Entanglement Wedge Reconstruction of Infinite-dimensional von Neumann Algebras using Tensor Networks

Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are infinite-dimensional. We define a von Neumann algebra of type II$_1$ acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II$_1$ acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative entropies in infinite-dimensional Hilbert spaces.


Introduction
The study of entanglement entropy has utilized results in the mathematical field of operator algebras [2][3][4]. In quantum field theory, von Neumann algebras are associated with causally complete subregions of spacetime [19]. Since AdS/CFT implies that information in the bulk is encoded redundantly in the boundary, quantum error correction is a natural framework in which to elucidate the connection between holographic quantum field theories and their gravity duals [1,6,23,25]. Quantum error correction with finite-dimensional Hilbert spaces has been used to argue that entanglement wedge reconstruction is identical to the Ryu-Takayanagi formula and the equivalence of bulk and boundary relative entropies [7,23]. In order to study a more realistic toy model where boundary subregions are characterized by infinite-dimensional von Neumann algebras, we should consider quantum error correcting codes defined on infinite-dimensional Hilbert spaces.
The purpose of this paper is to construct a Quantum Error Correcting Code (QECC) where the physical Hilbert space and the code subspace are infinite-dimensional and admit the action of infinite-dimensional von Neumann algebras. We describe a toy model that allows us to see how a von Neumann algebra on the code subspace is reconstructed on the physical Hilbert space. Von Neumann algebras acting on finite-dimensional Hilbert spaces must be of type I. Our toy model contains an example of an infinite-dimensional von Neumann algebra, namely a type II 1 factor, which is defined and explained in Section 2.4. 1 Furthermore, we show that in the context of operator-algebra quantum error correction, this QECC satisfies the following two statements: • Entanglement wedge reconstruction [29][30][31], • Relative entropy equals bulk relative entropy (JLMS formula [8]).
In particular, we first show that our QECC satisfies entanglement wedge reconstruction for a particular choice of von Neumann algebras acting on the code and physical Hilbert spaces, and then we invoke Theorem 1.1 in [17] to argue that our QECC also satisfies the JLMS formula. We finally show that the relative entropies defined with respect to the infinitedimensional von Neumann algebras we consider can be expressed as limits of the relative entropies defined with respect to finite-dimensional subalgebras. Thus, another way to see that our QECC satisfies the JLMS formula is to note that our QECC satisfies the JLMS formula with respect to finite-dimensional von Neumann algebras. The JLMS formula for finite-dimensional algebras is studied in [7].
The technical assumptions that connect entanglement wedge reconstruction and the JLMS formula are presented in Theorem 1.1 of [17], which we repeat below. Theorem 1.1 (Kang-Kolchmeyer [17]). Let u ∶ H code → H phys be an isometry 2 between two Hilbert spaces. Let M code and M phys be von Neumann algebras on H code and H phys respectively. Let M ′ code and M ′ phys respectively be the commutants of M code and M phys . Suppose that the set of cyclic and separating vectors with respect to M code is dense in H code . Also suppose that if Ψ⟩ ∈ H code is cyclic and separating with respect to M code , then u Ψ⟩ is cyclic and separating with respect to M phys . Then the following two statements are equivalent:
Tensor networks with a finite number of nodes have been used to construct QECC for finite-dimensional Hilbert spaces, which have yielded physical insights into holography [6,7]. One such example is the HaPPY code which demonstrates the kinematics of entanglement wedge reconstruction [15]. In particular, the code subspace of the HaPPY code consists of states where the areas of the extremal surfaces are not quantum fluctuating [27]. Furthermore, some aspects of entanglement wedge reconstruction have also been studied using random tensor networks, where the dimension of each tensor index is finitely large [25]. Given the utility of tensor networks for preparing holographic states [28] and the fact that actual holographic CFTs have infinite-dimensional Hilbert spaces, we expect that infinite-dimensional tensor networks provide additional insights. In particular, an infinite-dimensional tensor network can illustrate the connection between the Reeh-Schlieder theorem and quantum error correction. Furthermore, the modular operator of Tomita-Takesaki theory plays a central role in bulk reconstruction in the continuum limit [9,10]. By linking holographic QECC with infinite-dimensional operator algebras, an infinite-dimensional tensor network might allow one to perform explicit computations relevant to holography that involve the modular operator. An infinite holographic tensor network also has the potential to make boundary locality manifest. In this paper, we demonstrate that the use of tensor networks in quantum error correction can be generalized to the case of infinite-dimensional Hilbert spaces. In our toy model, the infinite-dimensional code and physical Hilbert spaces are constructed by tensoring together the Hilbert spaces of a countably infinite number of qutrits and then restricting to a countably infinite-dimensional subspace. Finite collections of qutrits are related by a tensor network as represented in Figure 1.1. Each connected graph defines an isometry from the state of two code qutrits (denoted as black nodes) to the state of four physical qutrits (denoted as white nodes). Our toy model explores how tensor networks with a repeated pattern can be generalized to define a QECC with infinite-dimensional Hilbert spaces. This model does not capture the negatively curved geometry of AdS; however, we believe that our construction can be generalized to encapsulate the holographic setup. A more detailed summary of our construction is given as follows.
• The code pre-Hilbert space pH code is defined to be the Hilbert space of a countably infinite collection of qutrit pairs, where all but finitely many qutrit pairs are in the maximally entangled state λ⟩ = 1 √ 3 ( 00⟩ + 11⟩ + 22⟩). Each code qutrit pair is represented by two vertically-aligned black nodes in Figure 1.1. The code Hilbert space H code is the completion of pH code . The physical pre-Hilbert space pH phys and physical Hilbert space H phys are constructed the same way. Each qutrit pair in the physical Hilbert space is represented by two vertically-aligned white nodes in Figure 1.1.
• We construct a bulk-to-boundary isometry from pH code to pH phys using the tensor network in Figure 1.1. The tensor network is comprised of infinite copies of connected diagrams, where a single connected diagram is represented in Figure 4.1. Each trivalent vertex is associated with the rank-four perfect tensor 3 of the three qutrit code T iãbc . Our tensor network maps the states of the black qutrits to the states of the white qutrits. Using these notations, the isometry associated with a connected diagram is explicitly given by where the qutrits are labeled as in Figure 1.1. The indices p, q,x,ỹ,z,w,c are all valued in {0, 1, 2}. The isometry from pH code to pH phys may be naturally extended to an isometry that maps H code to H phys .
• Using the code and physical Hilbert spaces and an isometry relating them, we define von Neumann algebras M code and M phys . The ⋆-algebra A code is defined to be the algebra of operators that only act nontrivially on a finite number of qutrits in the top row of black qutrits in Figure 1.1. The double commutant of A code defines M code , the von Neumann algebra acting on the top row of black qutrits. We explicitly show that the commutant M ′ code is the analogously defined algebra that acts on the bottom row of black qutrits. We also define M phys and M ′ phys which respectively act on the top and bottom row of white qutrits in Figure 1.1. To show that M code is a type II 1 factor, we define a linear function T ∶ M code → C, which is given by where λ⋯⟩ ∈ pH code is the state where all black qutrit pairs are in the state λ⟩. We demonstrate that T (O) is a trace and invoke Theorem 2.27 to prove that M code is a type II 1 factor. Likewise, M ′ code , M phys , and M ′ phys are also type II 1 factors. • We determine a map from M code to M phys that explicitly shows how operators that act on black qutrits may be reconstructed as operators that act on white qutrits. First, note that an operator O that acts on a black qutrit i in Figure 4.1 may be expressed as an operatorÕ that acts on the white qutritsã,b. The relation between O andÕ is given byÕ where Uãb is a unitary matrix that acts only on white qutritsã,b. By applying the above formula finitely many times, we may construct a map from A code into M phys which we call the tensor network map. We then show that there is a natural way to extend the tensor network map to a map from M code into M phys . We demonstrate that the image under the tensor network map of an operator O ∈ M code acts on the code subspace in the same way as O. The same statement holds for the commutant M ′ code . This demonstrates that our QECC satisfies statement 1 of Theorem 1.1.
• To show that our QECC satisfies the assumptions of Theorem 1.1, we find a dense subset of pH code that consists of cyclic and separating vectors with respect to M code . For example, a state in pH code where each black qutrit pair is in a pure state with maximal Schmidt number (such as λ⟩) is cyclic and separating with respect to M code . We also prove that any cyclic and separating state with respect to M code is mapped via the bulk-to-boundary isometry to a cyclic and separating state with respect to M phys . Thus, our QECC satisfies all assumptions and statements of Theorem 1.1.
An outline of this paper is given as follows. First, we review aspects of infinite-dimensional von Neumann algebras in Section 2 and Tomita-Takesaki theory in Section 3. In particular, we explain why type III 1 factors are relevant for quantum field theory. Then, we describe in detail our construction of an infinite-dimensional QECC in Section 4. Tensor networks play an important role in our toy model. In Section 5 we define von Neumann algebras M code ⊂ B(H code ) and M phys ⊂ B(H phys ). 4 In Sections 6 and 7, we show that our example satisfies the properties of bulk reconstruction in Theorem 1.1. In Section 8 we show that cyclic and separating vectors with respect to M code (M phys ) are dense in H code (H phys ). We also show that cyclic and separating vectors with respect to M code are mapped via the bulkto-boundary isometry u to cyclic and separating vectors with respect to M phys . It follows that our tensor network model satisfies both statements in Theorem 1.1. In Section 9, we prove that M code and M phys are type II 1 factors. In Section 10, we demonstrate that the relative Tomita operator defined with respect to M code or M phys may be bounded or unbounded, depending on the choice of states. In quantum field theory, the Tomita operators defined with respect to local operator algebras are generically unbounded [18]. In section 11, we show that the relative entropy of two cyclic and separating states may be computed by tracing over the entire Hilbert space except the Hilbert space of the first N qutrit pairs, computing the relative entropy of the reduced density matrices with the finite-dimensional relative entropy formula, and taking the limit as N → ∞.

