Holographic duality from random tensor networks

Journal of High Energy Physics, Nov 2016

Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

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Holographic duality from random tensor networks

Received: April Holographic duality from random tensor networks Patrick Hayden 0 Sepehr Nezami 0 Xiao-Liang Qi 0 Nathaniel Thomas 0 Michael Walter 0 Zhao Yang 0 Open Access 0 c The Authors. 0 0 Stanford Institute for Theoretical Physics, Department of Physics, Stanford University ity because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We nd that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Renyi entropies in our models and contrast it with AdS/CFT. Moreover, we nd that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk eld, we nd that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk eld. Increasing the entanglement of the bulk eld ultimately changes the minimal surface behavior topologically, in a way similar to the e ect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article de ne a more general form of bulk-boundary correspondence which could be useful for extending AdS-CFT Correspondence; Black Holes in String Theory; Gauge-gravity 1 Introduction 2 General setup 3 Ryu-Takayanagi formula entanglement Code subspace 2.1 4 Random tensor networks as bidirectional holographic codes Entanglement wedges and error correction properties 4.3 Gauge invariance and absence of local operators 5 Higher Renyi entropies 6 Boundary two-point correlation functions 7 Fluctuations and corrections for nite bond dimension The general bound on uctuations 7.2 Improvement of the bound under a physical assumption Possible e ects of even smaller bond dimension 8 Relation to random measurements and the entanglement of assistance 9 Random tensor networks from 2-designs 10 Conclusion and discussion A Analytic study of the three phases for a random bulk state B Derivation of the error correction condition C Uniqueness of minimal energy con guration for higher Renyi models D Calculation of C2n in section 6 E Partition function of Ising model on the square lattice F Average second Renyi entropy for 2-designs G Contractions of stabilizer states De nition of random tensor networks Calculation of the second Renyi entropy Ryu-Takayanagi formula for a bulk direct-product state Ryu-Takayanagi formula with bulk state correction Phase transition of the e ective bulk geometry induced by bulk Tensor networks have been proposed [1] as a helpful tool for understanding holographic duality [2{4] due to the intuition that the entropy of a tensor network is bounded by an area law that agrees with the Ryu-Takayanagi (RT) formula [5]. In general, the area law only gives an upper bound to the entropy [1], which for particular tensor networks and choices of regions has been shown to be saturated [6]. Tensor networks can also be used to build holographic mappings or holographic codes [6{8], which are isometries between the Hilbert space of the bulk and that of the boundary. In particular, some of us have recently proposed bidirectional holographic codes built from tensors with particular properties, socalled pluperfect tensors [8]. These codes simultaneously satis es several desired properties, including the RT formula for a subset of boundary states, error correction properties of bulk local operators [9], a kind of bulk gauge invariance, and the possibility of sub-AdS locality. The perfect and pluperfect tensors de ned in refs. [6] and [8], respectively, have entanglement properties that are idealized version of large-dimensional random tensors, which is part of the motivation why it is natural to study these tensor networks. In this work, we will show that by directly studying networks of large dimensional random tensors, instead of their \idealized" counterpart, their properties can be computed more systematically. Speci cally, we will assume that each tensor in the network is chosen independently at random. We nd that the computation of typical Renyi entropies and other quantities of interest in the corresponding tensor network states can be mapped to the evaluation of partition functions of classical statistical models, namely generalized Ising models with boundary pinning elds. When each leg of each tensor in the network has dimension D, these statistical models have inverse temperature / log D. For large enough D, they are in the long-range ordered phase, and we nd that the entropies of a boundary region is directly related to the energy of a domain wall between di erent domains of the order parameter. The minimal energy condition for this domain wall naturally leads to the RT formula.1 Besides yielding the RT formula for general boundary subsystems, the technique of random state averaging allows us to study many further properties of a random 1. E ects of bulk entanglement. Using the random tensor network as a holographic mapping rather than a state on the boundary, we derive a formula for the entropy of a boundary region in the presence of an entangled state in the bulk. As a special example of the e ect of bulk entanglement, we show how the behavior of minimal surfaces (which are minimal energy domain walls in the statistical model) is changed qualitatively by introducing a highly entangled state in the bulk. When the state is su ciently highly entangled, no minimal surface penetrates into this region, so that the topology of the space has e ectively changed. This phenomenon is analogous to the change of spatial geometry in the Hawking-Page transition [11, 12], where the bulk geometry changes from perturbed AdS to a black hole upon increasing temperature. 1In our models, the RT formula holds for all Renyi entropies, which is an important di erence from AdS/CFT [10]. We will discuss this point in more detail further below. 2. Bidirectional holographic code and code subspace. By calculating the entanglement entropy between a bulk region and the boundary in a given tensor network, we can verify that the random tensor network de nes a bidirectional holographic code (BHC). When the bulk Hilbert space has a higher dimension than the boundary, we obtain an approximate isometry from the boundary to the bulk. When we restrict the bulk degrees of freedom to a smaller subspace (\code subspace", or \low energy subspace") which has dimension lower than the boundary Hilbert space dimension, we also obtain an approximate isometry from this bulk subspace to the boundary. This bulk-to-boundary isometry satis es the error correction properties de ned in ref. [9]. To be more precise, all bulk local operators in the entanglement wedge of a boundary region can be recovered from that boundary region.2 3. Correlation spectrum. In addition to entanglement entropies, we can also study properties of boundary multi-point functions. In particular, we show that the boundary two-point functions are determined by the bulk two-point functions and the properties of the statistical model. When the bulk geometry is a pure hyperbolic space, the boundary two-point correlations have power-law decay, which de nes the scaling dimension spectrum. We show that in large-dimensional random tensor networks there are two kinds of scaling dimensions, those from the bulk \low energy" theory which do not grow with the bond dimension D, and those from the tensor network itself which grow / log D. This con rms that the holographic mapping de ned by a random tensor network maps a weakly-interacting bulk state to a boundary state with a scaling dimension gap, consistent with the expectations of AdS/CFT. The use of random matrix techniques has a long and rich history in quantum information theory (see, e.g., the recent review [13] and references therein). Previous work on random tensor network states has originated from a diverse set of motivations, including the construction of novel random ensembles that satisfy a generalized area law [14, 15], the relationship between entropy and the decay of correlations [16], and the maximum entropy principle [17]. The relation between the Schmidt ranks of tensor network states and minimal cuts through the network has been investigated in [18]. While the primary motivation for this work is to better understand holographic duality, its methods and even the nature of many of its conclusions place it squarely in this earlier tradition. In the holographic context, it was in fact previously shown that using a class of pseudo-random tensors known as quantum expanders in a MERA tensor network would reproduce the qualitative scaling of the Ryu-Takayanagi formula [19]. The remainder of the paper is organized as follows. In section 2 we de ne the random tensor networks. We show how the calculation of the second Renyi entropy is mapped to the partition function of a classical Ising model. In section 3 we investigate the RT formula in the large dimension limit of the random tensors, and discuss the e ect of bulk entanglement. As an explicit example we study the minimal surfaces for a highly entangled (volume-law) bulk state and discuss the transition of the e ective bulk geometry as a 2In this work, the entanglement wedge of a boundary region refers to the spatial region enclosed by the boundary region and the minimal surface homologous to it, rather than to a space-time region. function of bulk entropy density. In section 4 we study the properties of the holographic mapping de ned by random tensor networks, including boundary-to-bulk isometries and bulk-to-boundary isometries for the code subspace, and we discuss the recovery of bulk operators from boundary regions. In section 5 we generalize the calculation of the second Renyi entropy to higher Renyi entropies. We show that the n-th Renyi entropy calculation is mapped to the partition function of a statistical model with a Symn permutation group element at each vertex. The same technique also enables us to compute other averaged quantities involving higher powers of the density operator. In section 6 we use this technique to study the boundary two-point correlation functions. We show that the boundary correlation functions are determined by the bulk correlations and the tensor network, and that a gap in the scaling dimensions opens at large D in the case of AdS geometry. In section 7 we bound the uctuations around the typical values calculated previously and discuss the e ect of nite bond dimensions. Section 8 explains the close relationship between the random tensors networks of this paper and optimal multipartite entanglement distillation protocols previously studied in the quantum information theory literature. In section 9 we consider other ensembles of random states. We nd that the RT formula can be exactly satis ed in tensor networks built from random stabilizer states, which allows for the construction of exact holographic codes. Finally, section 10 is devoted to conclusion General setup De nition of random tensor networks We start by de ning the most general tensor network states in a language