Space-time random tensor networks and holographic duality

Space-time random tensor networks and holographic duality

Xiao-Liang Qi,    and Zhao Yang

In this paper we propose a space-time random tensor network approach for understanding holographic duality. Using tensor networks with random link projections, we define boundary theories with interesting holographic properties, such as the Renyi entropies satisfying the covariant Hubeny-Rangamani-Takayanagi formula, and operator correspondence with local reconstruction properties. We also investigate the unitarity of boundary theory in spacetime geometries with Lorenzian signature. Compared with the spatial random tensor networks, the space-time generalization does not require a particular time slicing, and provides a more covariant family of microscopic models that may help us to understand holographic duality.

holography, tensor networks, entanglement
institutetext: Stanford Institute for Theoretical Physics,
Physics Department, Stanford University, CA 94304-4060, USA

1 Introduction

Holographic duality proposes that a dimensional quantum field theory can be equivalently described by a dimensional quantum gravitymaldacena1998 ; witten1998 ; gubser1998 . The role of quantum entanglement in the holographic duality manifests itself in the Ryu-Takayanagi (RT) formula ryu2006holographic and its generalizationshubeny2007covariant ; faulkner2013quantum ; lewkowycz2013generalized ; dong2016deriving ; dong2016gravity . The RT formula and other properties of the holographic duality have motivated the tensor network approaches as an effort to develop a microscopic framework for holographic dualityswingle2012 ; qi2013 ; pastawski2015 ; yang2015 ; hayden2016 ; qi2017holographic ; donnelly2016living . Most tensor networks define states on a time slice. In appropriate limits they reproduce the RT formula, where the bulk geometry is replaced by the graph geometry in the tensor network approach. Thus these tensor networks resemble the static geometry (i.e. a geometry with a time-like Killing vector) where the RT formula applies. However, for more general dynamical geometries, the minimal surface and RT formula is not well-defined, and the entanglement entropy of a boundary region is dual to the area of an extremal co-dimension- surface in the bulk space-time, known as the Hubeny-Rangamani-Takayanagi (HRT) formulahubeny2007covariant . In the (spatial) tensor network picture, it is often imagined that the dynamical space-time is described by a spatial tensor network with a time-dependent geometry. However, there is an intrinsic problem with this picture. In contrast to the case of static space-time, in general, HRT surfaces of different boundary regions at the same boundary time cannot be embedded into a single Cauchy surface. Therefore it is impossible to find a “proper" Cauchy surface and describe the boundary state as a tensor network satisfying RT formula on this surface. Besides, the spatial tensor network description requires the choice of a time direction, which can not manifest the general covariance. Motivated by these problems, we develop a more covariant space-time tensor network description to holographic duality.

In this paper, we propose a new approach to the holographic duality based on space-time tensor networks. The tensor network we consider is defined in the bulk space-time, with random projections applied to each link (more details will be described later). Instead of describing a many-body wavefunction, the space-time tensor network defines the boundary parition function with arbitrary insertions. In other words, it defines the generating function of all (time-ordered) multi-point functions of the boundary theory. A time-ordered multi-point function is simply obtained by insertion of operators in the boundary links of the tensor network. We show that the Renyi entropy of a boundary region, after averaging over random projections in the bulk, can be mapped to the partition function of a discrete gauge theory. In the large bond dimension limit the gauge theory is in the classical limit, and the Renyi entropy is determined by the classical energy of the minimal action gauge field configuration. With the boundary condition determined by the boundary region, we show that the Renyi entropies in this limit are determined by the area of extremal surface bounding the boundary region. This result implies that the von Neumann entropy of our tensor network agrees with the HRT formula of holographic theories (while the Renyi entropies generically do not agreedong2016gravity ; dong2016deriving ). This approach can also be generalized to include bulk quantum fields, and compute bulk-boundary correlation functions. We show that the duality defined by this setup satisfies the properties of the bidirectional holographic codeyang2015 ; hayden2016 , but has the advantage that correlation functions can be studied for general space-time points, rather than being restricted to a time slice.

The remainder of the paper is organized as follows. In Sec.2, we describe the general setup of the space-time random tensor network. Starting from a bulk parent theory, we show how the boundary partition function and correlation functions are obtained by introducing random projections in the bulk. In Sec.3, we study the second Renyi entropy of a boundary region and show its relation to the partition function of a discrete gauge theory. We discuss how HRT formula is reproduced in proper large bond dimension limit. In Sec.4, we generalize the discussion to -th Renyi entropy. In Sec.5, we discuss the operator correspondence between bulk and boundary. We discuss how to define code subspace operators in the bulk and show the entanglement wedge reconstruction of such operators on the boundary. We also discuss the behavior of generic bulk-boundary and boundary-boundary correlation functions. As a special case of correlation structure, we also studied how the unitarity of the boundary theory depends on properties of the bulk theory. In Sec.6, we discuss how to introduce proper gauge fixing in the definition of random tensor network, which is essential for bounding fluctuations and justify the semi-classical approximation. In Sec.7, we summarize this paper and discuss several open questions.

2 General setup

2.1 An overview of space-time tensor networks

A tensor network is mathematically equivalent to a Feynman diagram. Each vertex that is adjacient to links is a rank- tensor with each label defined on a link connecting to the vertex. When two neighboring vertices are connected by one link, the corresponding labels are contracted, leading to a new tensor


To define the contraction, one shall specify a metric on each link. Without loosing generality, one can always assume the metric to be non-singular, since a singular metric can be viewed as a contraction in a lower dimensional subspace with a non-singular metric. Furthermore, one can always transform a non-singular metric to the standard form by a transformation on the vertex tensors. Therefore in the following we will always take the link metric to be the -function, such that the information about the network is all encoded in the vertex tensors and the geometry of the network.