Infinite-dimensional von Neumann algebras
In this section, we provide background information on operator algebras, including the definitions of type I, II 1 , II ∞ , and III factors, and elucidate their relevance to physics. We also prove theorems that are useful for constructing our infinite-dimensional QECC. First, we review the notion of Hilbert space and bounded operators in Section 2.1. With these notions, we recall basic theorems about Hilbert spaces, operators, and boundedness. Based on those, we explain the operator topologies in Section 2.2. Then we introduce relevant theorems and present definitions of von Neumann algebras in a physics-friendly manner in Section 2.3. We review the different types of von Neumann algebra factors in Section 2.4. This section mainly draws upon [13], [14], and [21]. Theorem 2.6 (Bounded Linear Transformation (BLT) Theorem [21]). Suppose O is a bounded linear transformation from a pre-Hilbert space pH to a Hilbert space H. Then O can be uniquely extended to a bounded linear operator (with the same norm) from the completion of pH to H.

Hilbert Space and Bounded Operators
One can also define an isometry more generally as a norm-preserving map from one Hilbert space to a different Hilbert space. An example is the isometry from H code to H phys considered in Theorem 1.1.
We will evaluate the norm of the above equation and use the triangle inequality on the right hand side. We need the inequality where K > 0 is some constant. This inequality follows from the fact that the limit converges for all ψ⟩ ∈ H, which implies that the set Given any > 0, there exists an M ∈ N such that for m > M , There also exists an N ∈ N such that for n > N , However we note that Theorem 2.9 is nontrivial; we demonstrate this by the following counter example where the statement holds where the equation (2.1) does not hold. Consider the double-sequence a n,m ∈ R, indexed by n, m ∈ N, which is defined as a n,m ∶= 1 m n + n m . (2.10) One can check that lim n→∞ lim m→∞ a n,m = lim m→∞ lim n→∞ a n,m = 0. (2.11) However, we get a nonzero limit of a n,n such that lim n→∞ a n,n = 1 2 . (2.12) This demonstrates that the Theorem 2.9 is not a simple consequence of the definition of a limit. Our proof of Theorem 2.9 makes use of Theorem 2.4, which is demonstrated above in its proof.  A sequence of bounded operators {O n } converges strongly if and only if lim n→∞ O n ψ⟩ converges for all ψ⟩ ∈ H. Note that the hermitian conjugates O † n need not converge strongly. We will sometimes use s-lim to denote a strong limit. Definition 2.13. The weak operator topology is the smallest topology that contains the following basic neighborhoods:

Topologies on B(H)
A sequence of bounded operators {O n } converges weakly if and only if lim n→∞ ⟨χ O n ψ⟩ converges for all χ⟩ , ψ⟩ ∈ H. We will sometimes use w-lim to denote a weak limit. Definition 2.14. The ultraweak operator topology is the smallest topology that contains the following basic neighborhoods:  [11], the norm operator topology is stronger than the strong operator topology and the ultraweak operator topology, which are both stronger than the weak operator topology.

