Holographic duality from random tensor networks
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.
Keywords:holography, black holes, tensor networks, scaling dimensions, quantum error correction, entanglement of assistance
Physics Department, Stanford University, CA 94304-4060, USA
Tensor networks have been proposed swingle2012 () as a helpful tool for understanding holographic duality maldacena1998 (); witten1998 (); gubser1998 () 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 ryu2006 (). In general, the area law only gives an upper bound to the entropy swingle2012 (), which for particular tensor networks and choices of regions has been shown to be saturated pastawski2015 (). Tensor networks can also be used to build holographic mappings or holographic codes qi2013 (); pastawski2015 (); yang2015 (), 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, so-called pluperfect tensors yang2015 (). These codes simultaneously satisfies several desired properties, including the RT formula for a subset of boundary states, error correction properties of bulk local operators almheiri2014 (), a kind of bulk gauge invariance, and the possibility of sub-AdS locality.
The perfect and pluperfect tensors defined in Refs. pastawski2015 () and yang2015 (), 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. Specifically, we will assume that each tensor in the network is chosen independently at random. We find that the computation of typical Rényi 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 fields. When each leg of each tensor in the network has dimension , these statistical models have inverse temperature . For large enough , they are in the long-range ordered phase, and we find that the entropies of a boundary region is directly related to the energy of a domain wall between different domains of the order parameter. The minimal energy condition for this domain wall naturally leads to the RT formula.111In our models, the RT formula holds for all Rényi entropies, which is an important difference from AdS/CFT dong2016gravity (). We will discuss this point in more detail further below. 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 tensor network:
Effects 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 effect 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 sufficiently highly entangled, no minimal surface penetrates into this region, so that the topology of the space has effectively changed. This phenomenon is analogous to the change of spatial geometry in the Hawking-Page transition hawking1983 (); witten1998b (), where the bulk geometry changes from perturbed AdS to a black hole upon increasing temperature.
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 defines 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 satisfies the error correction properties defined in Ref. almheiri2014 (). To be more precise, all bulk local operators in the entanglement wedge of a boundary region can be recovered from that boundary region.222In 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.
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 defines 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 , and those from the tensor network itself which grow . This confirms that the holographic mapping defined 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 collins2016random () 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 collins2010random (); collins2013area (), the relationship between entropy and the decay of correlations hastings2015random (), and the maximum entropy principle collins2013matrix (). The relation between the Schmidt ranks of tensor network states and minimal cuts through the network has been investigated in cui2015quantum (). 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 swingle2012b ().
The remainder of the paper is organized as follows. In Section 2 we define the random tensor networks. We show how the calculation of the second Rényi 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 effect 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 effective bulk geometry as a function of bulk entropy density. In Section 4 we study the properties of the holographic mapping defined 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 Rényi entropy to higher Rényi entropies. We show that the -th Rényi entropy calculation is mapped to the partition function of a statistical model with a 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 in the case of AdS geometry. In Section 7 we bound the fluctuations around the typical values calculated previously and discuss the effect of finite 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 find that the RT formula can be exactly satisfied in tensor networks built from random stabilizer states, which allows for the construction of exact holographic codes. Finally, Section 10 is devoted to conclusion and discussion.
2 General setup
2.1 Definition of random tensor networks
We start by defining the most general tensor network states in a language that is suitable for our later discussion. A rank- tensor has components with . We can define a Hilbert space with dimension for each leg of the tensor, and consider the index as labeling a complete basis of states in this Hilbert space. In this language, (with proper normalization) corresponds to the wavefunction of a quantum state defined in the product Hilbert space .
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 Fig. 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 . Here, denotes all vertices in the network, and is the state corresponding to the tensor at vertex . Each leg of a tensor corresponds to a Hilbert space. We will denote the Hilbert space corresponding to a leg from to another vertex by , and its dimension by . If a leg is dangling, i.e., not connected to any other vertex, we will denote the corresponding Hilbert space by and its dimension by . (Without loss of generality we can assume there is at most one dangling leg at each vertex.) Connecting two tensors at by an internal line then corresponds to a projection in the Hilbert space onto a maximally entangled state . Here denotes a state in the Hilbert space and similarly for . By connecting the tensors according to the internal lines of the tensor network, we thus obtain the state
in the Hilbert space corresponding to the dangling legs, . We note that is in general not normalized. Tensor network states defined in this way are often referred to as projected entangled pair states (PEPS) verstraete2004 ().