A tensor network in space-time is a discrete version of path integrals, which can be used to define the partition function of a statistical model, as is shown in Fig. 1. (As examples of recent works on space-time tensor networks, see Ref. levin2007tensor ; gu2008tensor ; jiang2008accurate ; evenbly2015tensor .) For simplicity we draw two-dimensional networks, but all our discussions apply to general dimensions. For concreteness, in Fig. 1 we wrote the explicit definition of the tensor which corresponds to the two-dimensional Ising model. The degrees of freedom are defined on links of the graph, which has dimension . ( for the Ising model.) Physical properties correspond to multipoint correlation functions, which can be computed by inserting operators (yellow boxes in Fig. 1 (a)) in links of the tensor networks. For example in the Ising model if one wants to compute the multipoint spin correlations, the operator to insert is the Pauli matrix , which has the matrix element . A tensor network can be defined on an arbitrary graph. For a generic curved space, one can introduce a triangulation of the space and define a tensor network on it, which is a discrete version of a quantum field theory on curved space.

Figure 1: (a) Illustration of a space-time tensor network which defines a statistical model, with the degrees of freedom defined on each link. With operator insertions (yellow boxes), evaluating this tensor network computes multipoint correlation functions in this system. (b) The definition of tensor that corresponds to the two-dimensional ferromagnetic Ising model.

2.2 The holographic space-time tensor network

Now we consider a different tensor network, which we propose to describe a holographic theory. We would like to define a tensor network on a -dimensional space with a boundary, as is shown in Fig. 2. The main difference from the tensor networks in Fig. 1 is a random projection defined at each link in the bulk, as is shown by the red arrows.111As will be discussed in Sec. 2.3 and Sec. 6, more precisely one should define the random projection on all links except those on a subgraph chosen by gauge fixing. However we decide to keep the simpler but imprecise definition here since it does not affect any result except for the discussion on fluctuations in Sec. 6. To be more precise, the random projection is an operator acting on the link Hilbert space with the form


with a random state in the Hilbert space of the link. The tensor at the vertex is not random. At this moment we will leave the vertex tensor general, and discuss different choices of it later.

The boundary links of the tensor network are not acted by the random projection. Instead, generic operators can be inserted to these links. Our proposal is to take the tensor network with boundary insertions and bulk random projections at each link as the definition of the multipoint function . Since the random projections can also be considered as operator insertions in the bulk theory, we can summarize this proposal as


where denotes the multipoint function in the bulk theory defined by the vertex tensors.

Before studying the properties of this theory, we first provide some physical intuition why random projections in the bulk are a necessary ingredient. If we remove the random projections, Fig. 2 defined multi-point correlation functions of an ordinary quantum many-body system in dimensions, with all operators supported on the boundary. It should be clarified that such a network clearly does not define a holographic duality. The discrepancy can be seen by considering equal time correlation functions, which are determined by the density matrix of the boundary. Since the boundary is simply a subsystem of the -dimensional system, the boundary reduced density matrix is generically a mixed state, while a stand-alone -dimensional system should have a pure state density matrix. The random projections are therefore the key feature that converts the tensor network in Fig. 2 to a holographic theory, the properties of which will be studied in the rest of the draft.

Figure 2: (a) The definition of multi-point functions of the boundary theory by a tensor network with one higher dimension. The red arrow along each link defines a random projector, as is illustrated in (b). (c) The random average over the direct product of two random projectors (denoted by the blue dashed line) results in the superposition of two operators, the identity operator and the swap operator, acting on the doubled Hilbert space. It should be noted that this figure has not taken into account the gauge redundancy subtlety discussed in Sec. 2.3 and Sec. 6.

2.3 Two conceptual subtleties

Before studying the properties of the holographic tensor networks, there are two conceptual points we need to discuss. First of all, the random projections on different links of the space-time network are all independent, so that for a given random realization, the system after applying random projections does not have any space-time symmetry. In particular, consider a bulk theory which has time reflection symmetry before applying the random projections, so that the forward time evolution from to gives the same state as the backward time evolution from to . For such a network, operator insertions at time slice computes equal-time correlation functions, which are determined by the density matrix . Now with random projections that are independent at time and , the two boundary states we obtained are generically different, which we denote by and , respectively. Now if we insert an operator on the boundary, the network computes with


With these two states being different, is generically not a density matrix since it is not Hermitian. Therefore our definition of the boundary theory (3) seems to be problematic. However, similar to the case of spatial random tensor networkshayden2016 , we are interested in certain limits where the tensors have large bond-dimensions, and different random configurations induce little fluctuation on entanglement properties. For such large dimension tensor networks, we argue that the space-time symmetry is restored in the sense of random average. This is similar to how disorder breaks translation symmetry in an electronic material, but all physical quantities that are self-averaged (such as conductivity) appear translation invariant. With this understanding in mind, in the following we will treat the boundary theory as a theory with the space-time symmetry of the network before random projections.

Secondly, one apparent inconsistency in this theory is that, after the random projection the bulk vertices become disconnected with each other, so that their contribution to the partition function seems to be simply an overall numerical factor independent from the operator insertions. Therefore the bulk geometry seems to have trivial effect on the boundary theory, which is obviously not what we want. As we will see in next section, the random average of quantities such as Renyi entropy is mapped to a partition function of discrete gauge theory in the bulk, which does have a nontrivial dependence to the bulk geometry. Therefore the apparent inconsistency indicates that the physical quantities do not self-average, and there is a large fluctuation between different random configurations. As we will discuss in detail in Sec. 6, this problem translates into a redundancy of the discrete gauge theory, which can be solved by a standard gauge fixing procedure. Translate back from the gauge theory to the random projections, the gauge fixing corresponds to removing the random projections on some bulk links which form a spanning tree of the bulk network. Since this gauge fixing only affects the fluctuation around large bond dimension saddle point, and it is much more natural to understand it after introducing the mapping to discrete gauge field, we opt to keep the imprecise definition of random projection on all links here, and ask the readers to keep in mind that actually the projections are applied to a subset of bulk links but the choice does not affect any result before Sec. 6.