Definition of von Neumann algebras
In this section, we define von Neumann algebras, factors, and hyperfinite von Neumann algebras.
• M is closed in the strong operator topology, • M is closed in the weak operator topology. Definition 2.18. A von Neumann algebra is an algebra that satisfies the statements in Theorem 2.17.
Given a ⋆-subalgebra of B(H) containing the identity, we can generate a von Neumann algebra by taking either the double commutant or the closure in the strong or weak topology. Definition 2.19. A factor is a von Neumann algebra M with trivial center. That is, where I denotes the identity operator. Note that the union of finitely many closed sets is also closed. However, the union of infinitely many closed sets need not be closed. In section 5, we define a hyperfinite von Neumann algebra by taking the closure of an infinite union of finite-dimensional von Neumann algebras. The closure introduces additional operators into the algebra. Definition 2.21. If M is a von Neumann algebra, a non-zero projection p ∈ M is called minimal if, for any other projection q, q ≤ p ⇒ (q = 0 or q = p). Definition 2.22. Let A be a ⋆-algebra that contains the identity operator I. Let T ∶ A → C be a linear function on A. The map T is • a state if T is positive and normalized, Given any normalized Hilbert space vector Ψ⟩ and a von Neumann algebra M ⊂ B(H), one can naturally define an associated state T Ψ ∶ M → C as For this reason, the term "state" is often used to refer to both Hilbert space vectors and positive, normalized linear functions of a von Neumann algebra.

Classification of von Neumann algebras
In this section, we review the classification of von Neumann algebra factors in a manner to have a direct consequence in physics. We first review type I factors, which are the only factors relevant for finite-dimensional Hilbert spaces. We then review type II factors. We explicitly construct type II 1 factors from our tensor network model in Section 5. We finally review type III factors. We explain why among type III factors we only expect type III 1 factors to arise as algebras in local quantum field theories.

Type I factors
Definition 2.23. A factor with a minimal projection is called a type I factor. Definition 2.24. A type I factor that is isomorphic 7 to the algebra of bounded operators on a Hilbert space of dimension n is a type I n factor. Definition 2.25. A type I factor that is isomorphic to the algebra of bounded operators on an infinite-dimensional Hilbert space is a type I ∞ factor.

Type II factors
Definition 2.26. A type II 1 factor is an infinite-dimensional factor M on H that admits a non-zero linear function tr ∶ M → C satisfying the following properties: • tr is ultraweakly continuous.

Type III factors
In order to define the von Neumann algebra of type III factor, we first recall from [13] the definition of the invariant S(M ) using the modular operator, which is presented in section 3. When 0 ∈ S(M ), every modular operator ∆ Ψ is not a bijection of D(∆ Ψ ) onto H. 8 It follows that the inverse of every modular operator is not defined on the entire Hilbert space. This is exactly desired for a local quantum field theory because the inverse of a modular operator is the modular operator defined with respect to the commutant: As shown in [18], ∆ ′ Ψ should not be bounded and thus should not be defined on the entire Hilbert space. If 0 ∉ S(M ), then there exists a state whose modular operator defined with respect to M ′ is bounded. Hence we expect the condition 0 ∈ S(M ) to be satisfied by the algebras arising from a physical local quantum field theory.
where I denotes the identity operator.
As explained in [18], we expect a local quantum field theory to have a continuous spectrum of the modular operator ∆ Ψ . Thus we see that the von Neumann algebra of type III 1 factor is the only factor that is relevant to physics among all possible type III factors.
We can also use S(M ) to characterize factors of types I or II. For such factors, S(M ) is given by the following theorem.

Relative Entropy from Tomita-Takesaki theory
In this section, we review aspects of Tomita-Takesaki theory that are relevant to Theorem 1.1. In particular, we need these definitions to show that our QECC satisfies the assumptions of Theorem 1.1. For a more thorough review, see Section 3 of [17] as well as [18]. For this definition to make sense, Ψ⟩ must be cyclic and separating with respect to M . Definition 3.4. Let S Ψ Φ be a relative Tomita operator. The relative modular operator is Definition 3.5 ( [12]). Let Ψ⟩ , Φ⟩ ∈ H and let Ψ⟩ be cyclic and separating with respect to a von Neumann algebra M . Let ∆ Ψ Φ be the relative modular operator associated with Ψ⟩ , Φ⟩ , and M . The relative entropy with respect to M of Ψ⟩ and Φ⟩ is Definition 3.6. Let M be a von Neumann algebra on H and Ψ⟩ be a cyclic and separating vector for M . The Tomita operator S Ψ is where S Ψ Ψ is the relative modular operator defined with respect to M . The modular operator ∆ Ψ = S † Ψ S Ψ and the antiunitary operator J Ψ are the operators that appear in the polar decomposition of S Ψ such that

The isometry between two infinite-dimensional Hilbert spaces
In this section, we show how a tensor network with infinitely many nodes can be used to define an isometry (i.e. a norm preserving map) from one infinite-dimensional Hilbert space to another. The isometry will be denoted by u ∶ H code → H phys . We first review some preliminary facts about the three qutrit code.

The three-qutrit code and a finite tensor network
The three-qutrit code is an example of a QECC. A code qutrit is isometrically mapped to a Hilbert space of three physical qutrits. The map is given by We can write this more succinctly as where i denotes an input leg andã,b,c denote output legs. We can apply successive isometries to create an isometry from two code qutrits to four physical qutrits. We illustrate this with a tensor network, represented in Figure   The isometry corresponding to Figure 4.1 is given by Throughout this paper, we use subscripts to associate qutrits with specific nodes in Let U be a unitary operator that acts on a two-qutrit state as and define λ⟩ ∶= where ψ⟩ãd is the same state as ψ⟩ ij , except on the white qutritsã,d. That is, starting with the state ψ ⟩ãbdẽ on the white qutrits, one can apply separate unitary transformations on white qutritsã,b andd,ẽ to recover ψ⟩ ij on white qutritsã,d and the maximally entangled state λ⟩ on qutritsb,ẽ.
Given an operator O that acts on qutrit i in Figure 4.1, we may define an operatorÕ that acts on the adjacent white qutritsã,b as follows: We say that O, which acts on the code Hilbert space, is reconstructed asÕ, which acts on the physical Hilbert space.