that is suitable for can de ne a Hilbert space Hk with dimension Dk for each leg of the tensor, and consider the index k as labeling a complete basis of states j ki in this Hilbert space. In this language, T 1 2::: n (with proper normalization) corresponds to the wavefunction of a quantum state jT i = P j ni de ned in the product Hilbert space Nn k=1 Hk. A tensor network is obtained by connecting tensors, i.e., by contracting a common index. For purposes of illustration, a small tensor network is shown in gure 1 (a). Before connecting the tensors, each tensor corresponds to a quantum state, so that the collection of all tensors can be considered as a tensor product state N x jVxi. Here, x denotes all vertices in the network, and jVxi is the state corresponding to the tensor at vertex x. Each leg of a tensor corresponds to a Hilbert space. We will denote the Hilbert space corresponding to a leg from x to another vertex y by Hxy, and its dimension by Dxy. If a leg is dangling, i.e., not connected to any other vertex, we will denote the corresponding Hilbert space by Hx@ and its dimension by Dx@ . (Without loss of generality we can assume there is at most one dangling leg at each vertex.) Connecting two tensors at x; y by an internal line then corresponds to a projection in the Hilbert space Hxy Hyx onto a maximally entangled and similarly for j yxi. By connecting the tensors according to the internal lines of the A tensor network that de nes a mapping from bulk legs (red) to boundary legs (blue). An arbitrary bulk state (orange triangle) is mapped to a boundary state. (For simplicity, we have drawn a pure state in the bulk. For a mixed state the map needs to be applied to both indices of the bulk density operator.) (c) The internal lines of the tensor network can always be combined with the bulk state and viewed as a state in an enlarged Hilbert space (enclosed by the dashed hexegon). In this view, each tensor acts independently on this generalized bulk state and maps it to the boundary state. tensor network, we thus obtain the state j i = @ in the Hilbert space corresponding to the dangling legs, N general not normalized. Tensor network states de ned in this way are often referred to as projected entangled pair states (PEPS) [20]. As has been discussed in previous works [6{8], tensor networks can be used to de ne not only quantum states but also holographic mappings, or holographic codes, which map between the Hilbert space of the bulk and that of the boundary. Figure 1 (b) shows a very simple \holographic mapping" which maps the bulk indices (red lines) to boundary indices (blue lines), with internal lines (black lines) contracted. A bulk state (orange triangle in gure) is mapped to a boundary state by this mapping. Such a boundary state can also be written in a form similar to eq. (2.1). Instead of viewing the tensor network as de ning a mapping, we can equivalently consider it as a quantum state in the Hilbert space H@ , which is a direct product of the bulk Hilbert space Hb and the boundary Hilbert space H@ . Denoting the bulk state as j bi, the corresponding boundary state is j i = @h bj From this expression one can see that the internal lines of the tensor network can actually be viewed as part of the bulk state. As is illustrated in gure 1 (c), one can view the maximally entangled states on internal lines together with the bulk state j bi as a state in the enlarged \bulk Hilbert space". This point of view will be helpful for our discussion. More generally, one can also have a mixed bulk state with density operator b, instead of the pure state j bi. The most generic form of the boundary state is given by the density = trP P = b Here the partial trace trP is carried over the bulk and internal legs of all tensors (i.e., over all but the dangling legs). In this compact form, one can see that the state function of the independent pure states of each tensor jVxihVxj. In this work, we study tensor network states of the form (2.3), where the tensors jVxi are unit vectors chosen independently at random from their respective Hilbert spaces. We will mostly use the \uniform" probability measure that is invariant under arbitrary unitary transformations. Equivalently, we can take an arbitrary reference state j0xi and function f (jVxi) of the state jVxi is then equivalent to an integration over U according to All nontrivial entanglement properties of such a tensor network state are induced by the projection, i.e., the partial trace with P . However, the average over random tensors can be carried out before taking the partial trace, since the latter is a linear operation. This is the key insight that enables the computation of entanglement properties of random Calculation of the second Renyi entropy We will now apply this technique to study the second Renyi entropies of the random tensor de ned in eq. (2.3). For a boundary region A with reduced density matrix write this expression in a di erent form by using the \swap trick", e S2(A) = Here we have de ned a direct product of two copies of the original system, and the operator FA is de ned on this two-copy system and swaps the states of the two copies in the region A. To be more precise, its action on a basis state of the two-copy Hilbert space is given by FA(jnAi1 denotes the complement of A on the boundary. jm0Ai2) = jn0Ai1 in base 2, Sn(A) = 1=(1 condensed matter and high energy literature. 3In the quantum information theory literature, the Renyi entropy is usually de ned with logarithm tr AnA)n . Here we use base e to keep the notation consistent with the We are now interested in the typical values of the entropy. Denote the numerator and denominator resp. of eq. (2.5) by Z1 = tr [( Z0 = tr [ These are both functions of the random states jVxi at each vertex. We would like to average over all states in the single-vertex Hilbert space. The variables Z1 and Z0 are easier to average than the entropy, since they are quadratic functions of the single-site density matrix and Z0 = Z0 S2(A) = Z1 + Z1 = We will later show in section 7 that for large enough bond dimensions Dxy the uctuations are suppressed. Thus we can approximate the entropy with high probability by the separate averages of Z1 and Z0: Throughout this article we use ' for asymptotic equality as the bond dimensions go to in nity. In the following we will compute Z1 and Z0 separately and use (2.9) to determine the typical entropy. To compute Z1, we insert eq. (2.3) into eq. (2.6) and obtain Z1 = tr ( P In this expression we have combined the partial trace over bulk indices in the de nition of the boundary state and the trace over the boundary indices in eq. (2.6) into a single trace over all indices. In the expression it is now transparent that the average over states, one at each vertex, can be carried out independently before couplings between di erent sites are introduced by the projection. The average over states can be done by taking an an integration over Ux 2 SU (Dx) with respect to the Haar measure. The result of this integration can be obtained using Schur's lemma (see, e.g., ref. [21]): jVxihVxj jVxihVxj = Ux (j0xih0xj j0xih0xj) Uxy Uxy = Here, Ix denotes the identity operator and Fx the swap operator de ned in the same way as FA described above, swapping the two copies of Hilbert space of the vertex x (which product of the dimensions corresponding to all legs adjacent to x, including the boundary dangling legs. It is helpful to represent eq. (2.11) graphically as in gure 2 (a) and (b). Carrying out the average over states at each vertex x, Z1 then consists of 2N terms if there are N vertices, with an identity operator or swap operator at each vertex. We can then introduce an Ising spin variable sx = choice of Ix and Fx, respectively. In this representation, Z1 becomes a partition function of the spins fsxg: x with sx= 1 Z1 = hx = For each value of the Ising variables fsxg, the operator being traced is now completely factorized into a product of terms, since Fx acts on each leg of the tensor independently. This fact is illustrated in gure 2 (c). The trace of the swap operators with P simply exp [ S2 (fsx = 1g ; P )] with S2 (fsx = in the Ising spin-down domain de ned by sx = 1g ; P ) the second Renyi entropy of P 1. The trace on boundary dangling legs gives a factor that is either Dx2@ or Dx@ , depending on the Ising variables sx and whether x is in A. To be more precise, we can de ne a boundary eld A [fsxg] = S2 (fsx = log Dx@ (3 + hxsx) + X log Dx2 + Dx : Then the trace at a boundary leg x@ gives Dx21@(3+hxsx). Taking a product of these two kinds of terms in the trace, we obtain the Ising action The form of the action can be further simpli ed by recalling that P has the direct product form in eq. (2.4). Therefore the second Renyi entropy factorizes into that of the bulk state b and that of the maximally entangled states at each internal line xy. The latter is a standard Ising interaction term, since the entropy of either site is log Dxy while the entropy of the two sites together vanishes. Therefore A [fsxg] = log Dxy (sxsy log Dx@ (hxsx + S2 (fsx = Here we have omitted the details of the constant term since it plays no role in later discussions. Eq. (2.13) is the foundation of our later discussion. The same derivation applies to the average of the denominator Z0 = tr [ ] in eq. (2.5), which leads to the same Ising there is no swap operator FA applied. One can de ne F1 = log Z1, F0 = log Z0, such that F1 and F0 are the free energy of the Ising model with di erent boundary conditions.4 Then eq. (2.9) reads 4The standard de nition of free energy should be 1 log Z1 but it is more convenient for us to de ne it without the temperature prefactor. the tensor network shown in gure 1. (b) The average over the state jVxihVxj Hilbert space (see eq. (2.11)). On the right side of the equality, the dashed line connected to the black dot stands for a sum over an Ising variable sx = 1. When sx = 1 (sx = 1), each green rectangle represents an operator Ix (Fx), respectively. (c) The state average of Z1 in eq. (2.10) for the simple tensor network shown in gure 1. We consider a region A consisting of a single site, and the green rectangle with X represents the swap operator FA. After contracting the doubled line loops one obtains the partition function of an Ising model, with the blue arrows representing the Ising variables. The dashed lines in the right of last equality represent three di erent terms in the Ising model contributed by the links, the bulk state (middle triangle) and the choice of boundary region A. That is, the typical second Renyi entropy is given by the di erence of the two free energies, i.e., the \energy cost" induced by ipping the boundary pinning eld to down ( 1) in region A, while keeping the remainder of the system with a pinning eld up (+1). In summary, what we have achieved is that the second Renyi entropy is related to the partition function of a classical Ising model de ned on the same graph as the tensor network. Besides the standard two-spin