As has been discussed in previous works qi2013 (); pastawski2015 (); yang2015 (), tensor networks can be used to define 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. Fig. 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 the figure) is mapped to a boundary state by this mapping. Such a boundary state can also be written in a form similar to Eq. (1). Instead of viewing the tensor network as defining a mapping, we can equivalently consider it as a quantum state in the Hilbert space , which is a direct product of the bulk Hilbert space and the boundary Hilbert space . Denoting the bulk state as , the corresponding boundary state is
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 Fig. 1 (c), one can view the maximally entangled states on internal lines together with the bulk state 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 , instead of the pure state . The most generic form of the boundary state is given by the density operator
Here the partial trace 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 is a linear function of the independent pure states of each tensor .
In this work, we study tensor network states of the form (3), where the tensors 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 and define with a unitary operator. The random average of an arbitrary function of the state is then equivalent to an integration over according to the Haar probability measure , with normalization .
All nontrivial entanglement properties of such a tensor network state are induced by the projection, i.e., the partial trace with . 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 tensor networks.
2.2 Calculation of the second Rényi entropy
We will now apply this technique to study the second Rényi entropies of the random tensor network state defined in Eq. (3). For a boundary region with reduced density matrix , the second Rényi entropy is given by .333In the quantum information theory literature, the Rényi entropy is usually defined with logarithm in base , . Here we use base to keep the notation consistent with the condensed matter and high energy literature. It is helpful to write this expression in a different form by using the “swap trick”,
Here we have defined a direct product of two copies of the original system, and the operator is defined on this two-copy system and swaps the states of the two copies in the region . To be more precise, its action on a basis state of the two-copy Hilbert space is given by , where denotes the complement of on the boundary.
We are now interested in the typical values of the entropy. Denote the numerator and denominator resp. of Eq. (5) by
These are both functions of the random states at each vertex. We would like to average over all states in the single-vertex Hilbert space. The variables and are easier to average than the entropy, since they are quadratic functions of the single-site density matrix . The entropy average can be expanded in powers of the fluctuations and :
We will later show in Section 7 that for large enough bond dimensions the fluctuations are suppressed. Thus we can approximate the entropy with high probability by the separate averages of and :
Throughout this article we use for asymptotic equality as the bond dimensions go to infinity. In the following we will compute and separately and use (9) to determine the typical entropy. To compute , we insert Eq. (3) into Eq. (6) and obtain
In this expression we have combined the partial trace over bulk indices in the definition of the boundary state and the trace over the boundary indices in Eq. (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 different sites are introduced by the projection. The average over states can be done by taking an arbitrary reference state and setting . Then the average is equivalent to an integration over with respect to the Haar measure. The result of this integration can be obtained using Schur’s lemma (see, e.g., Ref. harrow2013 ()):
Here, denotes the identity operator and the swap operator defined in the same way as described above, swapping the two copies of Hilbert space of the vertex (which means all legs connecting to ). The Hilbert space dimension is , the product of the dimensions corresponding to all legs adjacent to , including the boundary dangling legs. It is helpful to represent Eq. (11) graphically as in Fig. 2 (a) and (b).
Carrying out the average over states at each vertex , then consists of terms if there are vertices, with an identity operator or swap operator at each vertex. We can then introduce an Ising spin variable , and use () to denote the choice of and , respectively. In this representation, becomes a partition function of the spins :
For each value of the Ising variables , the operator being traced is now completely factorized into a product of terms, since acts on each leg of the tensor independently. This fact is illustrated in Fig. 2 (c). The trace of the swap operators with is simply with the second Rényi entropy of in the Ising spin-down domain defined by . The trace on boundary dangling legs gives a factor that is either or , depending on the Ising variables and whether is in . To be more precise, we can define a boundary field
Then the trace at a boundary leg gives .