3 The second Renyi entropy calculation

3.1 The second Renyi entropy of a boundary region

Figure 3: (a) The network representation of the boundary density matrix , with open legs at the cut representing the two indices of . (b) For a boundary region , (see text), which is represented by two copies of the same tensor network with an insertion of boundary operator (defined in Eq. (5)). (c) Carrying the random average over the random projectors at each link following Fig. 2 (c), one obtains the partition function of a gauge theory. The grey arrow is defined in (d) as the average of .

To relate the space-time tensor network to entanglement properties that we are familiar with, we consider equal time correlation functions at time . Since in the computation of equal time correlation there is no insertions at other time steps, we can simply write for any operator, with the density matrix given by the network in Fig. 3 (a) with open boundary links in region . 222As is discussed in Sec. 2.3, we are treating as the density matrix although in a particular random realization it may be non-Hermitian. In general, we can also consider space-time geometries without time reflection symmetry, so that is non-Hermitian even after the random average. All our results apply to such general situations. We now compute the entanglement entropy of an arbitrary boundary region . (Although we have been representing the boundary as a single line for simplicity, it should be remembered that the boundary is generically a -dimensional system with spatial dimensions extending perpendicular to the paper plane.) The second Renyi entropy can be computed by


with the partial swap operator that permutes the two copies of systems in region, but preserves the rest of the system hayden2016 . This expression of corresponds to Fig. 3 (b). Now if we take the random average over the link random projectors, we obtain Fig. 3 (c), in which each link projector in the doubled system is replaced by its average value given in Fig. 2 (c). Since the average is a sum over two operators, the whole tensor network can now be viewed as a partition function of Ising-like spin variables defined on each link, with and labeling the choice of identity channel and swap channel at this link, respectively.

Interestingly, independent from the details of vertex tensors, the bulk statistical model of obtained by random averaging is always a gauge theory with gauge invariance. The gauge transformation at a given site is defined by changing for all neighboring links , which is equivalent to interchanging the label of the two vertex tensors at , which therefore preserves the partition function. The choice of boundary region defines a boundary condition of the gauge vector potentials. Due to the gauge symmetry, the only gauge invariant information in this boundary condition is the location of flux. Since is defined as the swap operator in region and identity elsewhere, the flux at the boundary rests at the boundary , which is a co-dimension surface at the boundary. This situation is illustrated in Fig. 4 for a system with bulk dimension .

Figure 4: An illustration of the gauge field configuration . For a boundary region (the boundary of the organge region in the bulk), the boundary condition of is for the red links across , and elsewhere on the boundary. This boundary condition induces a flux co-dimension- surface bounding the boundary of , which is the purple dashed line. A gauge choice can be made by choosing in the bulk for all links crossing a co-dimension surface which bounds , and elsewhere.

With this boundary condition, the bulk partition function describes quantum fluctuations of the gauge field, the action of which is determined by integrating out the bulk “matter field" given by the vertex tensors. By adjusting the vertex tensors, one can obtain different dynamics of gauge field. Before specifying the vertex tensor, one can already see some interesting property of the Renyi entropy. Let us assume that a bulk matter field was chosen such that the gauge field is in the weakly coupled limit. For example, this can be achieved by taking flavor of bulk fields (so that the vertex tensor is a direct product of tensors, each with a finite dimension) and considering the large limit. The effective action of will be with induced by a single flavor. Therefore in the large limit the gauge field is weakly coupled, and the electric flux induced by the boundary condition (the purple dashed line in Fig. 4) does not fluctuate. With this assumption, the electric flux will rest at the co-dimension surface which minimizes the classical action of the bulk. Although the detail of this energy depends on the choice of bulk matter field, a general observation is that this classical action is actually the second Renyi entropy of a bulk region. To see that, one can take a gauge choice in the bulk by choosing a surface bounding , which we denote as in Fig. 4. can be chosen to be for all links crossing this surface, and everywhere else. In this choice, the bulk tensor network evaluates , where is the reduced density matrix of bulk region , and is the state of the bulk defined by contracting all tensors while leaving the indices open at a spatial surface (the disk with the blue and orange regions). For example, if we take the vertex tensors to be copies of the Ising tensors in Fig. 1 (b), and take the network to be infinite and translation invariant along the imaginary time direction , is the ground state of independent copies of the Ising model.

In summary, without specifying the vertex tensor, we obtain the following general equation


as long as the gauge field is in the weakly coupled limit. In the following we will pick an simplest choice of the vertex tensors, for which the action of gauge field can be explicitly obtained, and the Renyi entropy satisfies Ryu-Takayanagi formula asymptotically.

3.2 The bulk valence bond solid state and the HRT formula

As a specific example of the general results in the previous section, we consider a very simple tensor network defined in Fig. 5 (a). A closed loop is assigned to each -dimensional plaquettes of the network, which can be viewed as the world line (in the Euclidean space-time) of a qudit with dimension . 333Obviously, to make this state well-defined, one needs to specify not only the vertices and links in the network, but also plaquettes. If a link is adjacient to plaquettes, there are loops passing through link , so that the dimension of the link is taken to be . The vertex tensor simply passes each qudit along the direction of the loop. One can denote all plaquettes adjacient to a vertex by , and label the states of the two qudits at the two links which adjacient to both the plaquette and the vertex by and , as is shown in Fig. 5 (b). Then the explicit definition of the vertex tensor is


To understand the physical meaning of such a network, we can consider a network with translation symmetry in the imaginary time direction (such as the one in Fig. 5 (a)), in which case the network defines an imaginary time evolution . It should be reminded that we are now talking about the network without the random projections, which defines a bulk QFT. Denoting the time difference of two neighboring steps as , we see that (up to normalization) is a projection operator to maximally entangled EPR pairs at each link:


Equivalently, one can write and take . In this tensor network, at any spatial cut one obtains a state , which consists of a fixed configuration of EPR pairs. We follow the convention in the literature rokhsar1988superconductivity ; ran2007projected ; chen2012symmetry and call such a state a valence bond solid (VBS) state.

