysics + SD SBD meter Institute for Theoretical (Dated: April 12, 2016) [Theorem] Average value (21 (21) of OTO (Dated: April 12, 2016) • Average of OTO over local operators A and D at T=infty X OTO figures ⌦ ⌦ ↵ ⌦ X 1 ↵ ⌦ 1 † † ↵ ↵ † † ↵ † † † † X ⌦ A(0)D(t)A ⌦A(0)D(t)A(0)D ↵ 1 A(0)D(t)A (t) = A(0)D(t)A (0)D (t) = (0)D (t) (t) (22 † † † †(0)D a+d a+d A(0)D(t)A (0)D (t) = 44a+d A(0)D(t)A (0)D (t) (22) A,D 4 Beni A,D A,D YoshidaPauli operators (unitary 1-design) (2) (2) n a d S(2) BD n a d S average over A, D =2 (23 n a d SBD BD Perimeter Institute for Theoretical Physics =2 =2 (23) X (t) = e iHt BeiHt A = input A† (0)D† (t) ↵ output • If B Renyi-2 entropy ↵ j Pj (Dated: April 12, 2016) A j X⌦ PDF for 1U equations = 4a+d C A(0)D(t)A† (0)D† (t) DA,D (2) a d SBD is large 0 n then, OTO '=2 (2) (2) ↵ (2) This implies the mutual information IBD = SB + SD (1) D (1) (2) (2) (2) SBD is small B and D are not correlated, so the system is scrambling. • For finite T, we will consider the so-called Thermofield double state. B] = 0 Xthen ⌦ OTO =1 E /2 † † A(0)D(t)A (0)D (t) iE t ↵ = 1 X⌦ a+d (3) ↵ † † [Hosur-Qi-Roberts-BY] A(0)D(t)A (0)D (t) 39]), and then an explicit tensor network model was proposed in [15] (see also [40]).18 Before we begin, let us review the proposal of [12]. The tensor network representation of the thermofield double state is shown in Fig. 10. At the left and right ends, we have a hyperbolic network, representing the two asymptotically AdS boundaries. This network extends infinitely from the UV into the IR thermal scale at the black hole horizon. Then, the middle is flat representing the black hole interior. The entire network grows as t grows This captures key properties of scrambling for “large-N” theories. by adding more layers in the middle flat region. We would like to Ballistic further elaborate on this proposal of tensor network Eg) propagations of entanglement, RTrepresentation formula in of a wormhole geometry… the black hole interior. We will study networks of perfect tensors and demonstrate chaotic dynamics Operator by finding ballistic local unitary operators theitlinear size growth growsof linearly, OTO will and pick up. growth of the tripartite information until the scrambling time. For the rest of discussion, we take the infinite temperature = 0 limit so we can ignore the hyperbolic part scrambling and focus in on time is log(n) [Cleve et al 2006] For a non-local random quantum circuit, the the planar tiling of tensor networks representing the interior. Scrambling phenomena in a black hole AdS Figure 10: Tensor network representation of the Einstein-Rosen bridge. A four-leg tensor ives at each node. We will consider a network of perfect tensors. (46) (45) i1 ,...,i2n 25 8A s.t |A| = n. Ti1 ,...,i2n |i1 , . . . , i2n i = SA = |A| · log v ERB r normalization. We call a tensor T perfect if it is a associated with a i, called a perfect state, which is maximally entangled along any bipartition. AdS time input op Random unitary (maximally entangled) | i= Right CFT X Left CFT by briefly reviewing a definition of perfect tensors. Consider a tensor T with ond dimension v. A tensor can be represented as a pure state • stic propagation of entanglement • sor network representation of the Einstein-Rosen bridge. We will consider a rfect tensors. finding ballistic growth of local unitary operators and the linear growth of information until the scrambling time. For the rest of discussion, we take mperature = 0 limit so we can ignore the hyperbolic part and focus in on ing of tensor networks representing the interior. • [Hosur-Qi-Roberts-BY] OTO figures Beni Yoshida Scrambling in AdS black hole Perimeter Institute for Theoretical Physics (Dated: April 12, 2016) RT surface PDF for equations A TFD(0) B OTO figures mutual information I(A, B) is 'large exp( const · T ) Beni Yoshida Perimeter Institute for Theoretical Physics (Dated: April 12, 2016) PDF for equations TFD(T) A (2) (2) (2) IBD = SB + SD B (2) SBD OTO correlators will be small. ~T ⌦ information ↵ ' (almost) ⌦ · T) 1 X mutual zero exp( const † I(A, † B) is A(0)D(t)A (0)D (t) = 4a+d =2 † † A(0)D(t)A (0)D (t) A,D (2) n a d SBD ↵ ( What did we learn ? Lesson 1 The AdS/CFT correspondence is a quantum error-correcting code. Bulk quantum information is encoded in boundary like a hologram. very entangled tensor (eg random tensor) nds infinitely from the UV into the IR thermal scale at the black hole horizon. Then, middle is flat representing the black hole interior. The entire network grows as t grows dding more layers in the middle flat region. We would like to further elaborate on this proposal of tensor network representation of lack hole interior. We will study networks of perfect tensors and demonstrate chaotic mics by finding ballistic growth of local unitary operators and the linear growth of OTO correlator is the of space-time ripartite information until the scrambling time.probe For the rest of discussion, we take nfinite temperature = 0 limit so we can ignore the hyperbolic part and focus in on planar tiling of tensor networks representing the interior. (46) 8A s.t |A| = n. (45) Ti1 ,...,i2n |i1 , . . . , i2n i with a proper normalization. We call a tensor T perfect if it is a associated with a pure state | i, called a perfect state, which is maximally entangled along any bipartition. Namely, re 10: Tensor network representation of the Einstein-Rosen bridge. A four-leg tensor at each node. We will consider a network of perfect tensors. 5.1 his model has the additional nice property of implementing the holographic quantum error correction sal of [41]. | i= X i1 ,...,i2n 25 OTO correlator detects scrambling/ chaos SA = |A| · log v AdS Let us begin by briefly reviewing a definition of perfect tensors. Consider a tensor T with 2n legs and bond dimension v. A tensor can be represented as a pure state ERB Ballistic propagation of entanglement AdS Right CFT Figure 8: Tensor network representation of the Einstein-Rosen bridge. We will consider a network of perfect tensors. Left CFT time dynamics by finding ballistic growth of local unitary operators and the linear growth of the tripartite information until the scrambling time. For the rest of discussion, we take the infinite temperature = 0 limit so we can ignore the hyperbolic part and focus in on the planer tiling of tensor networks representing the interior. Lesson 2 (Dated: April 11, 2016) Lesson 3 OTO correlators are the probes of space-time (seeing the interior of a black hole) OTO = hA(0)B(t)C(0)D(t)i (1) B A t 0 A B ~T 1970s Black hole = Black hole = 2010s [1] Holographic quantum error-correcting codes [2] Chaos in quantum channel, Thank you ! [3] Complexity by design, arXiv:1609:xxxxx String Daniel Harlow (Harvard MIT) Fernando Pastawski (Caltech Berlin) John Preskill (Caltech) String CMT Xiao-liang Qi (Stanford) QI QI Daniel Roberts (MIT IAS) Simons collaborations