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 simplified by recalling that has the direct product form in Eq. (4). Therefore the second Rényi entropy factorizes into that of the bulk state and that of the maximally entangled states at each internal line . The latter is a standard Ising interaction term, since the entropy of either site is while the entropy of the two sites together vanishes. Therefore
Here we have omitted the details of the constant term since it plays no role in later discussions. Eq. (13) is the foundation of our later discussion. The same derivation applies to the average of the denominator in Eq. (5), which leads to the same Ising partition function with a different boundary condition for all boundary sites, since there is no swap operator applied. One can define , such that and are the free energy of the Ising model with different boundary conditions.444The standard definition of free energy should be but it is more convenient for us to define it without the temperature prefactor. Then Eq. (9) reads
That is, the typical second Rényi entropy is given by the difference of the two free energies, i.e., the “energy cost” induced by flipping the boundary pinning field to down () in region , while keeping the remainder of the system with a pinning field up ().
In summary, what we have achieved is that the second Rényi entropy is related to the partition function of a classical Ising model defined 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 Rényi entropy of the bulk state , and the Ising spins at the boundary vertices are coupled to a boundary “pinning field” determined by the boundary region . If the bulk contribution from 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 and determine the effective temperature of the Ising model. For simplicity, in the following we assume for all internal legs and boundary dangling legs. In this case we can take as the inverse temperature of the classical Ising model.
3 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 limit, the Ising model is in the low-temperature long-range ordered phase (as long as the bulk has spatial dimension ), so that the Ising action can be estimated by the lowest energy configuration. The boundary pinning field leads to the existence of an Ising domain wall bounding the boundary region , 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.
3.1 Ryu-Takayanagi formula for a bulk direct-product state
We first consider the simplest situation with the bulk state a pure direct-product state . 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” defined in Ref. pastawski2015 (). The second Rényi entropy of such a tensor network state is given by the partition function of Ising model in Eq. (13) without the term. Omitting the constant terms that appears in both and , the Ising action can be written as
In the large limit, the Ising model is in the low temperature limit, and the partition function is dominated by the lowest energy configuration. As illustrated in Fig. 3 (a), the “energy” of an Ising configuration 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 field . For the calculation of denominator , everywhere, so that the lowest energy configuration is obviously for all , with energy . For , the nontrivial boundary field for requires the existence of a spin-down domain. Each link with spins anti-parallel leads to an energy cost of . Therefore the Rényi entropy in large limit is
The minimization is over surfaces such that form the boundary of a spin-down domain, and denotes the area of , i.e., the number of edges that cross the surface. Therefore the minimal area surface, denoted by , is the geodesic surface bounding region. Here we have assumed that the geodesic surface is unique. More generally, if there are degenerate minimal surfaces (as will be the case for a regular lattice in flat space), is modified by .
With this discussion, we have proved that Ryu-Takayanagi formula applies to the second Rényi 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 Rényi entropies take the same value in the large 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 Rényi 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 Rényi entropy amounts to evaluating the Euclidean action of the two-fold replica geometry, which satisfies the Einstein equation everywhere in the bulk. Thus, in general, the second Rényi 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 -fold replica geometry is in general different from the geometry constructed by simply gluing copies of the original geometry around the minimal surfaces, the discrepancy between which can be seen manifestly from the -dependence of the holographic Rényi- entropy. We will see in Section 5 that our random tensor model can reproduce the correct Rényi entropies for a single boundary region if we replace the bond states 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 defined on a continuous manifold, one can consider a triangulation of a given spatial manifold and define a random tensor network on the graph of the triangulation. (See (bao2015cone, , Appendix A) for further discussion of the construction of the triangulation graph.) Denoting by the length scale of the triangulation (the average distance between neighboring triangles), the area in our formula is dimensionless and the area defined on the continuous Riemann manifold is given by (when the spatial dimension of bulk is ). Therefore , and we see that corresponds to the gravitational coupling constant .
Compared to previous results about the RT formula in tensor networks pastawski2015 (); yang2015 (), our proof of RT formula has the following advantages: Firstly, our result does not require the boundary region to be a single connected region on the boundary. Since the entropy in the large limit is always given by the Ising spin configuration 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 satisfied 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.
3.2 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. (13) is nonzero. If we still take the limit, the Ising model free energy is still determined by the minimal energy spin configuration, which is now determined by a balance between the area law energy for a domain wall , and the energy cost from bulk entropy. We can define the spin-down region in such a minimal energy configuration as , which bounds the boundary region , and corresponds to the region known as the entanglement wedge in the literature hubeny2007 (); heemskerk2012 (). The second Rényi entropy is then given by
The bulk contribution has two effects. First it modifies the position of the minimal energy domain wall , and thus modified 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 fields contribute corrections to the RT formula in AdS/CFT faulkner2013quantum ().