For the VBS state, the bulk action of the gauge field can be explicitly computed. When we consider the doubled theory and take the random average as is shown in Fig. 3 (c), each plaquette contributes a term which is determined by , i.e., the flux of the gauge field in that plaquette. If , the contraction of loops around this plaquette gives the statistical weight (), respectively. Therefore we obtain


which is the standard action of gauge theory. One should be reminded that has the boundary condition set by the choice of boundary region, as was illustrated in Fig. 4. In the denominator, the partition function has the same expression but with the boundary condition everywhere on the boundary. Therefore Eq. (9) tells us that the Renyi entropy of a region is given by the action cost of adding an electric flux threading the boundary of .

Figure 5: (a) The tensor network that corresponds to the loop state we consider in Sec. 3.2. The red arrows stand for random projectors, the same as in Fig. 2. (b) A more explicit definition of the vertex tensor, for a generic vertex that is adjacient to plaquettes. Each plaquette are adjacient to two links, which are labeled by , . . (c) The generalization of loop state by introducing bulk entanglement, as is discussed in Sec. 3.4. Now each link is labeled by the loops passing through it, and an additional index labeling the remaining “quantum field theory" degrees of freedom in the bulk. (d) An example of a tensor network with loops and additional bulk degrees of freedom.

In this action, plays the role of coupling constant, and the gauge field is weakly coupled in the large limit. In this limit, the electric flux induced by the boundary condition is heavy and classical. The classical action of an electric flux at a surface is simply with the area of . Therefore the lowest energy configuration is given by the minimal area surface (which we will also denote as ), and the entropy is given by


In other words, we have explicitly proved that the second Renyi entropy satisfies the covariant HRT formula (in Euclidean space-time) when the bulk is in the VBS state in the large limit. This result is the space-time analog of the RT formula in spatial random tensor networks discussed in Ref. hayden2016 , but the space-time approach is covariant, so that the entanglement entropy can now be computed for any boundary region in the space-time, rather than those restricted to a fixed spatial surface. If we consider a bulk which is a discretization of hyperbolic space , the corresponding boundary theory will have the full conformal symmetry (in Euclidean signature) at scales much larger than the discretization, which was not possible in the spatial random tensor network approach.

Furthermore, we would like to point out that even with Euclidean signature, where the HRT surface is a minimal surface, the space-time tensor network satisfying HRT is still not a trivial reformulation of the spatial tensor networks satisfying RT formula. For a generic geometry, different boundary regions can correspond to HRT surfaces that do not belong to a co-dimensional spatial slice of the bulk, which is by construction always true for a spatial tensor network. The space-time formalism includes the spatial tensor network as a special case, when the bulk geometry has time reflection symmetry and all minimal surfaces bounding regions on the boundary time-reflection-symmetric slice lie in the bulk time-reflection-symmetric surface.

3.3 Some more comments on Lorenzian time

For concreteness we have focused on Euclidean time, where the VBS state allows us to explicitly obtain the bulk gauge field action. However, physically our approach applies also to Lorenzian space-time geometry. The signature of the bulk theory is completely determined by the properties of vertex tensors. For a Lorenzian theory with unitarity, the vertex tensors should be unitary quantum gates. For example a unitary tensor network can be obtained by Trotter-Suzuki decomposition of a Hamiltonian time evolutiontrotter1959product ; suzuki1976generalized ; suzuki1976relationship . After introducing the random projection on links and compute the random average of for a boundary region, one still obtain a gauge field coupled to bulk matter. The effective action of the gauge field is obtained by integrating out the bulk matter, except that the path integral (i.e. contraction of bulk tensors) is now carried in a Lorenzian theory. The Lorenzian analog of the VBS state is a short-range correlated theory, such as a boson with a large mass. For example if we consider flavors of massive bosons , the double-copied theory contains and the gauge field couples to bosons by permuting the replica index. Integrating out the bosons, we expect to obtain a (discretized) Maxwell action for the gauge field, which is weakly coupled in the large limit. Once the Maxwell action is obtained, the calculation of still reduces to evaluating the effective action of an electric flux pinned to . With a Maxwell action in the weakly coupled limit, we expect


where the sum is over all flux configurations that terminates at , is the area of , and is some coupling constant. In the large limit, the sum will be dominated by the saddle point surface , although is not a minimal surface any more. Therefore we expect the HRT formula to apply for general geometries as long as the bulk theory is independent copies of a short-range correlated theory, in the large limit.

In the following discussion we will still use Euclidean signature and the VBS state as an explicit example, except in Sec. 5.4, but all discussions can be carried in parallel in Lorenzian time.

3.4 Bulk entanglement corrections to the second Renyi entropy

In Ref. hayden2016 , it was shown that the random tensor networks can be used to define a holographic mapping, which a network with both bulk indices and boundary indices that maps a bulk state to a boundary state. This is more generic than a random tensor network that directly defines a state on the boundary. For a given holographic mapping network and a given bulk state , the boundary region Renyi entropy in the large limit are shown to satisfy RT formula with bulk entanglement corrections. The bulk state contribution is a subleading term in , but it is essential for many things, such as finite mutual information between two far-away boundary regions. Here we will show that the space-time tensor network approach can also take into account a more general bulk state, which gives similar corrections to the Renyi entropies.