interaction term, the Ising model also has an additional term in its energy contributed by the second Renyi entropy of the bulk state b, and the Ising spins at the boundary vertices are coupled to a boundary \pinning eld" hx determined by the boundary region A. If the bulk contribution from b is small (which means major part of quantum entanglement of the boundary states is contributed by the tensor network itself), one can see that the parameters log Dxy and log Dx@ determine the e ective temperature of the Ising model. For simplicity, in the following we assume Ryu-Takayanagi formula Once the mapping to the classical Ising model is established, it is easy to see how the Ryu-Takayanagi formula emerges. In the large D limit, the Ising model is in the lowtemperature long-range ordered phase (as long as the bulk has spatial dimension that the Ising action can be estimated by the lowest energy con guration. The boundary pinning eld hx leads to the existence of an Ising domain wall bounding the boundary region A, and in the absence of a bulk contribution the minimal energy condition of the domain wall is exactly the RT formula. In this section we will analyze this emergence of the Ryu-Takayanagi formula and corrections due to bulk entanglement in more detail. Ryu-Takayanagi formula for a bulk direct-product state We rst consider the simplest situation with the bulk state a pure direct-product state b = N x j xih xj. In this case one can contract the bulk state at each site with the tensor of that site, which leads to a new tensor with one fewer legs. Since each tensor is a random tensor, the new tensor obtained from contraction with the bulk state is also a random tensor. Therefore the holographic mapping with a pure direct-product state in the bulk is equivalent to a purely in-plane random tensor network, similar to a MERA, or a \holographic state" de ned in ref. [6]. The second Renyi entropy of such a tensor network state is given by the partition function of Ising model in eq. (2.13) without the b term. Omitting the constant terms that appears in both Z0 and Z1, the Ising action can be written as A [fsxg] = In the large D limit, the Ising model is in the low temperature limit, and the partition function is dominated by the lowest energy con guration. As illustrated in the \energy" of an Ising con guration is determined by the number of links crossed by the domain wall between spin-up and spin-down domains, with the boundary condition of the domain wall set by the boundary eld hx. For the calculation of denominator Z0, the existence of a spin-down domain. Each link hxyi with spins anti-parallel leads to an energy cost of log D. Therefore the Renyi entropy in large D limit is 1 for x 2 A requires S2(A) = F1 The minimization is over surfaces [ A form the boundary of a spin-down domain, and j j denotes the area of , i.e., the number of edges that cross the surface. Therefore the minimal area surface, denoted by A, is the geodesic surface bounding A region. Here we have assumed that the geodesic surface is unique. More generally, if there are k degenerate minimal surfaces (as will be the case for a regular lattice in at space), F1 is modi ed by With this discussion, we have proved that Ryu-Takayanagi formula applies to the second Renyi entropy of a large dimensional random tensor network, with the area of geodesic surface given by the graph metric of the network. As will be discussed later in section 5, the higher Renyi entropies take the same value in the large D limit, and it can also be extended to the von Neumann entropy, at least if the minimal geodesics are unique (see section 7). However, the triumph that the second Renyi entropy is equal to the area of the minimal surface in the graph metric is in fact a signature that the random tensor construction deviates from the holographic theory. The holographic calculation of the second Renyi entropy amounts to evaluating the Euclidean action of the two-fold replica geometry, which satis es the Einstein equation everywhere in the bulk. Thus, in general, the second Renyi entropy does not exactly correspond to the area of the minimal surface in the original geometry. Due to the back-reaction of the gravity theory, the n-fold replica geometry is in general di erent from the geometry constructed by simply gluing n copies of the original geometry around the minimal surfaces, the discrepancy between which can be seen manifestly from the n-dependence of the holographic Renyi-n entropy. We will see in section 5 that our random tensor model can reproduce the correct Renyi entropies for a single boundary region if we replace the bond states jxyi by appropriate short-range entangled states with non-trivial entanglement spectrum. However, this does not resolve the problem for multiple boundary regions, for which we will have a more detailed discussion in section 5. To compare with the RT formula de ned on a continuous manifold, one can consider a triangulation of a given spatial manifold and de ne a random tensor network on the graph of the triangulation. (See [22, appendix A] for further discussion of the construction of the triangulation graph.) Denoting by lg the length scale of the triangulation (the average that lg1 d log D corresponds to the gravitational coupling constant 4G1N . distance between neighboring triangles), the area j Aj in our formula is dimensionless and Compared to previous results about the RT formula in tensor networks [6, 8], our proof of RT formula has the following advantages: rstly, our result does not require the boundary region A to be a single connected region on the boundary. Since the entropy in the large D limit is always given by the Ising spin con guration with minimal energy, the result applies to multiple boundary regions. Secondly, our result does not rely on any property of the graph structure, except for the uniqueness of the geodesic surface (if this is not satis ed then the entropy formula acquires corrections as discussed above; cf. section 9). If we obtain a graph by triangulation of a manifold, our formula applies to manifolds with zero or positive curvature, even when the standard AdS/CFT correspondence does not apply. In addition to these two points, we will also see in later discussions that our approach allows us to study corrections to the RT formula systematically. Notice that we are not limited to two-dimensional manifolds. One can consider a higher dimensional manifold and construct a graph approximating its geometry. It follows from our results that the entropy of a subregion of the boundary state is given by the size of the minimum cut on the graph, i.e., the area of the minimal surface in the bulk homologous to the boundary region. Ryu-Takayanagi formula with bulk state correction If we do not assume the bulk state to be a pure direct-product state, the bulk entropy term in eq. (2.13) is nonzero. If we still take the D ! 1 limit, the Ising model free energy is still determined by the minimal energy spin con guration, which is now determined by a balance between the area law energy log D j j for a domain wall , and the energy cost from elds down (hx = A region and up (hx = +1) elsewhere. A is the boundary of minimal energy spin-down domain (black dashed line) is an example of other domain wall con gurations with higher energy. The spin-down domain EA is called the entanglement wedge of A. (b) The minimal surfaces bounding two far-away regions A and B, which are also the boundary of the entanglement wedge of the completement region CD. (c) The e ect of bulk entanglement in the same con guration as panel (b). The entanglement wedges are deformed. bulk entropy. We can de ne the spin-down region in such a minimal energy con guration as EA, which bounds the boundary region A, and corresponds to the region known as the entanglement wedge in the literature [23, 24]. The second Renyi entropy is then given by S2(A) ' log D j Aj + S2 (EA; b) : The bulk contribution has two e ects. First it modi es the position of the minimal energy domain wall j Aj, and thus modi ed the area law (RT formula) term of the entropy. Second it gives an additional contribution to the entanglement entropy of the boundary region. This is similar to how bulk quantum elds contribute corrections to the RT formula in To understand the consequence of this bulk correction, we consider an example shown in gure 3 (b) and (c), where A and B are two distant disjoint regions on the boundary. If the bulk entanglement entropy vanishes, the RT formula applies and the entanglement wedges EA and EB are disjoint. Therefore we nd that S2(A) + S2(B) = S2(AB) and so vanishes in the large D limit.5 When the bulk state is entangled, if we assume the entanglement is not too strong, so that the entanglement wedges remain disjoint, the minimal energy domain walls A and B may change position, but remain disconnected. Therefore: S2(AB) ' log D (j Aj + j Bj) + S2(EA [ EB; b); I2(A : B) ' S2(EA; b) + S2(EB; b) S2(EA [ EB; b) = I2(EA : EB; b): From this equation, we see that even if a small bulk entanglement entropy may only lead to a minor correction to the minimal surface location, it is the only source of mutual in5The mutual information for Renyi entropy is generally not an interesting quantity, but it is meaningful in our case since it approaches the von Neumann mutual information for large D. formation between two far-away regions in the large D limit. (If we consider a large but nite D, and include spin uctuations of the Ising model, we obtain another source of mutual information between far-away regions, which vanishes exponentially with log D.) The suppression of mutual information between two far-away regions implies that the correlation functions between boundary regions A and B are suppressed, even if each region has a large entanglement entropy in the large D limit. In the particular case when the bulk geometry is a hyperbolic space, the suppression of two-point correlations discussed here translates into the scaling dimension gap of boundary operators, which is known to be a required property for CFTs with gravity duals [26{28]. A more quantitative analysis of the behavior of two-point correlation functions and scaling dimension gap will be postponed to section 6. Phase transition of the e ective bulk geometry induced by bulk We have shown that a bulk state with nonzero entanglement entropy gives rise to corrections to the Ryu-Takayanagi formula. In the discussion in section 3.2, we assumed that the bulk entanglement was small enough that the topology of the minimal surfaces remained the same as those in the absence of bulk entanglement. Alternatively, one can also consider the opposite situation when the bulk entanglement entropy is not a small perturbation compared to the area law term log D j Aj, in which case the behavior of the minimal surfaces may change qualitatively. In this subsection, we will study a simple example of this phenomenon, with the bulk state being a random pure state in the Hilbert space of a subregion