To understand the consequence of this bulk correction, we consider an example shown in Fig. 3 (b) and (c), where and are two distant disjoint regions on the boundary. If the bulk entanglement entropy vanishes, the RT formula applies and the entanglement wedges and are disjoint. Therefore we find that and so the “mutual information” between the two intervals vanishes in the large limit.555The mutual information for Rényi entropy is generally not an interesting quantity, but it is meaningful in our case since it approaches the von Neumann mutual information for large . 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 and may change position, but remain disconnected. Therefore:
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 information between two far-away regions in the large limit. (If we consider a large but finite , and include spin fluctuations of the Ising model, we obtain another source of mutual information between far-away regions, which vanishes exponentially with .) The suppression of mutual information between two far-away regions implies that the correlation functions between boundary regions and are suppressed, even if each region has a large entanglement entropy in the large 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 heemskerk2009 (); elshowk2012 (); benjamin2015 (). A more quantitative analysis of the behavior of two-point correlation functions and scaling dimension gap will be postponed to Section 6.
3.3 Phase transition of the effective bulk geometry induced by bulk entanglement
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 , 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 P93 (), which we will use as a toy model of a thermal state (i.e., of a pure state that satisfies the eigenstate thermalization hypothesis deutsch1991 (); srednicki1994 ()). The amount of bulk entanglement can be controlled by the dimension of the Hilbert space of each site. We will show that the topologies of minimal surfaces experience phase transitions upon increasing which qualitatively reproduces the transition of the bulk geometry in the Hawking-Page phase transition hawking1983 (); witten1998b (). 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 field correction. However, above a critical value of , 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 identified conclusively in the absence of causal structure, however, so our conclusions in this section are necessarily tentative.)
We consider a tensor network which is defined on a uniform triangulation of a hyperbolic disk. Each vertex is connected to a bulk leg with dimension in addition to internal legs between different vertices. Then we take a disk-shaped region, as shown in Fig. 4 (a). We define the bulk state to be a random state in the disk region, and a direct-product state outside:
The second Rényi entropy of a boundary region is determined by the Ising model partition function with the action (13).666For 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 modified graph.
The bulk contribution 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 sites is given by LL93 ()
in which 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 fine-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 , as in previous subsections, we obtain
Here is a spin-down region bounding a boundary region , and is the boundary of this region in the bulk (which does not include ). is the total volume of the disk region in the bulk.
Consider the Poincaré disk model of hyperbolic space, with the metric . The boundary is placed at with a small cutoff parameter. The disk region is defined by . Choose a boundary region , with 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 to be a curve described by (i.e., for each there is only one value), the volume and area of this curve can be written explicitly as
For fixed , when we gradually increase , 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 Fig. 4 (b). In the calculation, we fix , which means that the radius of the disk in proper distance is (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 (Fig. 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 that cannot be accessed by the minimal surfaces of any boundary regions (Fig. 5 (b)). Qualitatively, the minimal surfaces therefore behave like those in a black hole geometry, which always stay outside the black hole horizon. As increases, increases until it fills the whole disk (). Further increase of 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 (Fig. 5 (c)).
More quantitatively, the two phase boundaries in Fig. 4 (b) are fitted by (blue line) and (red line), respectively. The square root behavior of the blue line can be understood by taking the maximal boundary region of half the system size . At the critical , the diameter of the Poincaré 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 work is required to obtain the correct coefficient , as we show in Appendix A. In Fig. 6 (a), we present the evolution of the black hole size when increases and is fixed.
