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Holographic quantum error-correcting code - exactly solvable toy models for the AdS/ CFT correspondence arXiv:1503.06237 Beni Yoshida (Caltech) joint work with Daniel Harlow Fernando Pastawski John Preskill Elementary introduction to our paper !! Introduction I. Classical Limit AdS/CFT correspondence (Maldacena 98) A. Flattening a hyperbolic spac (d+1)-dim string theory I. Classical Lim B. Rotational symmetry A. Flattening bulk O(x) (x, r) boundary B. Rotational O(x) (x, r) • Powerful framework to study stronglyinteracting systems • Advanced our understanding of quantum gravity d-dim CFT Holographic entanglement entropy Ryu-Takayanagi formula (06) Holographic entanglement entropy the boundary field theory, find bulk bulk To compute entropy of region A in minimal Bulk/Boundary duality to Geometry/Entanglement duality • surface minimal bulk surface minimal area of the bulk surface with the same boundary: To compute entropy of region A in 1 field theory, find S (the A)boundary = min ∂m =∂A area(m ) +! minimal4G area N of the bulk surface with the same boundary: bulk Ryu and Takayanagi 2006 1 S ( A) =Recover,min ) +! ∂m =∂A area(m for example, in 1+1 Entanglement 4 G N dimensional conformal field boundary theory: Ryu and Takayanagi 2006 c SRecover, ( A( L)) for = example, log( L / ain) 1+1 +! 3 boundary Geometry conformal (Space-time) dimensional field theory: c S ( A( L)) = log( L / a) +! 3 MERA (Vidal 07) • Powerful numerical method to study strongly-correlated systems. Entanglement Renormalization and Holography In AdS/CFT, the emergent dimension of space can be remove short-range regarded as a entanglement renomalization scale. Entanglement renormalization, run backwards, prepares a region of length L in circuit depth O(log L). View the bulk space as a prescription for building up the boundary state MERA = Multiscale entanglement renormalization (Swingle, 2009). ansatz AdS/CFT as a tensor network (Swingle 09) AdS/CFT correspondence can be explained by a tensor network ? MERA$ AdS$metric$ S(⇢A ) |`A | Class$of$Quantum$Many6Body$States$That$Can$ Be$Efficiently$Simulated$Guifre$Vidal$(2008)$ Part I : A simple toy model The bulk-locality paradox A note on discrete, Abelian and non-Abelian lattice gauge theory, b Rindler-wedge reconstruction A bulk operator can be represented by some integral of local d reconstruction in the causal wedge. On the left is a spacetime diagram, to perturbatively construct operators in the CFT which obe boundary operators A if and is contained inside cetime extent of the causal wedge C[A] supported associated withon a boundary sub-only if thin a boundary time slice ⌃. The point x lies within C[A] and thus any motion, with the boundary conditions set by the dictionary ( the causal wedge of A. reconstructed on A. On the right is a bulk time slice containing x and ⌃, that allx can bulk interactions are suppressed by inver y similar to that of ourwill tensorassume networks. The point simultaneously wedges, so (x) has multiple representations in theset CFT. N , which will also the AdS radius in Planck units. At lea p L results in a straightforward prescription for the CFT procedure ure of AdS/CFT, that a bulk local observable can be represented p field we simply have operators in multiple ways, (x); is illustrated in figure 11. The idea L me CFT subregion A defines a subregion in the bulk, the causal Z geodesic y point x 2 C[A], bulk quantumline field theory ensures that any bulk (x) = dY K(x; Y )O(Y ), can be represented in the CFT as some nonlocal operator on A. Sd 1 ⇥R is called the AdS-Rindler reconstruction of the operator [34, 35]. i +distinct |the integral i + ·wedges · ·is over ulk point x can| lie within causal associated with where the conformal boundary and K(x, Y ) regions, the bulk operator (x) can have distinct representations in function”. The smearing function obeys the bulk wave equation rent spatial support. causal wedge tooperator (1.1) corresponding as we taketoxthe tobulk theoperator boundary. It can be chosen to on uniqueness of A the CFT ed as indicating thatand (x) Yis aare logical operator separated, preserving a code spacelike which we illustrate for AdS3 in t lbert space of the CFT. This code subspace is protected against 1; the point the x isboundary represented by a boundary integral over the g arts of the boundary are|0̃i “erased.” |If operator corre1̃i | i+| i + ··· Bulk locality paradox All the bulk operators must correspond to identity operators on the boundary. If so, the AdS/CFT correspondence seems boring ... AdS/CFT is a quantum code ? Solution: The AdS/CFT correspondence can be viewed as a quantum error-correcting code. Lattice gauge theory Lattice Lattice gauge theory gauge theory Beni Yoshida Beni Yoshida Beni Yoshida (Dated: April 12, 2015) (Dated: April 12, 2015) (Dated: April 12, 2015) note on discrete, Abelian andAbelian non-Abelian