The code and physical Hilbert spaces
Our general setup is depicted in Figure 1.1. In our construction of an infinite-dimensional QECC, the code and physical Hilbert spaces, H code and H phys , are each defined as the completions of pre-Hilbert spaces, pH code and pH phys . As Figure 1.1 shows, we may intuitively think of either the code or physical pre-Hilbert space as an infinite tensor product of two black qutrits or four white qutrits. From now on, whenever we say collection we are referring to the qutrits in a single connected diagram in Figure 1 The pre-Hilbert space pH code is defined to include states of black qutrits where all but finitely many pairs of black qutrits are in the state λ⟩, defined in equation (4.5), which we sometimes also refer to as the code reference state. Any vector in pH code is a finite linear combination of vectors in an overcomplete basis, where each basis vector may be written as where each p k or q k index (for k ∈ {1, 2, . . . , M }) is valued in {0, 1, 2} and specifies an orthonormal basis vector of a black qutrit. The index M can be any natural number. The qutrits in each collection are contained in square brackets. To shorten notation, we will refer to the above basis vector as M, {p, q}⟩. The ⊗ λ⟩ ⋯ means that all the black qutrit pairs in the (M + 1)th collection and beyond are in the reference state λ⟩. Note that these basis vectors are not all linearly independent.
Given two basis vectors M 1 , {p, q} 1 ⟩ and M 2 , {p, q} 2 ⟩, their inner product is calculated by ignoring all collections beyond the collection max (M 1 , M 2 ) and then taking the usual inner product on the remaining 9 max (M 1 ,M 2 ) -dimensional Hilbert space. Note that the basis vectors M, {p, q}⟩ are not all mutually orthogonal, but they are all normalized. With an inner product, we can define Cauchy sequences. The Hilbert space H code is defined as the closure of pH code so that all Cauchy sequences in H code converge. We start from pH code and include all Cauchy sequences to define H code . If the difference of two Cauchy sequences converges to zero, then we identify the two Cauchy sequences for the purposes of defining H code .
The physical pre-Hilbert and Hilbert spaces are defined in a completely analogous way. Each collection consists of four white qutrits. The physical reference state for four white qutrits is given by λλ⟩ ∶= λ⟩ãd λ⟩bẽ where we are referring to Figure 4.1 to label the qutrits. We choose this reference state for the white qutrits because it is the image of λ⟩ ij under the isometry given by equation (4.3).

The tensor network of isometries
The bulk-to-boundary isometry u is given by a linear norm preserving map u ∶ H code → H phys . First, we define its action on pH code and then use Theorem 2.6 to extend its domain to H code . Each vector in pH code is mapped to a vector in pH phys . The isometry u acts on the basis vector M, {p, q}⟩ by applying the isometry given in equation (4.3) to each collection separately. The state of each black qutrit pair is mapped to a state of four white qutrits. The code reference state is mapped to the physical reference state. Because the map u is linear and norm-preserving, a Cauchy sequence in pH code is mapped to a Cauchy sequence in pH phys . Thus, we can define u on all of H code .

Defining von Neumann algebras
Now that we have defined H code and H phys and the isometry u ∶ H code → H phys , we want to define von Neumann algebras on these Hilbert spaces.

Definition of M code
We now define M code ⊂ B(H code ). First, we define a ⋆-algebra called A code which acts on pH code . Referring to Figure 4.1 for qutrit labels, every operator a ∈ A code may be written as where a p 1 p 2 ⋯p N q 1 q 2 ⋯q N are the matrix elements of the operator. Each p k , q k index (k ∈ {1, 2, ⋯, N }) is valued in {0, 1, 2} and specifies an orthonormal basis vector of one black qutrit. The ⊗I⋯ means that a (N ) acts as the identity on all collections beyond the N th collection. Each collection is represented by square brackets. The label N may be any natural number. The (N ) superscript reminds us of the value of N for this operator. The operator a (N ) maps pH code → pH code . Because there exists a K > 0 such that a (N ) ψ⟩ ≤ K ψ⟩ for all ψ⟩ ∈ pH code , a (N ) is bounded. Thus, a (N ) maps Cauchy sequences in pH code into Cauchy sequences in pH code , and Theorem 2.6 implies that a (N ) is uniquely defined as a bounded operator acting on H code . The ⋆-algebra A code is closed under hermitian conjugation and contains the identity.
A sequence of operators {a n } ∈ A code converges strongly to an operator in B(H code ) if and only if lim n→∞ a n Ψ⟩ converges for all Ψ⟩ ∈ H code . The ⋆-algebra A code is not closed under strong limits. The von Neumann algebra M code is defined to be the closure of A code in the strong operator topology. We construct M code from all strongly converging limits of sequences in A code . In topology, to construct the closure of a set, it is necessary, but generally not sufficient, to include limits of converging sequences [21]. We must also include limits of nets, which are more general than sequences. However, it is possible to show that every operator in M code can be written as a strong limit of a sequence in A code . In the next section, we show that the set S ⊂ B(H code ) of bounded operators that are strong limits of sequences in A code is the smallest strongly closed subset of B(H code ) that contains A code , which implies that M code = S. This is because • S is equal to the commutant of a ⋆-algebra that contains the identity, which is a von Neumann algebra [18]. Because S is a von Neumann algebra, S is strongly closed.
• Any strongly closed subset of B(H code ) that contains A code must contain S because S is defined to only contain all strongly convergent sequences in A code .
We provide explicit details in the next subsection.