Fig. 6 (b) provides another diagnostic to differentiate the geometry with and without the black hole. The entanglement entropy is plotted as a function of the boundary region size. In the perturbed AdS phase (blue curve), is a smooth function of , just like in the pure AdS space. In the small black hole phase (black curve) and the maximal black hole phase (red curve), there is a cusp in the function at , as a consequence of the discontinuity of the minimal surface. For , 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 is qualitatively consistent with the behavior of a thermal state (more precisely a pure state with finite 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 different 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 Poincaré disk), but there is no hard cutoff. 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 sufficiently high. From this simple example we see how the bulk geometry defined 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 thermofield double state maldacena2003 (), 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
Here is a pure state density matrix while is a mixed state with finite entropy. This density matrix described a bulk state in which all qudits in the disk region are entangled with some thermal bath. The behavior of the geometry can be tuned by the entanglement entropy of for each site, which plays a similar role as 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 differences between the single-sided and two-sided case:
Because the bulk state is not a pure state, the entropy profile of the boundary system with respect to the different 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 profile. This phase transition corresponds to the transition between thermal AdS geometry and AdS black hole geometry witten1998b ().
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 different value . When the bulk entropy exceeds this value, the boundary state is a mixed state with entropy , 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.777We thank Aron Wall for bringing this point to our attention. 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.
4 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. yang2015 (), the concept of a bidirectional holographic code (BHC) was introduced, which is a holographic mapping with two different kinds of isometry properties. A BHC is a tensor network with boundary legs and bulk legs. We denote the number of boundary legs as and the number of bulk legs (i.e., the number of bulk vertices) as , and denote the dimension of each boundary leg as and that of each bulk leg as . The first isometry is defined from the boundary Hilbert space with dimension to the bulk Hilbert space with dimension . The physical Hilbert space is identified 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 defined 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 different 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” almheiri2014 (); pastawski2015 (). 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 limit and moreover have properties that are even better than the BHC constructed using pluperfect tensors in Ref. yang2015 ().
4.1 Code subspace
We start from the holographic mapping in the low energy subspace, or “code subspace” in the language of quantum error correction almheiri2014 (). Physically, the code subspace is a subspace of the Hilbert space which corresponds to small fluctuations around a classical geometry in the bulk. More precisely, the criterion of “small fluctuations” states that these states are described well by a bulk quantum field theory with the given geometrical background. In other words, in the code subspace the bulk fields (operators) at different 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 locality in the bulk fails if we consider the entire Hilbert space, is the essential feature of a theory of quantum gravity (defined as the holographic dual of a boundary theory), as compared to an ordinary quantum field 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 define the code subspace to be the tensor product of lower-dimensional subspaces at each vertex of the graph: . Here, is a -dimensional space at site in the bulk. The holographic mapping restricted to this subspace is simply a tensor network with a smaller bond dimension 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 that makes such a subspace an eligible code subspace.
When we view the tensor network as a linear map from the bulk to the boundary, the isometry condition means is the identity operator. To apply the results we obtained for the second Rényi entropy, it is more convenient to view the tensor network as a pure state. Choose an orthonormal basis of the bulk and a basis for the boundary. The linear map with matrix element can then be identified with the pure quantum state
In terms of the state, the requirement that is equivalent to the statement that the bulk reduced density matrix is maximally mixed. Therefore, the isometry condition can be verified by an entropy calculation.
For that purpose we calculate the second Rényi entropy of the whole bulk. In the large 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 field everywhere in the bulk, in addition to the boundary:
For computation of the bulk-boundary entanglement entropy, we should take for all , and for all boundary sites. (We have written Eq. (19) in this general form because other configurations of will be used in our later discussion.)
In this action, the effect of the bulk pinning field competes with the boundary pinning field . The relative strength of these two pinning fields is determined by the ratio . If , the lowest energy configuration will be the one with all spins pointing up. In the opposite limit , all spins point down. For the purpose of defining a code subspace with isometry to the boundary, we consider the limit . In that case all spins are pointing up, and the only energy cost in the Ising action (19) comes from the last term, leading to the entropy
which is the maximum possible for a state on the bulk Hilbert space since its dimension is . In the limit , the bulk is therefore in a maximally mixed state, so the corresponding holographic mapping from the bulk to the boundary is isometric. The isometry condition is equivalent to the condition that the lowest energy configuration of the Ising model has all spins pointing up.