lattice theory, based on Kogut’ A note gauge on discrete, non-Abelian lattice gauge theory, based o n and A non-Abelian lattice theory, based onand Kogut’s 1979gauge review. OAB OBC OCA OAB OABOCA OBC OBC OCA They are different operators, but act in the same manner in a low energy subspace. cf. Quantum secret-sharing code Entanglement wedge reconstruction Operators in the entanglement wedge can be reconstructed (?) * Entanglement wedge may extend over the horizon (vs firewall). * Along the spirit of the Ryu-Takayanagi formula. Holographic code Minimal model Let us construct the simplest toy model ! • 5 qubits on the boundary • 1 qubit on the bulk Minimal model Let us construct the simplest toy model ! • 5 qubits on the boundary • 1 qubit on the bulk Causal wedge reconstruction implies A bulk operator must have representations on ABC, BCD, .... Minimal model Let us construct the simplest toy model ! • 5 qubits on the boundary • 1 qubit on the bulk Entanglement wedge reconstruction implies A bulk operator must have representations on ABD, BCE, .... Desired properties A bulk operator must have representations on any region with three qubits. Beni Yoshida Five qubit code (Dated: April 14, 2015) a|0i + b|1i Abelian and non-Abelian lattice gauge theory, based on Kogut’s 1979 review. Encode one logical qubit into five physical qubits. a|0̃i + b|1̃i input a|0i + b|1i a|0̃i + b|1̃i Pauli X, Z OAB OBC output OAB OBC OCA logical Pauli X, Z OCA Logical Pauli X,Z have representations on any region with three qubits. p L “Distance 3” quantum code Six-qubit tensor Isometry from the tensor injecting a Pauli operator into one tensor leg pushing into three tensor legs X Y = X Y The tensor gives rise to an isometry along any bipartition. Holographic Code Causal wedge reconstruction D Input A A Entanglement wedge reconstruction D C Input A B Part II : Generic properties Perfect tensors : (C ) ! (C ) is anA isometry since B U (x0 , r, t = 0)U † = O(xoutput ris0 Nrik+1 = ,...,i ct. 2n . This map(1.3) 0 , t) Perfect state to(a)orthogonal. Such/ tensor an isometry (b) can be viewed as the boundary CFT operator O(x, t) can becode written as k -spin input states ar error-correcting where VB U A UB = = A pure state with maximal entanglement in any bipartition • O(x, t) = U O0 (t states where k input legs can be viewed as logical le † = 0)U (1.4) constructed from a perfect tensor, will be called a pe This contraction the map on B as depicted in Fig defines 1(b). A A state (2n spins) Perfect tensorBthe (2n legs) ator O0 (tPerfect = 0) depends only on O(x, t) where x is within EPR pair the bulk operator (x0 , r, t = 0) can be represented by boundary (a) (b) ausal wedge. UA or B B T A T U=B = T VB B UB duality A B A B A Figure 1. (a) DualityConsider of unitarya operators a perfect state. (b) Dist troduce the notion of perfect tensors. system ofin 2n number of states per spin. A wavefunction | i is said to be a We then introduce the notion of perfect tensors. A perfect stat duced density matrix ⇢A for all A such that |A| n is maximally by a tensor as follows: ⇢A / I A for all |A| n v 1 X v 1 2. X v(a) 1 Perfect state. (b) Figure X | i= i1 =0 i2 =0 · · · (2.1)Ti1 i2 ...in |i1 i2 . . . in i in =0 For readers from quantum information science, it y matrix on A. So, for any bipartition, the entanglement entropy Duality unitary a dual unitaryofoperator I ⌦operators VB that acts only on B such t • Given UA, there always exists VB such that UA ⌦ I| i = I ⌦ VB | i. This duality is often referred to as a gate teleportation wh state teleportation with EPR pairs. This duality between “gate teleportation” VB UA = Choi-Jamiolkowski isomorphism where a state is interpret An important consequence of the duality is that, in an B number A A a maximal of EPRB pairs. Here an EPR pair refe Pv 1 EPR pair 1 two-spin state: |EP Ri = pv j=0 |ji ⌦ |ji. Consider a pe in a bipartition into A and B such that |A| n. Then t will lead to bulk/ UA UB = T transformation T UB which VB acts exclusively on B such t = I⌦ boundary duality... duality of |A| decoupled copies of EPR pairs supported over spin B A decoupledB spinsA in B. Namely, | i = |EP Ri ⌦|A| ⌦ |0i ⌦2n 2|A| Distillation of EPR pairs UA VB = • By applying a unitary only on B, EPR pairs can be distilled B A A B EPR pair only on B ! UB = B A T B A UB distillation will lead to the RT formula... Almost perfect tensors • Perfect tensors are very rare ! • For qubits, there are only 2-leg and 6-leg perfect tensors. • In general, to increase the number of legs, we need to increase the spin (number of internal states). • But a random tensor does a similar job ! • Take a random quantum state and construct a tensor • It is almost maximally entangled along any cut (canonical typicality, i.e. Page’s theorem) Holographic state Holographic quantum state / code physical legs (a) (b) logical legs holographic state White dots representholographic Figure 3. (a) The