The commutant of A code and M code
In this section, we explicitly describe the commutant of A code , which is denoted by A ′ code . Then, we demonstrate that every operator in M code may be written as a strongly convergent sequence of operators in A code .
An orthonormal basis of pH code is an orthonormal basis of H code . To see this, let Φ⟩ ∈ H code . Let { φ n ⟩} ∈ pH code be a sequence that converges to Φ⟩. Suppose that Φ⟩ is orthogonal to every orthonormal basis vector of pH code . Using Definition 2.2, we need to show that Φ⟩ = 0. Indeed, ⟨φ n Φ⟩ = 0 ∀n ∈ N, so ⟨Φ Φ⟩ = 0. Hence, Φ⟩ = 0.
Thus, we may define an orthonormal basis of H code where each basis vector is a finite linear combination of the vectors given in equation (4.8). We will choose an orthonormal basis e i ⟩ , i ∈ N such that the first 9 orthonormal basis vectors in the sequence { e i ⟩} span the subspace of pH code where the qutrit pairs in the ( + 1)th collection and beyond are in the reference state λ⟩.
A consequence of Theorem 2.9 is that any operator O ∈ B(H code ) may be written as the following strong limit: Each operator O n acts as the projector onto λ⟩ on the qutrits in the (n + 1)th collection and beyond. Each O n may be written as where the coefficient of each term of the sum is defined as The ⊗ λ⟩ ⟨λ ⋯ means that in all collections past the nth collection, O n acts as the projector λ⟩ ⟨λ . Likewise, ⊗ λ⟩ ⋯ means that in every collection past the nth collection, the qutrits are in the state λ⟩. Each of the indices p k ,q k ,r k ,s k (k ∈ {1, 2, . . . , n}) are valued in {0, 1, 2} and denote an orthonormal basis vector of a single qutrit.
For each O n , define the following: The projector λ⟩ ⟨λ in equation (5.3) has been replaced by the identity operator. For any vector ψ⟩ ∈ pH code , we have lim  can be expressed as O = s-lim n→∞Ôn where eachÔ n may be written as above. Furthermore, every such strong limit is clearly in A ′ code . By comparing equation (5.6) with equation (5.1), it is clear the set of operators in A code together with strong limits of sequences in A code (which we called S in the previous subsection) is a von Neumann algebra. In fact, it is the smallest strongly closed subset of B(H code ) containing A code , which is M code by definition. This is because the strong closure of A code must at least contain all strongly convergent sequences of operators in A code . Hence, every operator in M code may be written as a strong limit of a sequence in A code .
Thus, we see that M ′ code may be constructed in the same way as M code , except operators in M ′ code only act nontrivially on the j qutrit in

Definition of the tensor network map
Having defined M code and M phys , we define a linear map from M code into M phys . An operator O ∈ M code is mapped toÕ ∈ M phys . We want the following to hold for all Ψ⟩ ∈ H code : We now describe how to construct this map (which we call the "tensor network map," not to be confused with the map u).

How the tensor network map acts on A code
We first define how the tensor network map acts on operators in A code before generalizing its definition to M code . The operator a (N ) in equation (5.1) is mapped toã (N ) , an operator that acts on H phys . The result is where U is defined in equation (4.4), and the subscripts refer to the specific white qutrits that U is acting on (see Figure 4.1). Given equation (6.2), which shows howã (N ) acts on vectors in pH phys , the domain ofã (N ) may be extended to all of H phys by demanding that a (N ) is a bounded operator and invoking Theorem 2.6. Becauseã (N ) acts trivially on the qutritsd,ẽ in each collection,ã (N ) ∈ M phys .
Equation (6.2) simply amounts to applying the map in equation (4.7) for a finite number of collections. It follows that for a, b ∈ A code , α, β ∈ C, and Ψ⟩ ∈ H code , the tensor network map has the following properties: We will prove these properties for all operators in M code in Section 7.

How the tensor network map acts on M code
Now that we specified how the tensor network map acts on A code , we need to specify how it acts on M code . Let {a n } ∈ A code be a strongly convergent sequence of operators. The image of each a n under the tensor network map isã n ∈ B(H phys ). We will show that {ã n } is a strongly convergent sequence. Then, we will extend the definition of the tensor network map by saying that the strong limit (s-lim n→∞ a n ) ∈ M code is mapped to the strong limit (s-lim n→∞ãn ) ∈ M phys . We will then prove that this map satisfies equation (6.1).
The fact that s-lim n→∞ a n converges means that the sequence of norms { ã n } is bounded from above because a n = ã n ∀n ∈ N. From Theorem 2.5, if lim n→∞ãn ψ⟩ converges for all ψ⟩ ∈ pH phys , then lim n→∞ãn Ψ⟩ converges for all Ψ⟩ ∈ H phys since pH phys is dense in H phys . The next theorem is necessary to show that lim n→∞ãn ψ⟩ converges for all ψ⟩ ∈ pH phys . Theorem 6.1. For any two vectors ψ 1 ⟩ , ψ 2 ⟩ ∈ pH phys , we may define a finite number of vectors η i ⟩ , χ i ⟩ ∈ pH code , (i ∈ {1, 2, . . . , Q} for some Q ∈ N) such that for any operator a (N ) ∈ M phys that may be written as the tensor network map image of some a (N ) ∈ A code , we have that Proof. Choose M ∈ N such that for both ψ 1 ⟩ and ψ 2 ⟩, the qutrits in the (M + 1)th collection and beyond are in the reference state λ⟩. Consider the following set of orthonormal vectors: where the labelsã k ,b k ,d k ,ẽ k (k ∈ {1, 2, . . . , M }) refer to the qutrits in the kth collection (see  Next, we define the following vectors in pH code It follows that so that ⟨ψ 1 ã (N ) ψ 2 ⟩ can be expressed as This demonstrates that we can express ⟨ψ 1 ã (N ) ψ 2 ⟩ as in equation (6.8) for Q = 9 M .
Given any ψ ⟩ ∈ pH phys , Theorem 6.1 asserts that we may choose a finite family of vectors ψ i ⟩ ∈ pH code (i ∈ {1, 2, . . . , Q} for some Q ∈ N ) such that, for any n, m ∈ N, This means that if {a n ψ i ⟩} is a Cauchy sequence for each i, (which it is by assumption) then {ã n ψ ⟩} is also a Cauchy sequence. This shows that lim n→∞ãn ψ ⟩ converges for any ψ ⟩ ∈ pH phys . Thus, the strong limit s-lim n→∞ãn exists and defines an operator, which is the definition of the image under the tensor network map of the strong limit s-lim n→∞ a n . By the definition of M phys , it follows that s-lim n→∞ãn ∈ M phys .
Suppose that the sequences {a n } ∈ A code and {b n } ∈ A code converge strongly to the same operator O. Suppose that s-lim n→∞ãn =Õ 1 and s-lim n→∞bn =Õ 2 . Then s-lim n→∞ (a n −b n ) = 0, which implies that s-lim n→∞ a n − b n = s-lim n→∞ (ã n −b n ) = 0. Hence,Õ 1 −Õ 2 = 0. Thus, the tensor network map is a well-defined map from M code into M phys .
6.3 How the tensor network map acts on M ′ code By construction, the tensor network map is a map from operators in M code into M phys . Due to the symmetry of the tensor network in Figure 4.1, we can also define the tensor network map on operators in M ′ code , which are mapped into M ′ phys in a completely analogous way.

Properties of the Tensor Network Map
In this section, we prove that equations (6.3) to (6.7) hold for all operators in M code .