in the bulk. As is well-known, a random pure state is nearly maximally entangled [29], which we will use as a toy model of a thermal state (i.e., of a pure state that satis es the eigenstate thermalization hypothesis [30, 31]). The amount of bulk entanglement can be controlled by the dimension of the Hilbert space Db of each site. We will show that the topologies of minimal surfaces experience phase transitions upon increasing Db which qualitatively reproduces the transition of the bulk geometry in the Hawking-Page phase transition [11, 12]. To be more precise, the entropy of the boundary region receives two contributions: the area of the minimal surfaces in the AdS background and the bulk matter eld correction. However, above a critical value of Db, the minimal surface tends to avoid the highly entangled region in the bulk, such that there is a region which no minimal surface ever penetrates into, and the minimal surface jumps discontinuously from one side of the region to the other side as the boundary region size increases to half of the system. This is qualitatively similar to how a black hole horizon emerges from bulk entanglement. (A black hole cannot be identi ed conclusively in the absence of causal structure, however, so our conclusions in this section are necessarily tentative.) We consider a tensor network which is de ned on a uniform triangulation of a hyperbolic disk. Each vertex is connected to a bulk leg with dimension Db in addition to internal legs between di erent vertices. Then we take a disk-shaped region, as shown in gure 4 (a). We de ne the bulk state to be a random state in the disk region, and a direct-product random pure state. We study the second Renyi entropy of a boundary region r = 1 . (b) The phase diagram of the boundary state as parametrized by the bond dimensions D and Db, corresponding to in-plane and bulk degrees of freedom, respectively. The blue line, obtained numerically, describes the phase boundary that separates the perturbed AdS phase and the small black hole phase. The red line distinguishes the small black hole phase and the maximal black hole phase. The three phases are discussed in more detail in the main text. j ibulk = @ The second Renyi entropy of a boundary region is determined by the Ising model partition function with the action (2.13).6 The bulk contribution S2 (fsx = 1g ; b) for a random state with large dimension only depends on the volume of the spin-down domain in the disk region, since all sites a play symmetric role. After an average over random states, the entropy of a bulk region with N sites is given by [32] S2 (N ) = log in which NT is the total number of sites in the disk region. Therefore the Ising action contains two terms, an area law term and the bulk term which is a function of the volume of spin-down domain. For simplicity, we can consider a ne-grained triangulation and approximate the area and volume by that in the continuum limit. If we denote the average distance between two neighboring vertices as lg, as in previous subsections, we obtain A [M#] = log D lg 1 j@M#j + log @ DjM#j=lg2 + D DVT =lg2 + 1 (VT jM#j)=lg2 A : 6For readers more comfortable with the graph theoretic description, here is a sketch in that language of the entropy calculation in the presence of a bulk random state. Because any vertex corresponds to projection to a random state, the insertion of a random bulk state amounts to connecting the bulk dangling legs to a single new vertex. Therefore, the study of entropies will be equivalent to the study of minimum cuts in the modi ed graph. Here M# is a spin-down region bounding a boundary region A, and @M# is the boundary the disk region in the bulk. r2d 2)=(1 r2)2. The boundary is placed at r = 1 > 0 a small cuto parameter. The disk region is de ned by r b. Choose a boundary region so that the boundary region is smaller than half the system size. (Boundary regions that exceed half the system size have the same entropy as their complement, since the whole system is in a pure state.) If we assume the minimal surface @M# to be a curve described by r = r( ) (i.e., for each there is only one r value), the volume and area of this curve can be written explicitly as S2(') = min Vr( ) = (r0( ))2 +r2( )+log VT = DVT =lg2 +1 DVr( )=lg2 +D(VT Vr( ))=lg2 ; r2)2 = For xed lg 1 log D, when we gradually increase lg 2 log Db, there are three distinct phases: the perturbed AdS phase, the small black hole phase, and the maximal black hole phase. The phase diagram can be obtained numerically, as shown in gure 4 (b). In distance is 1 (i.e., the AdS radius). In the perturbed AdS phase, although the minimal surfaces are deformed due to the existence of the bulk random state, there is no topological change in the behavior of minimal surfaces. As the size of the boundary region increases, the minimal surface swipes through the whole bulk continuously ( gure 5 (a)). In the small black hole phase, the minimal surface experiences a discontinuous jump as the boundary region size increases. There exists a region with radius 0 < rc < b that cannot be accessed by the minimal surfaces of any boundary regions ( gure 5 (b)). Qualitatively, the minimal surfaces therefore behave like those in a black hole geometry, which always stay outside the Further increase of log Db does not change the entanglement property of the boundary anymore, since the entropy in the bulk disk region has saturated at its maximum. This is the maximal black hole phase ( gure 5 (c)). square root behavior of the blue line can be understood by taking the maximal boundary region of half the system size ' = 2 . At the critical lg 2 log Db, the diameter of the Poincare disk goes from the minimal surface bounding the half system to a local maximum. For more detailed discussion, see appendix A. The second transition at the red line is roughly where the entanglement entropy of the bulk region reaches its maximum. However, more and lg 1 log D = 10 is xed. boundary regions in the three phases. The random pure state is supported at the orange region. to be 10. (b) Entropy pro le of the boundary system r = 1 2 [ '; '] with respect to the di erent boundary region size ' . The blue data points, black data points and the red data points correspond to the boundary entropy pro le in the perturbed AdS space, the small black hole phase and the maximal black hole case, respectively. The parameters are set as the same as the three phases in gure 5. Figure 6 (b) provides another diagnostic to di erentiate the geometry with and without the black hole. The entanglement entropy S2(') is plotted as a function of the boundary region size. In the perturbed AdS phase (blue curve), S2(') is a smooth function of ', just like in the pure AdS space. In the small black hole phase (black curve) and the as a consequence of the discontinuity of the minimal surface. For ' , S2(') shows a crossover from the AdS space behavior (which corresponds to the entanglement entropy of a CFT ground state) to a volume law. Such behavior of S2(') is qualitatively consistent with the behavior of a thermal state (more precisely a pure state with nite energy density) on the boundary. In summary, we see that a random state in the bulk region is mapped by the random tensor network to qualitatively di erent boundary states depending on the entropy density of the bulk. This is a toy model of the transition between a thermal gas state in AdS space and a black hole. In a more realistic model of the bulk thermal gas, the thermal entropy is mainly at the IR region (around the center of the Poincare disk), but there is no hard cuto . Therefore there is no sharp transition between small black hole phase and maximal black hole phase. The size of black hole will keep increase as a function of temperature. In contrast, the lower phase transition between perturbed AdS phase and the small black hole phase remains a generic feature, since the minimal surface will eventually skip some region in the bulk when the volume law entanglement entropy of the bulk states is su ciently high. From this simple example we see how the bulk geometry de ned by a random tensor network has nontrivial response to the variation of the bulk quantum state. Finding a more systematic and quantitative relation between the bulk geometry and bulk entanglement properties will be postponed to future works. At last, we comment on the case of two-sided black holes. As is well-known, an eternal black-hole in AdS space is the holographic dual of a thermo eld double state [33], which is an entangled state between two copies of CFTs, such that the reduced density matrix of each copy is thermal. As a toy model of the eternal black hole we consider a mixed bulk state with density matrix b = @ Here pxure is a pure state density matrix while x mix is a mixed state with This density matrix described a bulk state in which all qudits in the disk region jxj < b are entangled with some thermal bath. The behavior of the geometry can be tuned by the entanglement entropy of x mix for each site, which plays a similar role as log Db in the single-sided black hole case. The analyis of minimal surfaces for a boundary region in this state can be done exactly in parallel with the single-sided case. Therefore, instead of repeating the similar analysis, we only comment on two major di erences between the single-sided and two-sided case: 1. Because the bulk state is not a pure state, the entropy pro le of the boundary system with respect to the di erent boundary region size is not symmetric at half the system size. However, there is still a phase transition as a function of entropy density of the bulk, above which a cusp appears in the entropy pro le. This phase transition corresponds to the transition between thermal AdS geometry and AdS black hole geometry [12]. 