For this purpose, consider the definition of vertex tensor defined in Fig. 5 (c) and (d). In addition to loops running around plaquettes, there is another tensor (the blue dot and blue lines in Fig. 5 (c)) which describes some additional bulk degrees of freedom. Denote the dimension of each blue line as , the dimension of each link is now with the number of plaquettes adjacent to the link. Physically, if we take and keep finite, the loops contribute a large amount of short-range entanglement, while the remaining degrees of freedom labeled by the blue line can be viewed as a “bulk quantum fields", which made a smaller but possibly longer-ranged contribution to quantum entanglement between different bulk regions. In the following we will refer the degrees of freedom with dimension as the “bulk quantum fields".

After the random average in the second Renyi entropy calculation, the gauge field is now coupled to both the loops and the bulk quantum fields. The action of the gauge field is given by a sum of these two contributions:


Here denotes the second Renyi entropy of the co-dimension surface which crosses all links with , for the bulk quantum fields. 444To be more precise, each link is dual to a co-dimensional surface in the dual graph, and is the union of them. Consequently, in the large limit the gauge field is still weakly coupled, and the partition function is dominated by the flux configuration threading the minimal surface . In this limit, the last term simply gives the entanglement entropy of region in the state of bulk quantum fields, with is a Cauchy slice of the entanglement wedge of , as is illustrated in Fig. 4. In summary, in this limit we obtain


which is in agreement with previous resultsfaulkner2013quantum ; hayden2016 .

We would like to note that the formula also applies more generically if we take and simultaneously, except that now the two terms in action (12) may compete with each other, and the minimal surface is determined by minimizing the action, rather than the area.

4 Higher Renyi entropies

The generalization of the discussion above to higher Renyi entropies is straightforward. For the calculation of -th Renyi entropy of a boundary region one would like to calculate


with the cyclic permutation of the copies of systems in region. For the tensor network defined in Fig. 2, this calculation corresponds to taking copies of the network and insert an operator at the boundary, in the same way how is inserted in Fig. 3 (c). The random average of this tensor network is determined by that of copies of random projectors:


with the normalization constant , and are the elements of permutation group with objects, which act on the -copied system by permuting the indices of the copies.

Following the same derivation as the second Renyi entropy case, this random average is thus mapped to the partition function of a gauge theory. The gauge invariance comes from the symmetry of permuting the vertex tensor at any vertex . By choosing different bulk vertex tensors, one can obtain different dynamics of the gauge field.

If we choose the same VBS state that was studied in Sec. 3.2, the action of the gauge field has the standard form


Note that the permutation group elements is defined for an oriented link, with . The flux defined by the product should have oriented properly. (For example one can choose to be oriented such that the plaquette is always on the left of the link when one goes from to . ) The function denotes the number of loops in a permutation element . One can also write with the trace taken in the same representation of as in Eq. 15.

Similar to the case, in the large limit, the non-Abelian discrete gauge theory (16) is in the weakly coupled limit, with the electric flux classical. The minimal action configuration consists of an electric flux threading the minimal surface , with the flux the cyclic permutation . Therefore we obtain the HRT formula


The generalization to systems with bulk entanglement, given by networks in Fig. 5 (c) and (d), also directly applies to the -th Renyi entropy. In the limit and finite , the -th Renyi entropy of a boundary region is given by


The discussion of large limit here assumes that a single configuration (modulo gauge transformations that will be discussed in Sec. 6) dominates in the large limit. For a geometry with a unique minimal surface, this assumption should be correct. However, upon increasing , the number of low energy configurations increase quicklyTherefore, the large limit and large limit might be non-commuting. We also would like to note that the -independent Renyi entropy in large limit is a feature shared by spatial random tensor network stateshayden2016 , but different from large gauge theories with gravitational dualdong2016gravity . This difference is related to the absence of gapless gravitons in the bulk, and is a key ingredient of AdS/CFT correspondence that is missing in the tensor network models.

5 Operator correspondence between bulk and boundary

5.1 Bulk-boundary correlation functions

In the discussion so far we have focused on entanglement properties of the boundary state. Since different bulk states can be mapped to different boundary states, we can also discuss the mapping between bulk operators and boundary operators. We consider the network in Fig. 5 (d), with bulk states consisting of loops with dimension and bulk quantum fields with dimension . Instead of considering a boundary correlation function, we can consider a boundary-bulk correlation function, with operator insertions the boundary and in the bulk, as is shown in Fig. 6 (a). It should be noticed that we are inserting operators in the bulk which acts only on the “bulk quantum field" index. Physically one can think such operators as “low energy operators" in the bulk. If we consider the effect of bulk operators as a perturbation to the bulk state, this boundary-bulk correlation function is computing the effect of such perturbations to a generic boundary correlation function. Since all such perturbations can be viewed as modifying the boundary theory, one expects that each multipoint operator in the bulk corresponds to some boundary operator.

Figure 6: (a) The definition of bulk-boundary correlation function, with insertion of operators at the boundary (orange boxes) and bulk quantum field theory states (green boxes). (b) The choice of bulk region (yellow disk) and boundary regions (red), (blue), all of which lies in a spatial surface (the whole disk). and are spatial slices of the entanglement wedge of and , such that their interface is the minimal surface bounding .

5.2 Equal time correlations and the error correction property

The reconstruction of the bulk operators, which lie in the entanglement wedge of some boundary subregions, onto the corresponding boundary regions is interpreted as a quantum error correction codealmheiri2014bulk ; dong2016reconstruction ; harlow2016ryu ; dong2016bulk which explicitly illustrates the “subregion-subregion duality" in the context of AdS-CFT. Roughly speaking, for a boundary region , there exists a code subspace of the bulk Hilbert space, and a subalgebra of operators acting on this subspace. For any operator in this subalgebra, and any state , there exists an operator acting on boundary region such that


where is the holographic mapping from the bulk Hilbert space to the boundary Hilbert space. Physically, the code subspace is the Hilbert space of the effective field theory living in the boundary’s entanglement wedge. Explicit examples are shown in pastawski2015 ; yang2015 ; hayden2016 ; qi2017holographic .