Instead of requiring , we can write down more precisely the isometry condition by requiring that the all-up configuration has the lowest energy. Consider a generic spin configuration with a spin-down domain . The energy of this configuration is . Here and are the volume and the surface area of , respectively. In order for the all-up configuration to be stable, we need for all nontrivial , which requires
For example, if the bulk is a (triangulation of) hyperbolic space (with curvature radius ), a disk with boundary area has volume . Here we have measured both area and volume by the triangulation scale . Therefore the isometry condition requires
There is a finite range of which satisfies the isometry condition, which is a consequence of the fact that the area/volume ratio is finite in hyperbolic space. For comparison, the same discussion for a disk in flat space with boundary area will require . Therefore the ratio must scale inversely with the size of the whole system .888In the pluperfect tensor work yang2015 (), the code subspace was defined by selecting some of the bulk sites, each having . In contrast, the properties of random tensor networks considered in this work enable us to make a uniform choice of small at every site, which is more convenient.
A useful remark is that the isometry condition (21) (or more precisely, a slightly weaker condition with replaced by ) is obviously necessary by a counting argument: In order for an operator defined in region to be mapped to the boundary isometrically, it needs to be first mapped to the boundary of , so that the dimension of the Hilbert space at the boundary must be at least as large as the dimension of the bulk Hilbert space . With this observation, what we see from the Ising model representation is that the large- random tensor network is an optimal holographic code, in the sense that an isometry is defined as long as the counting argument does not exclude it. Of course one should keep in mind that this optimal property is only true asymptotically in the large limit.
4.2 Entanglement wedges and error correction properties
Having shown that the holographic mapping defines an isometry from the bulk to the boundary degrees of freedom for suitable ratios , it is natural to ask whether this isometry has the error correction properties proposed in Ref. almheiri2014 (), i.e., whether operators in the bulk can be recovered from parts of the boundary instead of from the whole boundary. Specifically, consider an operator in the bulk which only acts nontrivially in a region . Denote the complement of in the bulk by . We say that can be recovered from a boundary region if there exists a boundary operator such that pastawski2015 ()
We note that condition (23) is composable: For example, if and can be recovered from and , respectively, then for the corresponding boundary obervables and . It follows that
for any bulk state and the corresponding boundary state . In the same way, an arbitrary -point function in the bulk can be obtained from a corresponding correlation function on the boundary.
In the language of quantum error correction, Eq. (23) states that the logical operator acting on the degrees of freedom in can be realized by an equivalent physical operator acting on the degrees of freedom in only. We are now interested in understanding when all operators in the region can be recovered from . That is, we would like the quantum information stored in subsystem to be protected against erasure of the degrees of freedom in , the complement of on the boundary. This amounts to another entropic condition, namely, that in the pure state defined in Eq. (18) there is no mutual information between and the region nielsen2007algebraic (), which ensures that the mutual information between and is maximal:
For the reader’s convenience, we recount a short proof of this fact in Appendix B.
In general it is important that Eq. (24) is evaluated in terms of von Neumann entropies rather than Rényi entropies. In the limit of large , however, both entropies are closely approximated by the Ryu-Takayanagi formula as long as the minimal surfaces are unique (see Section 7). What is more, we may even arrange for the Ryu-Takayanagi formula to be satisfied exactly, without any assumption on the uniqueness of minimal surfaces, by using ensembles of random stabilizer states instead of Haar random states (see Section 9). In the following we shall therefore evaluate the quantum error correction condition (24) in terms of second Rényi entropies and assume (for simplicity) that the RT formula holds exactly.
To understand when the error correction condition holds, we consider the configuration shown in Fig. 7. The calculation of is straightforward. Given the isometry condition (21), the whole bulk is in a maximally mixed state after tracing over the boundary, so that also takes the maximal value . In the calculation of , the pinning field is set to for and for . The boundary spin-down field in will pin a spin-down domain (orange region in Fig. 7). We consider the case when is in the spin-up (blue) domain, in which case the energy cost gives the entropy . Here is the domain wall bounding region , and is the spin-up domain, which is the entanglement wedge of . The first term is the area law energy cost of the domain wall, and the second term is the volume law energy cost. can be computed similarly by flipping the pinning field in to downwards. Due to the isometry condition (21), flipping the field in does not create new spin-down domains, so that the only difference between and is an additional energy cost in the region that is exactly . Therefore condition (24) holds, and the operators in can be recovered from . As a final note, observe that the domain wall is generally not the minimal surface, due to the presence of the bulk pinning field, but our conclusion holds as long as is in the spin-up domain and is disconnected from .
For comparison, we can consider the same configuration in Fig. 7 and ask whether operators in can be recovered from . This requires the calculation of . Following an analysis similar to the previous paragraph, one can obtain , and . Here is the complement of in the bulk, which is the entanglement wedge of . Therefore the mutual information , so that cannot be recovered from .