2. Similar to the single-sided case, there is a second phase transition where further increase of bulk entropy density does not change the boundary entanglement feature any more. The transition point for two-sided case occurs at a slightly di erent value state is a mixed state with entropy lg 1 log D 14 bb2 , which is given by the boundary area of the disk region in the bulk. The boundary of the disk plays the role of the black hole horizon. While the behavior observed here is consistent with black hole formation, it is important to stress that the conclusion is actually ambiguous. Geodesics can be excluded from regions of space even in the absence of a black hole.7 The presence of a black hole is ultimately a feature of the causal structure, so resolving the ambiguity would require introducing time into our model. Random tensor networks as bidirectional holographic codes In the previous section we discussed the entanglement properties of the boundary quantum state obtained from random tensor networks. In this section we will investigate the properties of random tensor networks interpreted as holographic mappings (or holographic codes). In ref. [8], the concept of a bidirectional holographic code (BHC) was introduced, which is a holographic mapping with two di erent kinds of isometry properties. A BHC is a tensor network with boundary legs and bulk legs. We denote the number of boundary legs as L and the number of bulk legs (i.e., the number of bulk vertices) as V , and denote the dimension of each boundary leg as D and that of each bulk leg as Db. The rst isometry is de ned from the boundary Hilbert space with dimension DL to the bulk Hilbert space with dimension DbV . The physical Hilbert space is identi ed with the image of this isometry from the boundary to the bulk, so that the full bulk Hilbert space is redundant in the sense that it contains many non-physical states. The condition identifying these physical states can be formulated as a gauge symmetry. The second isometry is de ned from a subspace of the bulk Hilbert space to the boundary. The physical interpretation of this subspace is as the low energy subspace of the bulk theory. The bulk theory is intrinsically nonlocal in the space of all physical states, but locality emerges in the low energy subspace. More precisely, the degrees of freedom at di erent locations of the low energy subspace are all independent, and a local operator acting in the low energy subspace can be recovered from certain boundary regions, satisfying the so-called \error-correction property" [6, 9]. For this reason, the low energy subspace is also referred to as the code subspace. In this section, we will investigate the properties of random tensor networks and show that they satisfy the BHC conditions in the large D limit and moreover have properties that are even better than the BHC constructed using pluperfect tensors in ref. [8]. Code subspace We start from the holographic mapping in the low energy subspace, or \code subspace" in the language of quantum error correction [9]. Physically, the code subspace is a subspace of the Hilbert space which corresponds to small uctuations around a classical geometry in the bulk. More precisely, the criterion of \small uctuations" states that these states are described well by a bulk quantum eld theory with the given geometrical background. In other words, in the code subspace the bulk elds (operators) at di erent spatial locations are independent and the Hilbert space seems to factorize with respect to the bulk position. The fact that one cannot take the code subspace to be the entire Hilbert space, i.e. that 7We thank Aron Wall for bringing this point to our attention. locality in the bulk fails if we consider the entire Hilbert space, is the essential feature of a theory of quantum gravity (de ned as the holographic dual of a boundary theory), as compared to an ordinary quantum eld theory in the bulk. In general, the choice of code subspace is not unique. However, the random tensor network approach allows for a simple and explicit choice. We de ne the code subspace to be the tensor product of lower-dimensional subspaces at each vertex of the graph: Hcode = N x Hx (Db). Here, Hx (Db) is a Db-dimensional space at site x in the bulk. The holographic mapping restricted to this subspace is simply a tensor network with a smaller bond dimension Db for each bulk leg. In the following, we investigate the condition for the bulk-to-boundary map to be an isometry, which thus determines the value of Db that makes such a subspace an eligible code subspace. When we view the tensor network as a linear map M from the bulk to the boundary, obtained for the second Renyi entropy, it is more convenient to view the tensor network as a pure state. Choose an orthonormal basis fj ig of the bulk and a basis fjaig for the with the pure quantum state bulk reduced density matrix b = tr@ (j M ih M j) = D V I is maximally mixed. Therefore, b the isometry condition can be veri ed by an entropy calculation. For that purpose we calculate the second Renyi entropy of the whole bulk. In the large D limit, this is mapped to an Ising model partition function in the same way as in the RT formula discussion, except that there is now a pinning eld everywhere in the bulk, in addition to the boundary: A [fsxg] = because other con gurations of hx; bx will be used in our later discussion.) In this action, the e ect of the bulk pinning eld bx competes with the boundary eld hx. The relative strength of these two pinning elds is determined by the ratio log Db= log D. If log Db log D, the lowest energy con guration will be the one with all spins pointing up. In the opposite limit log Db log D, all spins point down. For the purpose of de ning a code subspace with isometry to the boundary, we consider the limit log D. In that case all spins are pointing up, and the only energy cost in the Ising action (4.2) comes from the last term, leading to the entropy S2;bulk = V log Db; ent boundary conditions. In the large D limit, these Ising partition function are dominated by the contribution of the lowest energy spin con guration, which gave rise to the Renyi entropies Sn(A) are approximated by the Ryu-Takayanagi formula. Before going into the details, we rst present our conclusion: 1. For a system with volume (i.e., number of bulk vertices) V , for an arbitrary small > 0, one can de ne a critical bond dimension Dc = and c2n constants independent from the volume. The meaning of the exponent c2n will be explained below. In the limit D Dc the deviation satis es SnRT(A)j < ; with a high probability P ( ) = 1 is the RT formula for the n-th Renyi entropy, including the bulk correction. We will always assume that the bulk dimension Db is nite, so that in the large D limit the minimal surface A is determined by minimizing the area. 2. We subsequently show that under a plausible physical assumption on the free energy of the statistical models, the bound given in eq. (7.1) can be improved by reducing the critical bond dimension to Dc = 0 2V 2= 2n ; with 0 a non-universal constant. The meaning of the exponent 2n will be explained The general bound on To start, we denote the Ising action (5.6) of the minimal energy spin con guration with We shall assume throughout this section that the minimal energy con guration is unique (otherwise see section 9 and appendix F). In the large D limit, Z0(n;1) approaches Z0(n;1);1 e A(mni)n[h0;1], with h0;1 denoting the boundA(mni)n[h1] = (n eld con guration for the calculation of Z(n) and Z1(n), respectively. We note that 0 1)Sn(EA; b) + log Cn;x and A(mni)n[h0] = log Cn;x, with Cn;x de ned in eq. (5.3) and the text below it. Thus the RT formula (7.2) can also be SnRT(A) = 1 A(mni)n[h1] = 4 where we have used that Z1(n) Z(n);1 since at nite temperature the partition function receives contributions from all spin con gurations, not just the minimal energy con guration. The key insight now is that the second moment of Z1(n), (Z1(n))2, can be interpreted as limit, the lowest energy spin con guration is given by the same minimal energy surface as that in the Ising model for the Z(n) calculation, with corresponding energy A(m2inn)[fhxg] = 2A(mni)n[h1] + X log log (2n)! + (2n 1) log Db. By combining eqs. (7.4), (7.6) and (7.7) we The last term comes from the di erent normalization factors in the average Z(n) and . Thus the ground state energy of this Sym2n-spin model is essentially two times that of the Symn-Ising model. More precisely, it follows from (7.5) that 1 = 1 = To bound the right-hand side term, we use that by assumption the minimal energy conguration is unique; all other con gurations incur an additional energy cost of at least 1) log Db V (cf. the discussion before eq. (5.9)). Since there are (2n)! con gurations at each bulk site this leads to the conservative upper bound and jZ0(n)=Z0(n);1 1j < =4 with probability at least 1 32 ec2nV =D 2. In this case we can bound the deviation of the n-th Renyi entropy from the Ryu-Takayanagi formula (7.3) by SnRT(A)j = where we have used 2 such that log(1 =2. We have thus proved that the desired bound (7.1) holds with probability at least 1 DDc , where Dc = 32 Interestingly, the above results for the Renyi entropies can be used to show corresponding statements for the von Neumann entropy. For a bulk direct product state, this is easy to see: here, the Ryu-Takayanagi formula amounts to SnRT(A) Sn(A) for any quantum state and S(A) log Dj Aj in any tensor network state, we have essentially matching upper and lower bounds for the von Neumann entropy, and hence S(A) ' log Dj Aj with high probability. This result can be established more generally even in the presence of an entangled bulk state as long as Db adapting the techniques of [40] (cf. section 8). Improvement of the bound under a physical assumption The above results establish rigorously that the entropies approximate the Ryu-Takayanagi formula in the limit of large D. However, the technique lead to a rather conservative estimate of the nite D correction, since it only proves that the entropy is close to the RT value for exponentially large bond dimension D ec2nV . In this subsection we would like to argue based on a plausible physical assumption that actually the RT formula applies to a much larger range of D, as long as D is bigger than some power law function of V . To start, let us reinvestigate eq. (7.7), which was the basis of the general bound (7.1). In obtaining eq. (7.7) we replaced the energy of all higher energy spin con gurations by their minimum log DDV . This leads to a very conservative bound since most con gurab tions certainly have an energy much higher than that. Since the statistical model has a local action, the number of excitations with lowest energy is actually proportional to V rather than exponential of V . Although the number of slightly higher energy excitations are super-extensive, it is still true that the free energy of the spin model is extensive at nite temperature. Furthermore, the free energy approaches the ground state energy in the lo temperature (large D) limit exponentially, since the probability of lowest energy excitation with energy Eg is suppressed by the Boltzman weight e Eg = D Eg=2. Using these plausible physical observations we can write the asymptotic form of the free energy F = there is generically a power law term (log D)a multiplying the exponential factor in the free energy density. However, this power law correction is not important for our bound, C(log D)aD Eg=2V , with C a constant. Note that since we can choose an energy In the limit D V 1= 2n this implies that 2n=2V: D 2n=2 by choosing a constant C0 slightly larger than C. If we substitute this estimate for the conservative bound (7.7) then eq. (7.8) becomes D 2n=2 and likewise for Z0(n). We may now proceed as above and conclude that, under the assumption (7.10) on the Ising models, the Renyi entropies satisfy the RT formula to arbitrary precision and with arbitrarily high probability if D V 2= 2n. This improves the dependency of the bond dimension on the system size from an exponential function of V to a To illustrate the behavior of the free energy (7.10) in an explicit example, we consider an Ising model on