Theorems on strong and weak convergence
The following theorems will be useful in proving some properties of the tensor network map. Theorem 7.1. Suppose that for a sequence {a n } ∈ A code , lim n→∞ ⟨Ψ a n Φ⟩ = 0 for any Ψ⟩ , Φ⟩ ∈ H code . Suppose that the sequence of norms { a n } is bounded from above. Let a n be the image under the tensor network map of a n . Then lim n→∞ ⟨Θ ã n Φ ⟩ = 0 for any Θ ⟩ , Φ ⟩ ∈ H phys .
Proof. Let { θ ⟩}, { φ m ⟩} ∈ pH phys be Cauchy sequences that converge to Θ ⟩ , Φ ⟩ ∈ H phys respectively. We may compute where K 1 , K 2 are some positive real numbers and we used the fact that the sequence { ã n } is bounded from above. First, fix m, large enough so that the first two norms on the r.h.s. of equation (7.3) are each less than 3 . Due to Theorem 6.1 and the assumption that w-lim n→∞ a n = 0, we have that lim n→∞ ⟨θ ã n φ m ⟩ = 0. Hence, we can choose N ∈ N such that for n > N , the third norm on the r.h.s. of equation (7.3) is less than 3 . We conclude that lim n→∞ ⟨Θ ã n Φ ⟩ = 0.

The tensor network map is linear
We now demonstrate the linearity of the tensor network map. Consider two sequences of operators in A code , {a n } and {b n }, converging strongly to O 1 and O 2 respectively. Then for α, β ∈ C, s-lim n→∞ (αa n + βb n ) = αO 1 + βO 2 . The image of each a n isã n and the image of each b n isb n . The image of αO 1 + βO 2 under the tensor network map is thus given by αÕ 1 + βÕ 2 . Hence, the tensor network map is linear when acting on all operators in M code .

The tensor network map commutes with hermitian conjugation
If {a n } ∈ A code strongly converges to O, then w-lim n→∞ a † n = O † by Theorem 7.2. Each a n is mapped toã n under the tensor network map, and {ã n } strongly converges toÕ ∈ B(H phys ).
Each a † n is mapped toã † n =ã † n , and w-lim n→∞ã † n = (Õ) † . Since M code is defined from A code by taking strong limits, there must exist a sequence {b n } ∈ A code that converges strongly to O † . Then, s-lim n→∞bn = O † . Note that, for any two Ψ⟩ , Φ⟩ ∈ H code , lim n→∞ ⟨Ψ (a † n − b n ) Φ⟩ = 0. The sequence of norms { a † n − b n } is bounded above because a † n = a n ∀n ∈ N and {a n } and {b n } converge strongly. Furthermore, for any Ψ ⟩ , Φ ⟩ ∈ H phys , lim n→∞ ⟨Ψ (a It follows that

The tensor network map preserves the norm
Consider any ψ ⟩ ∈ pH phys . By Theorem 6.1, there exists a finite family of vectors ψ i ⟩ ∈ pH code , (i ∈ {1, 2, . . . , Q} for some Q ∈ N) such that for any a ∈ A code , Consider a sequence {a n } ∈ A code that strongly converges to O ∈ M code . Then we have lim n→∞ ⟨ψ i a n ψ i ⟩ , (7.10) In particular, for any O ∈ M code , we have that The norms of ψ ⟩ andÕ ψ ⟩ may be expressed as Proof. Let {a n } ∈ A code be a sequence that converges strongly to O ∈ M code . Letã n ∈ M phys be the image under the tensor network map of a n for every n ∈ N. By the definition of the tensor network map, s-lim n→∞ãn =Õ. It follows that Ou Ψ⟩ = lim This theorem demonstrates the bulk reconstruction property of the tensor network map. We can linearly map a given operator O ∈ M code to an operatorÕ ∈ M phys such that for all Ψ⟩ ∈ H code , uO Ψ⟩ =Õu Ψ⟩ , uO † Ψ⟩ = O † u Ψ⟩ =Õ † u Ψ⟩ .

Cyclic and separating vectors
In this section we identify a set of cyclic and separating vectors with respect to M code that is dense in H code . Then, we prove that all cyclic and separating vectors with respect to M code are mapped to cyclic and separating vectors with respect to M phys via the isometry u. This shows that our infinite-dimensional QECC satisfies the assumptions of Theorem 1. The vector ψ ′ ⟩ ⊗ λ⟩ ⋯ ∈ H code is cyclic with respect to M code because operators in A code ⊂ M code may act on it to obtain any vector in pH code , and pH code is dense in H code . Furthermore, ψ ′ ⟩ ⊗ λ⟩ ⋯ is certainly separating with respect to A code as one can see from the definition of A code in equation (5.1). To see that ψ ′ ⟩ ⊗ λ⟩ ⋯ is separating with respect to all of M code , note that the same logic as above implies that ψ ′ ⟩ ⊗ λ⟩ ⋯ is cyclic with respect to M ′ code . Hence, ψ ′ ⟩ ⊗ λ⟩ ⋯ is separating with respect to M code .
Alternative Proof. We now give an alternative and more explicit proof of the fact that ψ ′ ⟩ ⊗ λ⟩ ⋯ is separating with respect to all of M code . Given a sequence {a n } ∈ A code that strongly converges to O ∈ M code we need to show that O( ψ ′ ⟩ ⊗ λ⟩ ⋯) = 0 implies that O annihilates every vector in pH code (which would imply that O annihilates every Cauchy sequence and hence every vector in H code ).
First, we will construct a suitable (yet overcomplete) basis of pH code . Let us assume that ψ ′ ⟩ is a state of the black qutrits in the first M collections. Since ψ ′ ⟩ is a vector in a finite-dimensional factorized Hilbert space with maximal Schmidt number, we may write it as where α k are nonzero coefficients that satisfy ∑ 3 M k=1 α k 2 = 1, e k ⟩ i is an orthonormal basis of the i black qutrits in the first M collections and f k ⟩ j is an orthonormal basis of the j black qutrits in the first M collections.
We consider the following vectors in pH code , which form a basis. Assume that L ≥ M .
where k and k ′ each label a basis vector for their respective black qutrits in the first M collections, and p and q ( ∈ {1, 2, . . . , M }) each run over the three orthonormal basis vectors of their respective black qutrits in the ith collection. All black qutrit pairs past the Lth collection are in the reference state λ⟩.
We first consider the basis vectors that satisfy L = M . The vectors M, k, k ′ ⟩ and M,k,k ′ ⟩ are orthogonal for k ′ ≠k ′ . This is also true for the vectors O M, k, k ′ ⟩ and O M,k,k ′ ⟩ since O is a limit of operators which act as the identity on f k ′ ⟩ in equation (8.2). Since ∑ 3 M k=1 α k M, k, k⟩ = ψ ′ ⟩ ⊗ λ⟩ ⋯, then O( ψ ′ ⟩ ⊗ λ⟩ ⋯) = 0 implies that O M, k, k⟩ = 0 for all k. Let U ∈ A code be an operator that acts as the identity operator on every vector in the tensor product in equation (8.2) except that it may act arbitrarily on f k ′ ⟩. We can choose U to send f k ⟩ to f w ⟩ for w ≠ k. Because U commutes with O, we have that 0 = UO M, k, k⟩ = O M, k, w⟩ and hence O annihilates every basis vector with L = M . This argument can be repeated in a completely analogous way for the case L > M (since ψ ′ ⟩ ⊗ λ⟩ ⋯ = ( ψ ′ ⟩ ⊗ λ⟩) ⊗ λ⟩ ⋯ and ψ ′ ⟩ ⊗ λ⟩ has maximal Schmidt number) to show that O annihilates all vectors in pH code , and hence all of H code .
Recall that a vector is cyclic and separating for M code if and only if it is cyclic and separating for M ′ code [18]. Hence, cyclic and separating vectors for M ′ code are also dense in H code . Theorem 8.2 ([17]). If Ψ⟩ ∈ H code is cyclic and separating with respect to M code , then u Ψ⟩ ∈ H phys is cyclic and separating with respect to M phys .
Proof. To show that u Ψ⟩ is cyclic, we need to show that given any Φ ⟩ ∈ H phys and > 0, we can choose an operator P ∈ M phys such that Pu Ψ⟩ − Φ ⟩ < .
Choose φ ⟩ ∈ pH phys such that φ ⟩ − Φ ⟩ < 2 . Let λ ⋯⟩ ∈ pH phys denote the vector for which all boundary qutrit pairs are in the reference state λ⟩. Choose an operatorP ∈ M phys such thatP λ ⋯⟩ = φ ⟩. Choose O ∈ M code such that O Ψ⟩− λ⋯⟩ < 2 P , where λ⋯⟩ ∈ pH code is the vector for which all qutrit pairs are in the reference state λ⟩. LetÕ denote the image of O under the tensor network map.
We take P =PÕ. This shows that u Ψ⟩ is cyclic with respect to M phys . A completely analogous argument shows that u Ψ⟩ is cyclic with respect to M ′ phys , so it is also separating with respect to M phys .