holographic state. physical legs. code (b) The holographic code. Red dots connected by dotted lines represent logical legs. u-Takayanagi formula exactly. Ryu-Takayanagi formula, it’s exact sult 1. Consider a wavefunction | i characterized by !hyperboli sors. Then its entanglement entropy SA on a connected region A re [Claim] Entanglement entropy for A (connected region) is equal to the geodesic length. SA = log2 v · spin dim A A length is theAdS$metric$ length of the geodesic line of A. In fact, the above result holds S(⇢A )for any tiling of a space with nonp ong as the distance functions have no local minima. To obtain the result, |`A | we use the correspondence between local m ns of EPR pairs. Let us first consider a wavefunction which is obtain perfect tensors as depicted in Fig. 6(a). By applying local unitar lusively acting on A, one can decouple spins which are not entan ill EPR pairs bridging across A and B. In Fig. 6(a), we first dis lying some appropriate unitary transformation on spins in A wh Geodesic line from local moves Imagine an “algorithm” to find a geodesic line by local updates. local moves B geodesic local move A (Even-leg hyperbolic space does not have a local minima). Local move = distillation of EPR pairs Fig. 6(b). In general, local moves correspond to some distillation processes sin moves always bring a line over the tensor so that it reduces the number of cuts seen, in a distill perfectEPR statepairs in a and bipartition into“junks”. A and B such that |A| n, Local moves decouple •have distill |A| copies of EPR pairs by applying local unitary transformations only (a) distillation of EPR pairs B A (b) B A B A B local move A Figure 6. (a) Distillation of two EPR pairs. (b) The corresponding local mov entangled with A as depicted in Fig. 7. This leaves physical spins which are along the geodesic. We then apply similar procedure on A to obtain multiple map entanglement irs bridging AGeometric and B. This leaves pairs of of physical spins which are sitting along desic. The number of EPR pairs is exactly the number of cuts made by the , i.e. •the length line of the geodesic. is possible lines of A and B Geodesic between twoItregions A andthat B =geodesic EPR pairs er. Long-range B B A local unitary on B Short-range A local unitary on A 7. The geometric map of entanglement. White dots represent physical spins after tary transformations on A and B. depicted in Fig. 4(a). For all the open legs on the bulk, w egs ofatwo holographic states along somestates, circlesbetween atresulting the sames sents pure state i which has entanglement states. Since we| have injected mixed the Aeach hole and wormhole penWe legs from copy of holographic together. Thir ing out the second system call such ablack state simply a black holestates since its properties eometry with The two cutting AdS spaces areviewed connected a worm black hole. circlewhich may be as anby event hor T r (| ih |) = ⇢ (3.1) As a mixed state As a purified state esulting tensor network represents a pure state | i which has B any information inside it. wo systems A and B.view By tracing outhole theas second system of tw One may also the black a contraction (a) (b) on A which is identical to the one obtained by injecting sider two identical copies of holographic states denoted by A n legs of a single copy of holographic state. T r (| ih |) = ⇢ B legs of two holographic states along some circles at the sa A horizon open legs from each copy of holographic states together. ne obtains the mixed state ⇢ on A which is identical to the on black hole wormhole geometry with two AdS spaces which are connected by a w B maximally mixed states to open legs of a single copy of hologra resulting tensor network represents a pure state | i which two systems A and B. By tracing out the second system –8– T rB (| ih |) = ⇢ Figure 4. Wormhole geometry for a black hole. inject maximally mixed state one obtains the mixed state ⇢ on A which is identical to the Entanglement in a black hole • EPR pairs along the wormhole (ER=EPR ?) • RT formula with a black hole A B1 B2 re 8. The geometric map of entanglement for the wormhole geometry (color onl ws represent local moves and distillation of EPR pairs. Holographic state (multi-partition) rmula servescircuits, as a generic upper bound on does the bipartite entanglement of athe wavefunc om MERA so the wavefunction not necessarily saturates RT ising itfrom MERA circuits, so the wavefunction does notapproximately necessarily saturates Yet, is not difficult to construct a wavefunction that satisfiesthe rmula. Yet, it is not difficult to construct a wavefunction that approximately Takayanagi formula for bipartite entanglement along specific cuts by somesati Multi-partite entanglement he Ryu-Takayanagi entanglement specific cuts by s ensor network. For formula instance,for onebipartite may achieve this goal along by distributing EPR ERA•tensor network. For instance, one