the square lattice. (It should be noted that the Ising model case does not directly apply to the discussion above since Sym2n-spin models are used there. However the behavior of free energy is generic for gapped spin models.) Speci cally, we shall consider a cylindrical geometry given by an M N square lattice with periodic boundary conditions along the rst direction and open boundary conditions along the second one. In this setup, the minimal surface bounding a boundary region is unique. As the boundary region we using Onsager's solution [41, 42]. The asymptotic behavior for large D is given by presented in appendix E. Possible e ects of even smaller bond dimension When D does not satisfy the condition D (CV )1= , the deviation of the entropy from the RT value can be large. An interesting question is whether the correction to the RT formula is simply a renormalization of the coe cient of the area law, or if there is a qualitative change. For the second Renyi entropy, the quantity log Z1=Z0 is the free energy cost induced by the boundary pinning eld hx in the Ising model. The behavior of this free energy cost depends on the strength of the uctuations of the domain wall con guration. If the domain wall only uctuates mildly around the minimal energy con guration, one can naturally expect the energy cost of the domain wall is still proportional to its area, although the coe cient may be renormalized to be di erent from the bare value given by the lowest energy con guration. In contrast, if the domain wall is strongly energy of the domain wall may have a qualitatively di erent dependence in the minimal area j Aj. Interestingly, the behavior of the domain wall in the Ising model was studied a long time ago. If the bulk spatial dimension d 3, it was found that there is a critical temperature Tr (which is lower than the phase transition temperature Tc of the Ising model), below which the uctuations of a domain wall con guration have a range. The transition at Tr is known as the roughening transition [43, 44].12 It is natural to expect that the RT formula for second Renyi entropy applies for any log D > Tr 1, which nite even if the system size V goes to in nity. However, it is not clear how to bound the deviation of S2(A) from log Z1 , given by the uctuation terms in eq. (2.8). uctuation of its position is always strong. Consequently, the RT formula does not apply to any nite D if we take V ! 1 Relation to random measurements and the entanglement of assistance pairs Nhxyi jxyi for the internal edges and The average over random tensors that has played a central role in this work has appeared previously in the quantum information literature, but with a very di erent motivation. The de nition of the boundary state j i in eq. (2.2) involves contracting the random vertex x jVxi at the bulk vertices with a bulk state j bi as well as a collection of Bell x jx@xi connecting boundary vertices to their boundary connecting points @x. To obtain a new physical interpretation for the state j i one can start with the state j i = j bi and perform a random measurement at every bulk vertex x. The post-measurement state on the unmeasured boundary vertices will then have the same distribution as j i that the state j i in eq. (8.1) is supplemented by new bulk-boundary Bell pairs N as compared to eq. (2.2). The reason lies in the change of perspective; in section 2 the random vertex states were being projected to Bell pairs and the bulk state, but here the Bell pairs and the bulk state are being projected to the random states, and therefore we need a larger Hilbert space to get a non-empty Hilbert space after projection. See gure 11. These two perspectives are mathematically equivalent in our examples. The post-measurement state on the unmeasured boundary vertices will then have the same distribution as j i. From this point of view, boundary entanglement is being induced by performing a suitable measurement on a joint bulk-boundary state. 12We would like to thank Steven Kivelson for teaching us this result. " "Vx| |Φb! ⊗ " |xy! ⊗ " |x∂ x! bulk state and two boundary dangling legs @1 and @2. (b) The construction of the section 2. The state of the random tensors N x jVxi is contracted with Bell pairs and the bulk state to obtain the boundary state j i@1@2 on the Hilbert spaces of @1@2. Bell pairs are shown by thick lines. (c) The construction of section 8. Here we have a large background state j i and contract with random vertex states to obtain the state j i@1@2 on the boundary. Note that we need to add extra boundary One of the basic problems of quantum information theory is how to establish as much high-quality entanglement as possible between spatially separated parties. One scenario that had been considered was to start with a pure state j iABC of three systems and to ask how much entanglement could be induced on average between A and B upon measuring C, optimized over all possible C measurements. Because the party in possession of C is helping A and B establish entanglement, this quantity is known as the entanglement of assistance EA(A; B) [45]. Concavity of the entropy implies a trivial upper bound: S( A), and likewise for Y . In a remarkable paper, Smolin et al. showed that this upper bound was asymptotically achievable [46]: EA(Ak; Bk) k = min[S( A); S( B)]: Going further, one can imagine partitioning C into subsystems C1; C2; : : : ; Cm and allowing only local measurements of each Cj instead of joint measurements of the entire C system. From an engineering perspective, such a scenario could arise naturally if A and B are distant and the Cj represent intermediate \repeater" stations in a network [47]. The additional locality restrictions will reduce the amount of entanglement that can be induced between A and B. While the concavity upper bound still applies, it can be applied here with a bit more nesse. If we choose any subset S fC1; : : : ; Cmg, then the bound implies that this multipartite version of the entanglement of assistance, EAmulti(A; B) , will be bounded above by S( ASc ) since the total entanglement generated between A and B will be no more than the entanglement between ASc and B after measuring S but prior to measuring Sc. Likewise, EAmulti(A; B) S( BSc ). Therefore min[S( AS ); S( BS)] = where the equation follows from the fact that the entropies of two complementary subsystems of a pure state are always the same. The Smolin et al. result applied inductively gives EAmulti(Ak; Bk) k = Consider now the special case in which j i has the form of eq. (8.1) used in this paper. Set A to be any boundary region, B the complement Ac of A in the boundary and identify the di erent subsystems Cj with the bulk vertices x. The righthand side of eq. (8.4) is then nothing other than the Ryu-Takayanagi formula with corrections due to the bulk state j bi, since minimizing over subsets S amounts to minimizing over cuts in the tensor network: S( AS ) = j Aj log D + S(EA; b); where EA is the bulk region corresponding to the minimizing set S and j Aj is the size of the cut separating EA from its complement. This matches eq. (3.3) up to the substitution of the von Neumann entropy for the second Renyi entropy. (The reason for taking the 1 limit in (8.4) is essentially to make all Renyi entropies equal after suitable small perturbations to the state. For reasonable physical choices of j bi such as quantum on the righthand side of (8:4). This has been shown, for example, in the case that A is an interval in a 1+1 dimensional CFT [48].) While the original proof of the multipartite entanglement of assistance formula used classically-inspired random coding information theory techniques, subsequent proofs proceeded by performing appropriate isotropic measurements of the C subsystems [40, 49]. Because of the equivalence between contracting random tensors and performing random measurements, the analyses in the quantum information theory literature are mathematically very similar to the calculations in this article. The analog of the calculations justifying reconstruction of a bulk operator contained in the entanglement wedge of a boundary region A has even appeared, again with a di erent motivation, as the \split-transfer protocol" [40, 50]. One could go as far as to rename the one-shot multipartite entanglement of assistance formula of [50] the \fully-quantum Ryu-Takayanagi" formula, in that it captures the essence of Ryu-Takayanagi without making any prior assumptions about the geometrical interpretation of the bulk state. Aside from connecting to pre-existing literature, one virtue of this change of perspective is that it suggests a possible physical justi cation for the random tensor networks in our model. One could imagine taking the state in a quantum theory of gravity and measuring the Planckian degrees of freedom of a large \bulk" subset, leaving some \boundary" degrees of freedom and bulk elds unmeasured. If the Hilbert spaces are large and the measurements generic, then the measurements should reveal almost no information about the bulk, inducing a nontrivial mapping between non-Planckian bulk degrees of freedom and the boundary. In this way, xing the bulk Planckian degrees of freedom in the bulk-boundary state through measurement generically produces a holographic correspondence. Expanding around a particular background geometry in this picture amounts to choosing a bulk-boundary state with the correct area law entropy and randomly xing the Planckian degrees of freedom through projection. Random tensor networks from 2-designs In the construction of our random tensor network state (2.1), the tensors jVxi were chosen to be Haar-random, i.e., drawn from to the unitarily invariant ensemble of pure states. However, our calculations for the second Renyi entropy in section 2.2 made use only of the second moments of the Haar measure. This calculation led to the emergence of a classical Ising model and thereby to the Ryu-Takayanagi formula. It is therefore natural to consider other ensembles of pure states whose rst two moments agree with those of the Haar measure, known collectively as complex projective 2-designs [51]. It follows from the discussion in section 7 that for a tensor network state with Haarrandom tensors and bulk direct product state b, the Ryu-Takayanagi formula S(A) ' log Dj Aj will be satis ed with high probability in the limit of large D if the minimal geodesic is unique. This conclusion was obtained from considering higher moments of the Haar measure and therefore does not apply for a general 2-design. Another complication arises from the fact that the tensor network state can be zero (i.e., = 0) with nonzero probability, in which case its entropies are not well-de ned. In appendix F we show that for any 2-design the boundary state is nonzero with high probability and that, moreover, S2(A)6=0 second Renyi entropy conditioned on the boundary state being nonzero. We note that, since the lower bound in (9.1) matches the deterministic upper bound up to a constant, it follows that