M code is a hyperfinite type II 1 factor
In this section, we prove that M code satisfies the assumptions of Theorem 2.27, from which it follows that M code is a type II 1 factor. The same argument shows that M ′ code , M phys , and M ′ phys are also type II 1 factors.
For O ∈ M code , define the following linear function from M code → C: For any operator O 1 ∈ M code , it is possible to choose a neighborhood N of O 1 in the ultraweak operator topology such that T (O 2 ) − T (O 1 ) < for all O 2 ∈ N . We may pick the neighborhood to be Hence, T is ultraweakly continuous.
For a, b ∈ A code , it is easy to check that T (ab) = T (ba). Since operators in M code may be written as strong limits of operators in A code , for some > 0 and some choice of sequences Given O 1 ∈ M code , let {a n } ∈ A code be a sequence of operators that converges strongly to O 1 . We need to show that for any choice of and { ξ i ⟩}, { η i ⟩}, there exists an N ∈ N such that n > N ⇒ a n ∈ N . We calculate for some K > 0. We used the fact that the sequence of norms Then, choose N so that for all n > N , Hence for any > 0, it is possible to choose an N ∈ N such that for n > N , T (O 1 )−T (a n ) < .
Then we can conclude that lim n→∞ T (a n ) = T (O 1 ).

The relative Tomita operator
In this section, we study the relative Tomita operator defined on H code . See Section 3 of [17] for a review of Tomita-Takesaki theory. Given Ψ⟩ , Φ⟩ ∈ H code , the relative Tomita operator with respect to M code is denoted by The vector Ψ⟩ must be cyclic and separating with respect to M code , but Φ⟩ can be anything.
In this section, we show that S c Ψ Φ can be bounded or unbounded, depending on the choice of Ψ⟩ and Φ⟩. In Section 10.1, we compute the norm of the relative Tomita operator for a general, finite-dimensional Hilbert space. In Sections 10.2 and 10.3, we provide one example in our setup where S c Ψ Φ is bounded, and one example where S c Ψ Φ is unbounded.

Norm of the Tomita operator in a finite-dimensional Hilbert space
In this section we consider a Hilbert space H = H 1 ⊗ H 2 for finite-dimensional Hilbert spaces H 1 and H 2 with equal dimension D. We want to compute the norm of the relative Tomita operator S Ψ Φ defined with respect to the algebra of operators acting on H 1 . First, we perform Schmidt decompositions of Ψ⟩ and Φ⟩: where e k ⟩ and g k ⟩ (k ∈ {1, 2, . . . , D}) are orthonormal bases of H 1 and f k ⟩ and h k ⟩ are orthonormal bases of H 2 . All of the α k coefficients must be nonzero. The action of S Ψ Φ on any normalized state is given by ,j=1 c ij 2 = 1. The norm of S Ψ Φ is found by maximizing the norm of the right hand side above with respect to the coefficients c ij , subject to the normalization constraint. One finds that In this section, we show that it is possible to choose states for which the relative Tomita operator is bounded. We consider as a special case S c ψ φ for ψ⟩ , φ⟩ ∈ pH code . Suppose that for ψ⟩ (resp. φ⟩), the qutrit pairs in the n ψ th (resp. n φ th) collection and beyond are in the reference state λ⟩. We note that there are many choices of n ψ and n φ , but our argument is independent of the choice we make.
We consider a finite case of n by letting n = max (n Ψ , n Φ ). By considering equation (10.1) for the case that O can be written as a (N ) in equation (5.1) with N = n − 1, we may see how S c ψ φ acts on any vector in pH code for which the qutrit pairs in the nth collection and beyond are in the reference state λ⟩. Let us temporarily restrict our attention to the 9 n−1dimensional Hilbert subspace spanned by these vectors, which may be written as H i ⊗ H j , where H i and H j are the 3 n−1 -dimensional Hilbert spaces containing the states of the qutrits labeled by i and j respectively in n−1 copies of Figure 4.1. Doing the Schmidt decomposition as in equation (10.2) (where we set D = 3 n−1 ), we find that the maximum value of S c ψ φ χ⟩ for a normalized vector χ⟩ ∈ H i ⊗ H j is It is crucial that none of the α k coefficients vanish.
Let us now restrict our attention to the larger subspace of pH code where all qutrit pairs in the (n+1)th collection and beyond are in the reference state λ⟩. We want to do Schmidt decompositions of ψ⟩ and φ⟩ in this 9 n dimensional Hilbert subspace. Let α k ,β k , e k ⟩, g k ⟩, f k ⟩, h k ⟩ for k ∈ {1, 2, . . . , 3 n−1 } be defined as in equation (10.2) for the Schmidt decomposition in the 9 n−1 dimensional subspace considered in the previous paragraph. Next, define where 0⟩ , 1⟩ , 2⟩ are states of the nth black qutrit labeled i. The vectors ĝ p ⟩, f p ⟩, and ĥ p ⟩ are defined analogously. Furthermore, definê We defineβ p analogously. The Schmidt decomposition is then given by If χ⟩ is a normalized vector in the 9 n dimensional subspace, then the maximum value of S c ψ φ χ⟩ is Iterating the procedure of doing the Schmidt decompositions in larger subspaces of the code pre-Hilbert space, we see that for any vector η⟩ ∈ pH code , Choose any Θ⟩ ∈ H code . Let { θ ⟩} ∈ pH code be a sequence that converges to Θ⟩. Define a sequence of operators {a } ∈ A code such that θ ⟩ = a ψ⟩ ∀ ∈ N. Note that a † φ⟩ = S c ψ φ θ ⟩ ∀ ∈ N. For any , m ∈ N, we then have that Hence, lim →∞ a † φ⟩ exists. Thus, S c ψ φ is a bounded operator defined on all of H code .