In the space-time tensor formalism, we define the code space as the space of all bulk states sharing the same background geometry, in the limit . Thus, the states in the share the same entanglement wedge. In the following part, we will show that, within , the operators that act on the entanglement wedge of some boundary region can be reconstructed on .

In the previous sections, we treat the VBS states as the backbone of the bulk geometry, on top of which the bulk quantum fields live. However, from another perspective, we can treat both the VBS states and the bulk quantum fields together as the bulk states. In contrast with in Eq.18, which denotes the th Renyi entropy of only the bulk quantum fields, we use to represent the entropy of all the bulk states in . Thus in the limit, Eq.18 can be rewritten as . In this section, all notations with super-index refers to the direct product of the bulk quantum fields and the VBS states.

In order to prove that the space-time tensor formalism have the error correction property, one needs to prove that all operators within the code subspace, acting in the entanglement wedge of , can be reconstructed on the boundary . According to Ref. dong2016reconstruction ; harlow2016ryu ; dong2016bulk , we only need to show that the relative entropy in the compliment of region and (denoted by and , see Fig.6 (b)) satisfies , where and are the density matrices of two states , and . , where is the boundary state corresponding to the bulk state . Because , and (Eq. (18)), we only need to show that . To study this quantity we introduce a replica trick


Therefore if for all , by analytic continuation we have proved the relative entropy equality.

Figure 7: An illustration of in the space-time tensor network. The red and blue cylinders represent and , respectively, which share the same geometry and only differ in the bulk quantum field. The boundary condition is set by inserting a cyclic permutation operator in the boundary region . In the limit, the leading contribution is from the VBS states. Thus the dominating classical flux configuration contains a flux following the minimal surface , and the gauge field variables on the links, modulo gauge transformation, are fixed as for links crossing a codimension- surface that bounds and , and identity elsewhere.

After the random average, we obtain


The first term is the same as Eq. (16), and the second term shows the contribution from the bulk quantum fields. In the limit, the first term dominates. Minimizing the first term leads to the classical configuration that contains the flux of cyclic permutation winding around the co-dimensional two minimal surface (see Fig.7). For such a spin configuration, we see that


Thus we conclude


Details that derives Eq. (23) from Eq. (22) is shown in Appendix.A.

It should be noted that the terms on the r.h.s of the Eq.23 all have the super-index , which include both the loop states and the bulk quantum fields states. Therefore the error correction properties apply to not only the operators in the bulk quantum field theory, but also to more general bulk operators that can act on the entire bulk Hilbert space, as long as it does not change the entanglement too much to change the location of the minimal surface. For example, one can consider an operator acting on a short-range EPR pair across the minimal surface , which reduces its entanglement entropy by some amount . If the minimal surface is unique, the location of minimal surface remains the same after this change, so that such an operator can be reconstructed from the boundary region . In general, the subalgebra of operators that can be reconstructed on does not have a tensor factorization in real space.

As a side remark, we would like to mention that the above proof also directly applies to the spatial random tensor networks in Ref. hayden2016 .

5.3 More general correlators

The random average technique allows us to understand more refined structure of correlation functions. For this purpose we generalize the concept of correlation matrix defined in Ref. hayden2016 . For two space-like regions and (which does not need to be defined on the same time slice), each region defines a sub-algebra of operators. We denote as a orthonormal basis of Hermitian operators in , which satisfies


The correlation matrix between the two regions is defined as


Any two-point correlation function can be expressed as a bilinar form, where is the coefficient of expanded onto the basis , and similar for . Therefore contains complete information about correlation functions between and . To define a local-unitary invariant measure of the correlation, we study


is independent of the choice of operator basis , . for all determines all singular values of correlation matrix . The singular values of corresponds to a special basis of operators satisfying , so that this basis is the analog of primary fields in conformal field theories, and the singular values (which decays in power law in the CFT case) provide a complete measure of the correlation structure between and .

Using the orthonormal condition Eq. 24, one obtains


where and are two permutation operators


Therefore after random average, the calculation of is mapped to a partition function of discrete gauge theory in the same way as -th Renyi entropy, except for a different boundary condition. The boundary condition now consists of flux ending at and flux ending at , instead of a flux of cyclic permutation for both and in the case of Renyi entropy calculation. In the large bond dimension limit, if the two regions are far away enough such that the minimal surface bounding is disjoint (i.e. when the mutual information ), the flux surfaces for will also be disjoint, consisting of a () surface on minimal surface (), respectively. This leads to


where is the same quantity defined for the bulk QFT between the entanglement wedge regions and . If the bulk low energy degree of freedom is trivial, , in which case corresponds to a flat spectrum of singular values at . The factorized form suggests that there is no connected correlation functions. When there is a nontrivial , it contributes all nontrivial connected correlation functions. In other words, the correlation spectrum (singular value spectrum of two-point functions) of regions is identical to that of the bulk QFT in the large limit, up to an overall factor . It should be noted that the discussion here is independent from whether and are space-like separated, and only requires that the HRT surface bounding (which in our setup is well-defined even if and are not space-like separated) is disjoint.

The above discussion can also be generalized from boundary-boundary correlation functions to more general bulk-boundary correlation functions. For a given set of boundary points and bulk points, one can always define a “generalized density martrix" by contracting all other indices in the tensor network and leave these boundary points and bulk points open. For the network in Fig. 6 (a), this means to remove all boundary operators and the bulk operators, leaving the corresponding indices open. This “generalized density matrix" determines all multi-point correlations between these bulk and boundary points, in the same way how an ordinary density matrix determines equal time correlators. Specifically, we focus on the correlation function between two regions and calculate the , for all .