From the two cases studied above, we can see that operators in a bulk region can be recovered from a boundary region if and only if is included in the entanglement wedge of . It should be noted that this statement only applies to small bulk , or for sufficiently small regions if is larger, when the entanglement wedge (spin-down domain in the Ising model) is independent of the direction of the pinning field in .
4.3 Gauge invariance and absence of local operators
In the two subsections above, we showed how a large and small random tensor network defines bulk-to-boundary isometries with error correction properties. In this subsection we would like to investigate the other direction of the BHC, i.e., the boundary-to-bulk isometry. To define this isometry, we need to require that the boundary-bulk entanglement entropy be equal to , which is the maximum possible entropy for the boundary. This requires the opposite condition from Eq. (21):
To satisfy this condition, we can take as a single site in the bulk, for which the condition is reduced to , with the number of links connected to . If this condition is satisfied for each site, Eq. (25) also applies to other regions, since always holds. Therefore the condition ensuring a boundary-to-bulk isometry is
This is similar to the condition proposed in Ref. yang2015 (), with the difference that Ref. yang2015 () has because each tensor is required to be rigorously a unitary mapping from the in-plane legs to the bulk leg.
When this isometry condition is satisfied, the boundary-to-bulk isometry maps each boundary state isometrically to a bulk state in a larger Hilbert space with dimension . It should be clarified that the physical Hilbert space is always that of the boundary, and that the -dimensional Hilbert space, which is factorizable into a direct product of each bulk site, is just an auxiliary tool. The situation is very similar to a gauge theory, in which one can embed gauge invariant states into a larger auxiliary Hilbert space by treating the gauge vector potential as a physical field. In fact, it was shown in Ref. yang2015 () that the physical Hilbert space – the image of the boundary Hilbert space under the holographic mapping – can be defined by a gauge invariance condition. The discussion also applies to the random tensor network satisfying condition (26).
The main property of the boundary-to-bulk isometry is that the bulk theory is intrinsically nonlocal. To be more precise, consider an arbitrary region that disconnected from the boundary, as shown in Fig. 8. We would like to show that any operator supported in is mapped to the boundary trivially, i.e.,
Here, we have denoted the whole boundary as region , while is the identity operator on the boundary, and is a constant. This statement is equivalent to the statement , which means there is no mutual information between and the whole boundary. Following an argument similar to that of the previous subsection, and using condition (25) one can easily conclude that
as is illustrated in Fig. 8. Therefore all purely bulk operators are trivial, and only those in regions adjacent to the boundary contain nontrivial information about boundary physical operators. As was discussed in Ref. yang2015 (), this property is a consequence of the gauge symmetry of the tensor network. For all tensor networks, there is a gauge symmetry induced by acting unitarily on each internal leg while preserving the physical state after contraction. However, for tensor networks with the boundary-to-bulk isometry property, this gauge symmetry is isometrically mapped to constraints on the bulk legs.
In summary, we have shown that a BHC can be built from a large random tensor network with bulk leg dimension satisfying condition (26). The boundary theory is mapped isometrically to a nonlocal theory in the bulk, with the physical (boundary) Hilbert space defined by gauge constraints. A code subspace is defined by a local projection at every bulk site to a smaller subspace with dimension which satisfies condition (21). A bulk-to-boundary isometry is defined in the code subspace, and a bulk local operator in the code subspace can be recovered from a boundary region as long as the entanglement wedge of this region encloses the support of this bulk operator. In this way, random tensor networks can be used to define a bulk theory with intrinsic nonlocality and emergent locality in a subspace, as is desired for a theory of quantum gravity.
5 Higher Rényi entropies
In this section, we will generalize the second Rényi entropy calculation to higher Rényi entropies, and show that the higher Rényi entropies of a random tensor network are also mapped to partition functions of classical spin models, with the spin now living in a different target space, the permutation group of . For , the permutation group reduces to the target space of the Ising model.
Again we use the natural logarithm to define higher Rényi entropies. We now define:
Here denotes the direct product of -copies of , and is the permutation operator that permutes the copies cyclically in region. For a basis of region , a basis of the direct product space is given by , and the action of is given by