may achieve by ,where distributing di↵erent scales into acreate scale invariant way so that Sthis log(L) It length is not difficult a wavefunction with but... L is E A /goal airs length scales in a scale way so that SA /tensors log(L)with where h of at A.di↵erent Such distributions of EPR pairsinvariant may be possible by using he length A. Such of EPR pairs may be possible by using tensors ctures or of placing thedistributions so-called disentanglers at each layer of a MERA circuit. eg)do distribute in a circ ee structures or placing the so-called disentanglers at layer of MERA ever such simple constructions with EPR pairs noteach giveEPR rise pairs to awavefuncHowevermulti-partite such simple entanglement constructions which with EPR pairs give rise to wavef treedo not h genuine atensor holographic wavefunction must ons genuine multi-partite entanglement a holographic wavefunction To with see this, let us recall a result by Patrickwhich Hayden and his friends on the m ossess. see this, let us recallunder a result Patrick Hayden and his friends y of the To tripartite entanglement theby gauge/gravity correspondence. For on egativity of thesubsets tripartite entanglement the gauge/gravity correspondence. n-overlapping of spins A, B, C, under the tripartite entanglement is defined hree non-overlapping subsets of spins A, B, C, the tripartite entanglement is defi s follows • Negativity of tripartite information (any “holographic state”) I(A, B, C) = SA + SB + SC SAB SAC SBC + SABC . (1.1) I(A, B, C) = SA + SB + SC SAB SAC SBC + SABC . ( neric quantum state, this quantity can be negative, positive or zero. Let us -- EPR pair, then I(A,B,C)=0. or a generic quantum state, this can be negative, positive zero. Le ider a wavefunction | i which is quantity dual to some classical gravity with or negative owand consider a wavefunction i which isA, dual to some classical gravity with nega --three GHZconnected state, |then I(A,B,C)>0. e, consider regions B, C. We assume that a wavefuncurvature, andformula, considermeaning three connected A, B, C.entropy We assume that a wavef fies the RT that theregions entanglement is proportional Multi-partition and residual regions A |B| B large C A B C A B small C Multi-partition and residual regions A B C A B C A B D RC RA A |B| C B large small C Multi-partition and residual regions A B C A B C A D A |B| RD C B large C D RC RA B RB A B small C Multi-partition and residual regions A B C A D RC RA A |B| C B large B C A B D D R RD A C B C RB A B small C Multi-partition and residual regions • Identified segment of geodesic lines = EPR pairs • Residual regions = Multipartite entanglement ? A B C A D RC RA A |B| C B large B C A B D D R RD A C B C RB A B small C Residual regions in holographic state (a) (b) RC C C R RA B B A A D C C RC D R RA B B A A Residual regions in holographic state (a) (a) (b) (b) RC RC C C C C R R RA RA B B B B A A D D RA RA A A A A C C C C RC RC D D R R BB B B A A s upper bounded by the minimal number of cuts one needs to make to connect t dpoints of A, i.e. the length of the geodesic line of A. Here, the Ryu-Takayan mula serves as a generic upper bound on the bipartite entanglement of a wavefunct sing from MERA circuits, so the wavefunction does not necessarily saturates the mula. Yet, it is not difficult to construct a wavefunction that approximately satis Perfect tensor (state) is the key for negative tripartite • Ryu-Takayanagi formula for bipartite entanglement along specific cuts by so entanglement ! ERA tensor network. For instance, one may achieve this goal by distributing E Split invariant 2n-perfectway state rs at di↵erent length scales in a-- scale so into that four SA /subsets log(L) A, where L D. pairs may be possible by using tensors w length of A. Such distributionsB,ofC,EPR D e structures or placing the so-called disentanglers at each layer of a MERA circu However such simple constructions with 0< EPR do not -- Assume |A|,pairs |B|, |C|, |D| <give n+1rise to wavefu ns with genuine multi-partite entanglement which a holographic wavefunction m sess. To see this, let us recall a result by Patrick Hayden and his friends on Then the tripartite entanglement is always ativity of the tripartite entanglement under the gauge/gravity correspondence. F negative ! ee non-overlapping subsets of spins A, B, C, the tripartite entanglement is defin A C B follows Negativity of tripartite entanglement I(A, B, C) = SA + SB + SC SAB SAC SBC + SABC . (1 a generic quantum state, this quantity can be negative, positive or zero. Let Summary of the talk • A toy model of the AdS/CFT correspondence Causal wedge and entanglement wedge reconstruction • The RT formula is exact for a single connected region Negativity of tripartite entanglement • Toward the AdS/CFT limit ? • Dynamics, fast scrambling ? Von Neumann vs Peter Shor Entanglement entropy (1964) vs Quantum'Error'Correc.ng'Codes'(1995)' Quantum code (1995)