Figure 8: An example of the bulk-boundary correlation between bulk region and boundary regions and . The bulk theory contains a closed world line of a qudit that intersects with and the entanglement edge of but not with that of (see text).

As a concrete example, we consider a very simple network, illustrated in Fig. 8. The bulk state is defined as the direct product of a VBS state and a single pair of qudits with dimension . The qudits have a world line which forms a close loop. is a bulk small region that intersects with the world line . Two boundary regions and are chosen such that links with the closed co-dimension surface , but does not link with . For such a state, we analyze the correlation spectrum between and and that between and .

When we calculate between and , because the doesn’t link with the world line , in the large limit,


From this result, we know there is only one non-zero singular value of matrix in the large limit, . One can compare this result with the disconnected piece of the correlation function:


which is a rank 1 matrix. We can define


where is the swap operator acting in the region . After random average, this quantity corresponds to a gauge theory with the boundary condition of acting on the first replica and acting on the second replica. A straightforward calculation shows that in the large limit


Thus we conclude that there is no connected correlation function between boundary region and the bulk region in the large limit.

The situation is totally different for the correlation function between boundary region and bulk region . Similarly, in the large limit,


while the disconnected part is


Thus there is non-trivial connected correlation between and . Physically, the non-trivial correlation is a consequence of the fact that operators acting on the qubit with world line can be locally reconstructed on boundary region .

5.4 Comments on unitarity of the boundary theory

As we discussed earlier, our formalism applies to both Euclidean and Lorenzian space-time. In Lorenzian case an important question is whether the boundary theory defined by our tensor newtork is unitary. This question is closely related to the correlation function we discussed in previous subsections. In this section, we will show that boundary unitarity emerges when the bulk has enough degrees of freedom. In contrast, in the limit of small number of bulk degrees of freedom (which is the case if we restrict to the code subspace around a given classical geometry), the corresponding boundary theory only have unitarity in the code subspace.

In order to decide whether the evolution is unitary, we treat the evolution operator as a state in doubled system (Fig. 9 (a)) and measure the mutual information between the past subsystem (P) and the future subsystem (F). This mutual information is also the quantum channel capacity of lloyd1997capacity . If and only if the past state is maximally entangled with the future state, the evolution operator is unitary. To study the entanglement in , we first study the second Renyi entropy. It is obvious that


since is a pure state. Thus we only need to calculate the entropy of the past or the future state by inserting swap operators to the past or the future boundary. The corresponding bulk geometry is a bulk Lorenzian time evolution and its complex conjugate, connected at an initial time slice and a final time slice, as is shown in Fig.9(a). (One can glue the forward and backward time evolution at an arbitrary bulk Cauchy surface that bounds the boundary initial time slice. Different choices of the Cauchy slice gives the same partition function due to unitarity of the bulk quantum field theory.) For concreteness we consider the entropy of the future system. This calculation is similar to Sec.3 except that the boundary condition now is for all vertical links crossing the future boundary, which we denote as , and elsewhere on the boundary (Fig. 9 (b) and (c)). Therefore the only nontrivial flux is in the time circle. After random average, we obtain a gauge theory in the bulk. If we denote the set of all bulk links with as , the boundary of has to contain the future boundary . If in addition to there are other boundary of in the bulk (Fig. 9 (d)), the boundary is a flux line. In the semiclassical limit, the Renyi entropy is determined by


which is determined by the bulk region that has minimal second Renyi entropy in the bulk theory.

Figure 9: Illustration of the unitarity calculation. (a) The time evolution of the boundary in a time interval corresponds to a network with open legs on the past and future boundary, which can be viewed as a pure state in a doubled Hilbert space. The corresponding bulk geometry is a Lorentzian theory between two Cauchy surfaces and bounding the past and future boundary time slice, respectively. (b) The circuit that computes the second Renyi entropy of the future subsystem (see text). A swap operator is applied to the future subsystem of two replica, while identity operator acts on the past. (c) The bulk geometry corresponding to the circuit (b). There is a swap operator for each red link crossing the future boundary, and an identity operator for each blue link crossing the past boundary. After random average, the boundary condition of the gauge theory is for all red links, and elsewhere. (d) Illustration of possible semiclassical configuration of the gauge theory, with for all links crossing the green region . The bulk action determines the position of the boundary of (red closed curve).

The channel capacity of the boundary (37) can have different behavior depending on the bulk theory. We will discuss two cases below.

  1. Code subspace unitarity. The bulk theory contains short-range UV degrees of freedom (the VBS states) and the low energy quantum field theory degrees of freedom. We can restrict the bulk time-evolution to the latter, while requiring the UV degrees of freedom to stay in its ground state. In other words, the bulk time evolution operator before random projection looks like


    In this case, the second Renyi entropy is a sum of VBS and QFT contributions:


    Here is the complement region of , and is the net bulk volume, such that is the total dimension of the bulk low energy QFT subspace. Under the code subspace condition


    the minimal entropy is given by , which means covers the whole Cauchy surface in the bulk. It is interesting to note that the code subspace condition is the same as that in the spatial RTNhayden2016 . The second Renyi entropy in this case corresponds to

    In the semiclassical limit, the calculation here can be generalized to higher Renyi entropies, and all Renyi entropies converge to the same value . Therefore the channel capacity is , which tells us that the theory has unitarity in the code subspace.

  2. Unitarity in the whole boundary Hilbert space. If we consider a large , we can also reach the opposite situation of volume law entropy exceeding the area law entropy for all regions:


    In this case, the region with minimal second Renyi entropy is , in which case the boundary expands to the boundary itself. This corresponds to , which is the maximal value of the boundary. The large case corresponds to a boundary-to-bulk isometry discussed in the spatial RTN casehayden2016 . Physically, such a large means the bulk geometry is strongly fluctuating, since, for example, different states in this big bulk Hilbert space can have completely different configurations of extremal surfaces bounding the same boundary region.