S(A) is at most constantly away from SRT(A) with high probability. One random ensemble of particular interest is given by stabilizer states. Stabilizer states, de ned as common eigenvectors of generalized Pauli operators, are quantum states that can be highly entangled, but whose particular algebraic structure allows for e cient simulation and e ective reasoning [52]. It has been shown in [53] that pure stabilizer states Thus (9.1) applies to the entropies of the corresponding tensor network state (2.1) constructed from random stabilizer states. Such a state is again a stabilizer state, as we argue in appendix G. The particular algebraic structure of stabilizer states implies that their reduced density matrices not only have at spectrum (so that all Renyi entropies agree with the von Neumann entropy) but in fact that all their entropies are quantized in units of log p. It follows that for large p the Ryu-Takayanagi formula S(A) = SRT(A) = log Dj Aj will hold exactly with high probability (even in the presence of multiple minimal geodesics). In particular, we may use this construction to obtain random holographic codes and evaluate their error correcting properties by using (4.7) and the exact Ryu-Takayanagi formula (9.2) purely from the structure of the tensor network. In [6], holographic codes were constructed from perfect tensors, i.e., states that are maximally entangled across any bipartition, and it was shown that under certain circumstances this already implies the Ryu-Takayanagi formula (such as for single intervals in nonpositively curved space). Random stabilizer states are perfect tensors with high probability,13 and so the analysis and results of [6] can likewise be applied to our random tensor networks constructed from stabilizer states with high bond dimension. However, our tensors are not only perfect or pluperfect [8] but also generically so and therefore can achieve the Ryu-Takayanagi formula for arbitrary subsystems. Another consequence of (9.2) is that any entropy inequality that is valid for arbitrary quantum states, or even just for stabilizer states [54, 55], is also valid for the RyuTakayanagi entropy formula, thereby establishing a conjecture from [22]. This can be understood as consistency check of the Ryu-Takayanagi formula, generalizing [56], where the validity of strong subadditivity was veri ed for the Ryu-Takayanagi formula. We refer to [57] for a detailed analysis of the entanglement properties of tensor networks built from random stabilizer states. Conclusion and discussion In this work we have studied the quantum information theoretic properties of random tensor networks with large bond dimension. In the following we will revisit our method from a more general perspective and summarize our ndings. Viewing each tensor as a quantum state jVxi, the tensor network state (jVxihVxj) obtained by contracting these tensors is a linear function of each tensor. Denote by fn( ) an arbitrary function that is a monomial function of the state with degree n. Then the state average of fn over all possible choices of jVxi is exactly mapped to the partition function of an classical spin model, with degrees of freedom in the permutation group Symn, with the spins de ned on the vertices of the same graph that underlies the tensor network. Di erent physical quantities can be translated to di erent functions fn( ). When the tensor network is used as a quantum state of the boundary, one can consider tr trA A, which corresponds to the n-th Renyi entropy of A. When the tensor network is used as a linear map, it can be viewed as a \holographic mapping" between two parts of the degrees of freedom (boundary and bulk, respectively). In this case, in addition to the Renyi entropies one can study the entanglement entropy of a given region while another region is n for an arbitrary region 13This follows as a special case of our result for a tensor network with a single vertex. We thank Fernando Pastawski for explaining an alternative proof of this fact to us. projected to a certain quantum state. For example, one can project the bulk into a given quantum state and study the entanglement properties of the resulting boundary state. We can also de ne basis-independent measures of correlation functions and relate that to a calculation of monomial functions, which allows us to study the behavior of two-point functions in the boundary state. The mapping between the random state average and the spin-model partition function has rich consequences. For a random tensor network state, in the large D limit the Ryu-Takayanagi formula can be proven for all Renyi entropies, where the minimal surface area condition comes naturally from minimizing the energy of the spin model with given boundary conditions. The Ryu-Takayanagi formula also generalizes naturally to include bulk state corrections when there is nontrivial quantum entanglement in the bulk. As a particular example, we study the behavior of minimal surfaces in the presence of a bulk random state, and show how the minimal surface behavior can change topologically upon increase of the bulk entanglement entropy, in a way that is qualitatively consistent with black hole formation. In addition to entanglement entropy, we also studied the behavior of two-point correlation functions. The boundary correlation functions between two regions are directly determined by bulk correlation functions between two corresponding regions known as the entanglement wedges of the boundary regions. In the special case of hyperbolic space, our results on correlation functions imply that the boundary theory has power law correlations with a large scaling dimension gap. In the large D limit there are two types of scaling dimensions, those which does not scale with D coming from the bulk quantum state, and those which scale with D coming from the tensor network contribution. Such behavior of the scaling dimension gap is consistent with those of CFT ground states with a gravity dual, although the condition is necessary but not su cient. Random tensor networks provide a new framework for understanding holographic duality. Besides the properties studied in this paper, many other physical properties can be evaluated by the mapping to classical spin models. Compared to other tensor network models, properties of the random tensor networks can be studied much more systematically. The large dimension D limit is an analog of the large N limit in gauge theories. The fact that a random tensor network with large dimension automatically satis es many desired properties for holographic duality further supports the point of view that semi-classical gravity is deeply related to scrambling and chaos. There are a several open questions that shall be studied in future works. One question is whether it is possible to use a random tensor network to describe the ground state of a conformal eld theory. The underlying graph of random tensor networks on hyperbolic space is invariant under a subgroup of discrete isometries of the bulk which do not involve transformation in time. Therefore we expect the distribution of tensor network states on the boundary to remain invariant under the subgroup of boundary conformal transformations that correspond to the bulk discrete isometries, modulo complications arising from the cuto . It is an open question whether we can modify the tensor network state to preserve the whole conformal symmetry. Related to the discussion of Renyi entropies, this may require modi cation of the state on links between vertices. It would also be interesting to consider random tensor network models where the same tensor is placed at each vertex.14 Another question is how to generalize this formalism to include dynamics. What Hamiltonians of the boundary theory can be mapped to local Hamiltonians in the bulk \low energy" subspace? How to see that conserved currents on the boundary correspond to massless elds in the bulk? The answers to these questions will also be essential for understanding how the bulk gravity equation emerges. We would like to thank Xi Dong, David Gross, Daniel Harlow, Steve Kivelson, Juan Maldacena, Fernando Pastawski, Brian Swingle and Aron Wall for their helpful insights. PH and MW gratefully acknowledge support from CIFAR, FQXI and the Simons Foundation. SN is supported by a Stanford Graduate Fellowship. XLQ and ZY are supported by the National Science Foundation through grant No. DMR-1151786, and by the David and Lucile Analytic study of the three phases for a random bulk state In this appendix, we will provide an analytical explanation of when the transition happens between the perturbed AdS phase and the small black hole phase, and between the small black hole phase and the maximal black hole phase. In particular, we will show a) why at the transition between the perturbed AdS phase and the small black hole phase, lg 2 log Db scales as the square root of lg 1 log D, b) why in the large D limit, the transition between the small black hole phase to the maximal black hole phase happens at lg 2 log Db = lg 1 log D(1 + b2)=(2b). In fact, the problem we are going to solve has already been set up in eq. (3.4). The transition between the perturbed AdS phase to the small black hole phase is decided by the stability of the solution that covers half of the boundary system and goes through the center of the Poincare disk. Such a solution is the extremal solution of eq. (3.4), since it minimizes the area contribution from the domain wall and maximizes the volume contribution from the bulk random state. However, when this solution becomes a local maximum instead of a minimum, it means that the minimal surfaces of all the boundary regions would avoid the center of the Poincare disk. In other words, there exists a region in the bulk inaccessible to any measurements from the boundary smaller than half system size. For convenience, we use (x; y) coordinates instead of (r; ) in this problem. Thus what we care about is 2S2( =2) y(x1) y(x2) y=0 = lg 1 log D 14After the rst version of this manuscript had appeared, Matthew Hastings showed that for large D the entanglement spectra of reduced density matrices have the same limiting behavior in both models [58]. Therefore typical Renyi entropies in the model with identical tensors are also given by the Ryu-Takayanagi formula if D is su ciently large. hard to analytically diagonalize instability happens at lg 1 log D 2S2( =2) y(x1) y(x2) y=0 expression as a matrix, the rst term is always a positive de nite matrix after integrating by parts of the derivative term, and the second term is a negative de nite matrix, which and maximizes the volume contribution from the bulk random pure state. Although it is , it is straightforward to observe that the Now we turn to the second question, the transition between the small black hole phase and the maximal black hole phase. In order to understand the formation of the maximal black hole, we need a more detailed investigation of eq. (3.4). We rst focus on the random pure