6 Gauge fixing and finite D fluctuations

In Sec.2.3, we point out an apparent inconsistency in our theory, that is the boundary theory seems to be irrelevant to the bulk tensor geometry after random projection. In contrast, as we have seen in Sec. 3 to 5, the random average leads to a discrete gauge theory and gives RT formula, which clearly shows that the information of bulk geometry is encoded in the boundary theory. In this section, we will show that this apparent paradox is related to the gauge redundancy, which leads to an overall constant factor in partition function of the theory. We will show how to solve the redundancy problem by a gauge fixing, which corresponds to a refined definition of the random projections on a selected subset of links rather than all bulk links.

6.1 Gauge redundancy in the original space-time tensor

In this subsection, we will first demonstrate how the inconsistency arises, and then we will resolve the problem by gauge fixing.

The inconsistency can be easily seen in the following calculation. For simplicity, we calculate the second Renyi entropy of some boundary region . In all of the previous calculations, we have assumed the following approximations.


However as we mentioned in Sec.2.3, if we impose random projections on all bulk links, the l.h.s of the above equation do not depends on the bulk tensor geometry, because the isolated bulk vertices that appear both on the denominator and the numerator cancels before the random average. However, the r.h.s of the above equation obviously depends on the bulk tensor network geometry, because it calculates the free energy cost of the flux that bounds the region A, which is dominated by the extremal surface contribution in large limit. Thus the assumption that l.h.s and r.h.s in Eq.42 are approximately equal to each other is wrong. This is in sharp contrast with the situation in spatial random tensor networks in Ref. hayden2016 , where the same approximation is valid in the large limit.

To analyze the origin of the problem in Eq. (42), we analyze the fluctuations in the random average. The l.h.s. of Eq. (42) can be expanded as


where we have only written down one single term in the leading order of the expansion. Physically, we expect the fluctuation of to be small in the large bond dimension limit. However, if we literally express the random average in gauge theory partition function, on the numerator of the second term we have a term with gauge theory, while on the denominator we have gauge theory. Note that the partition function obtained from the random average is literally a sum over gauge vector potentials, which thus contains redundant copies of the properly gauge fixed partition function. If we denote the gauge theory partition function with proper gauge fixing (which means only gauge inequivalent configurations are summed) as , and that for the gauge theory as , in the large limit we have


In large limit, since they are all dominated by trivial configuration (as long as the minimal action configuration is unique modular gauge transformations). The exponentially large factor is the reason of the large flucutation which makes Eq. (42) invalid. Our goal is to introduce a gauge fixing procedure to remove the redundancy factor.

6.2 Gauge fixed space-time tensor

For discrete gauge theories, the simplest approach of gauge fixing is to fix the gauge vector potential on some links to identity. An appropriate gauge fixing is obtained by choosing a subset of links such that if we set the link variable for all , then 1) no gauge flux is fixed; 2) no gauge transformation can be carried while preserving the gauge fixing. These two requirements are equivalent to requiring to be a spanning tree of the network. Since the sum over permutation group element comes from average over random projections on that link, the gauge fixing is equivalent to selecting a spanning tree and introduce random projections on on links that are not on . More explicitly, we summarize the general procedure of gauge fixing below, which is also illustrated in Fig. 10. (In this figure, the red links represent the boundary system and all other links are the bulk.)

  1. Exclude the boundary tensor network (the red tensors in Fig.10(a)) as well as the bulk links that directly connect the bulk and the boundary (the blue links in Fig.10(a)) and we first only focus on the isolated bulk tensor network (the black tensor network in Fig.10(a)).

  2. Find a spanning tree in the isolated bulk tensor network. In graph theory, a spanning tree of an undirected graph is a subgraph that is a tree and includes all the vertices of . In addition, pick one link that directly connects the boundary with the spanning tree , as is shown in Fig. 10 (b) by the green link connecting the boundary. Denote the union of and this extra link as .

  3. The holographic theory is defined by introducing an independent random projector on all bulk links that do not belong to , i. e. all the black links in Fig. 10 (b).

Figure 10: (a) is a tensor network in which the red tensors are the boundary system and the black tensors are named as isolated bulk tensor network. Blue links directly connect the boundary system with the isolated bulk tensor network. Fig.(b) shows one particular choice of the links (green links) on which the random projectors are removed following the gauge fixing procedure.

From the gauge fixing procedure above one can see directly that the gauge redundancy is completely removed by this choice. In Appendix B we provide a counting argument to further verify that. With this modified definition of random projection, we have for example and with the gauge redundancy removed. Also from the definition above it is clear that the boundary theory obtained after random projection still has a nontrivial dependence on the bulk geometry and bulk theory since the bulk geometry is not completely detached.

6.3 Finite D fluctuation

In this subsection, we prove that in the limit, the fluctuation can be bounded. The proof is largely parallel to the discussion in Ref.hayden2016 , so we will only sketch the main steps here for completeness.

In the derivation of Renyi entropies, we have also made the assumption by taking separate average of the numerator and denominator:


Following the convention of Ref. hayden2016 we denote the numerator and denominator of the first line by and , which are functionals of the random projectors. The large value of these partition functions are dominated by the classical configuration, which we denote by and . The RT formula is given by


At finite , both the numerator and denominator are fluctuating. The finite deviation of the entropy from the RT value is given by


This deviation of entropy is bounded by directly using the following result from Ref. hayden2016 :

  • If we find an upper bound for


    and in the large limit, then in the limit ,


Here the left-hand side of the equation means the probability that the entropy deviation from the RT formula value is smaller or equal to .

If the minimal action field configuration is unique, then using the fact that all other field configurations have a statistical weight that is suppressed at least by a factor