state region r b, and assume the minimal surface enters this region at angle ' and '. The minimization problem in eq. (3.4) can be solved by asking 0 = lg 1 log D DVT =lg2 D2Vr( )=lg2 Z ' The above variational equation contains both the derivative and the integration (contained in Vr( )) of r( ). But in the large D limit, which indicates that the transition happens when Db is also big, as long as 2Vr( ) < VT , D2Vr( ) b DVT near transition point. Thus in this limit, the above equation can be simpli ed with only r( ) and its derivatives left. r2( ))2 = 0: The trick we use to solve this equation is to transform it back to a minimization problem, I[r( )] is the objective function to be minimized with respect to r( ). I[r( )] = Because I[r( )] does not explicitly contain , thus using a Legendre transformation, we only need to solve a rst order di erential equation. I[r( )] = lg 2 log Db = C; whose analytic solution is r2( ) = C = 1)2 + 4b2 sin2 ' b2 = r2(') lg 2 log Db cos(2') + lg 1 log D 2 cos2 ' which can be satis ed if What is interesting is that this condition is independent of ', the angle at which the minimal surfaces enters the random pure state region. In other words, when which means the minimal surfaces will enter the random pure state region. However, to the boundary of the random pure region, indicating that the formation of the single sided black hole is complete. Thus we have proved in the large D limit, the transition between the small black hole phase and the maximal black hole phase happens at lg 2 log Db = lg 1 log D(1 + b2)=2b. Derivation of the error correction condition In this appendix we give a short proof that the vanishing of the mutual information I(C : BC), eq. (4.7), implies that any operator OC in bulk region C can be recovered from the boundary region A. We do so for the reader's convenience as the proof will describe the construction of the boundary operator rather explicitly, but note that the result can be readily extracted from the literature [34, 59, 60]. In the following it will be crucial to distinguish the input systems C and C of the de ned in (4.1). We will thus denote the latter by C0 and C0, so that bulk-to-boundary isometry M from the corresponding subsystems of the pure state j M i illustration of the recovery equation (4.6). where j CC0 i and j C C0 i denote maximally entangled states between C and C0 and between trA( M ) = C0 inition, j M i is a puri cation of (B.1), but we can also nd a puri cation that respects the product structure j CC0 i j BC0E i, obtained by purifying C0 to a maximally entangled state and trAC0 ( M ) to an arbitrary pure state j BC0E i. If we choose the dimension of E to be su ciently large then the two puri cations can be related by an isometry V from A to CE: V j M i = j CC0 i It can now be readily veri ed that any bulk operator C can be recovered from A by using is a direct consequence of the following calculation: OA j M i = V y C V j M i = V y C j CC0 i = M C j CC0 i where we have used (B.2) and that C j CC0 i = for an illustration. Uniqueness of minimal energy con guration for higher Renyi models In this appendix we give a formal proof of the assertion made in section 5 that the spin model with action (5.6) has a unique minimal energy con guration, given by setting the entanglement wedge EA to the cyclic permutation C tity, provided the entanglement wedge EA is unique. (n) and its complement to the iden For simplicity, we assume that sions are powers of a xed integer), and we consider the equivalent spin model with energy E[fgxg] = where the gx are variables in Symn, with x and y ranging over both bulk and boundary vertices, subject to the boundary conditions gx = C (n) in A and gx = 1 in A (cf. section 8). The rst observation is that n (g) is equal to the minimal number of transpositions (i.e., permutations that exchange only two indices) required to write a permutation g. This d(gx; gy) := n de nes a metric. In particular, it satis es the triangle inequality. The second ingredient is that, by the integral ow theorem, we can decompose a maximal ow between A and A into edge-disjoint paths. Each path starts in A, ends in A, and by the max- ow/min-cut theorem there are j Aj many such paths P1; : : : ; Pj Aj Now consider an arbitrary con guration fgxg that satis es the boundary conditions. We can bound its energy by looking only at those edges that occur in one of the paths, resulting in the lower bound k=1 hxyi2Pk Along each path Pk, the rst spin is assigned the cyclic permutation C(n) and the last spin the identity permutation 1. Therefore, the triangle inequality (invoked once for each path) k=1 hxyi2Pk X d(C(n); 1) = (n Note that the right-hand side is just the energy cost of the con guration where we assign C(n) to the spins in EA and 1 to all other spins. We claim that this is the unique minimal energy con guration. To see this, suppose that fgxg is an arbitrary con guration that achieves this energy cost. where gx = C Case 1. The only permutations that appear in fgxg are C(n) and 1. Then the domain (n) is a minimal cut between A and A, i.e., an entanglement wedge for A. Since we have assumed that the entanglement wedge is unique, it must be equal to EA. Thus fgxg is the con guration described above. Case 2. The con guration fgxg contains some other permutations. Since it is a minimal energy con guration, both inequalities (C.1) and (C.2) above must be tight. The fact that the rst inequality is tight means that if an edge is not contained in any of the paths Pk then the con guration fgxg necessarily assigns the same permutation in Symn to its endpoints. It follows that the rst inequality remains tight if we modify the con guration fgxg by changing an entire domain from one permutation to another. For the second inequality, we can use the triangle inequality to see that the sequence of permutations in any path Pk must always be of the form C permutations that are neither C(n) nor 1. Indeed, if this were not the case then the energy cost of the corresponding path would be higher than (n 1). But this implies that by either changing all other permutations to C two distinct minimal energy con gurations that only contain C(n) and 1. By case 1, this is (n), or by changing all of them to 1, we obtain ; 1; : : : ; 1, where denotes a sequence of condition holds for OB. (b) The graph representing the matrix M = tr OAOB . We use red and blue dots to represent the basis operators OA and OB, respectively. We have drawn slightly assymetric shape to keep track of the di erence between A and B regions. (c) Using the orthonormality condition in sub gure (a), the quantity (M yM )n is tranformed to a contraction of 2n copies of . (d) A compact way of drawing tr (M yM )n , which corresponds to eq. (6.6). Calculation of C2n in section 6 In this appendix, we will present the derivation from eq. (6.5) to eq. (6.6) in section 6. We rst calculate M yM using the orthonormality condition (6.1). = M Similarly we can apply the orthonormality condition in the B region when we multiply M yM . For example, in which a; b; c; d are indices in the Hilbert space of A, and m; n; k; l are those in B. The best way of visualizing this calculation is by introducing a diagrammatic representation, as is shown in gure 13. In the trace of (M yM )n, there are 2n copies of the density matrix . The n contractions of A indices lead to pairwise permutations between pairs of density matrices 1 $ 2n. Similarly, the contractions of B indices lead to pairwise permutations between 2 $ 3, 4 $ 5, . . . , 2n $ 1. This concludes the proof of eq. (6.6). Partition function of Ising model on the square lattice In this appendix, we calculate the partition function of the Ising model in the large D (low temperature) limit on a 2D rectangular lattice of size M N , with periodic boundary conditions along the rst direction and open boundary conditions along the second one. We will use several results that can be found in [42]. As in the main text, we let boundary pinning eld pointing down everywhere, by Z0( ), and its zero-temperature limit by Z01 = Z0( ! 1). When system size is large, M; N sinh 2 (cos 1 + cos 2) 1 Z = 1 + D 4 + O(D 6) MN d log (1 W ( )) ( ) = W ( ) = 1( ) = tanh( ) 1 + tanh( ) ( ) = (1 + tanh( )2)2 2( ) = tanh( ) 1 + tanh( ) Thus the leading order correction in the large D limit is M N D 4. with boundary pinning eld down everywhere except for in a single interval of length L. in [42]. When L 1, we can expand Z1( )=Z11 to leading order in D and L, = 1 + D 4 + O(D 6) MN Thus the leading order correction is 2LD 1. Average second Renyi entropy for 2-designs In the following, we will show that (9.1) holds for an arbitrary 2-design in the limit of large bond dimension. We recall from section 7 that the inequality S2(A) S(A) holds for arbitrary quantum states, while S(A) log Dj Aj in any tensor network state. Therefore it remains to prove the lower bound on the average of the second Renyi entropy. T = 1= Q x Dx. Together with our calculation in sections 2 and 3 it follows that T =T as follows: 1 = 1 = where A0 refers to the Ising action in its original form (2.13) and A to the simpli ed form (3.1) with constants removed. But any nontrivial spin con guration incurs an energy cost of at least log D, so that we obtain the upper bound 2V =D: By Chebyshev's inequality, it follows that, for any " > 0, We now condition on the event that T =T we obtain the following bound using concavity of the logarithm, 1 ". Writing Xgood for corresponding averages, S2(A)6=0 Z1=pgood. On the other hand, T =T " implies that 6= 0. Thus, p6=0 where we have used (F.2) and S2(A) j Aj log D = O(log D), as follows from the deterministic upper bound in (9.1), which holds for an arbitrary tensor network state. The desired lower bound, S2(A)6=0 Contractions of stabilizer states In this appendix we will show that a tensor network state built by contracting stabilizer states is again a stabilizer state. More generally, let j iA and j iAB denote two stabilizer a fact that is certainly well-known to experts. To see this, we start by writing the contracted state as trA(gAhAB) '(gA; hAB); where we have introduced the function '(gA; hAB) K = f(gA; hAB) 2 G H : '(gA; hAB) 6= 0g is a subgroup of G H and that the restriction of ' to K is a group homomorhA and hB are elements of the generalized Pauli groups of A and B, respectively. Thus the latter case, hA = gA1 for some overall phase ; in particular, hA commutes with G. to verify that '(gAgA0; hABh0AB) = 0hBh0B = '(gA; hAB)'(gA0; h0AB): This implies both that K is a subgroup of G H and that ' K is a group homomorphism. '(K) is a (commutative) subgroup of the Pauli group; it follows that j B0ih B0j = '(gA; hAB) = jker 'j X gB = jKj 1 Thus j B0i is indeed a subnormalized stabilizer state, as we set out to show. 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Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi. Holographic duality from random tensor networks, Journal of High Energy Physics, 2016, 9, DOI: 10.1007/JHEP11(2016)009