Example where S c Ψ Φ is unbounded
In this section, we show that for a particular choice of Ψ⟩ , Φ⟩ ∈ H code , S c Ψ Φ is unbounded. Let Ψ⟩ be the vector for which all qutrit pairs are in the reference state λ⟩. Φ⟩ will be constructed as a limit of a sequence of vectors { φ n ⟩} ∈ pH code . Let {δ i } be a sequence of positive real numbers such that ∑ ∞ i=1 δ i is finite. For N ∈ N, let e N a ⟩, a ∈ {1, 2, . . . , 3 N }, denote an orthonormal basis vector of the qutrits labeled i (see Figure 4.1) in the first N collections. In particular, (10.14) Let f N a ⟩, a ∈ {1, 2, . . . , 3 N }, denote an orthonormal basis vector of the qutrits labeled j in the first N collections, defined in the same way as above. Each φ n ⟩ is defined by where c n ab is a 3 n × 3 n matrix to be specified. The ⊗ λ⟩ ⋯ indicates that all black qutrit pairs in the (n + 1)th collection and beyond are in the reference state λ⟩. Choose an arbitrary x ∈ R such that x > 0. Each c n ab is defined by . . .
Assuming n > m, we see that φ n − φ m ≤ ∑ n i=m+1 δ i . Thus, Φ⟩ ∶= lim n→∞ φ n ⟩ exists. To demonstrate that S c Ψ Φ is unbounded, we will construct a sequence of bounded operators {a n } ∈ A code such that lim n→∞ a n Ψ⟩ = 0 while lim n→∞ a † n Φ⟩ does not converge. For n ∈ N, define a n ∶= n √ 3 n ( e n 1 ⟩ ⟨e n 1 i 1 ⋯in ⊗ I j 1 ⋯jn ) ⊗ I⋯, (10.19) where { n } is a sequence of positive real numbers that we will specify later. Note that a n Ψ⟩ = n ( e n 1 ⟩ i 1 ⋯in ⊗ f n 1 ⟩ j 1 ⋯jn ) ⊗ λ⟩ ⋯, (10.20) a n Ψ⟩ = n .
Next, we will consider the sequence {a † n Φ⟩}. Note that, for n ∈ N, One can verify that a † n φ n ⟩ ≤ a † n Φ⟩ . Hence, We may set δ k = 1 k 2 . Then (x + ∑ n k=1 √ 3 k δ k ) grows without bound. We may choose n to go to zero slowly enough so that n (x + ∑ n k=1 √ 3 k δ k ) also grows without bound. Hence, a † n Φ⟩ grows without bound, so S c Ψ Φ is an unbounded operator.

Computing relative entropy for hyperfinite von Neumann algebras
While the definition of relative entropy for infinite-dimensional von Neumann algebras is elegant, it is difficult to use in practice. To compute the relative entropy, one in principle needs to explicitly perform a spectral decomposition of the relative modular operator. However, because our setup involves hyperfinite von Neumann algebras, we can show that there is a more practical method to compute relative entropy. Recall that a hyperfinite von Neumann algebra M may be written as M = (∪ ∞ n=1 M n ) ′′ where each M n denotes a finite-dimensional subalgebra of M and M n ⊂ M n+1 ∀n ∈ N. We will show that given a hyperfinite von Neumann algebra M and two cyclic and separating vectors, the relative entropy of the two vectors may be computed by computing their relative entropy with respect to M n and then taking the limit n → ∞. This result parallels the result of [12], but our explanation is better suited for studying our setup. 10 Computing the relative entropy with respect to M n intuitively amounts to performing a partial trace and using the finite-dimensional relative entropy formula on the reduced density matrices. In the next subsection, we precisely describe how to use the finitedimensional relative entropy formula to compute the relative entropy defined with respect to a finite-dimensional subalgebra of a hyperfinite algebra. In particular, we will write the entropy in a form that is convenient for taking the limit n → ∞. In section 11.2, we review the monotonicity of relative entropy, which we use later. In section 11.3, we fully explain why the limit of finite-dimensional entropies equals the infinite-dimensional entropy.

Defining relative entropy with respect to a finite-dimensional subalgebra
The purpose of this section is to describe the relative entropy defined with respect to a finitedimensional subalgebra of a hyperfinite algebra in a way that will be useful when we consider the limit of larger and larger subalgebras. Let M be a hyperfinite von Neumann algebra on H, and let M n be a finite-dimensional subalgebra of M . Let Ψ⟩ , Φ⟩ ∈ H be cyclic and separating with respect to M . Suppose that we want to compute the relative entropy of Φ⟩ and Ψ⟩ with respect to M n . Note that while Φ⟩ and Ψ⟩ are separating with respect to M n , they need not be cyclic. However, they may still be thought of as cyclic if we restrict our attention to subspaces of H denoted by M n Ψ⟩ and M n Φ⟩. We now explain how to compute the relative entropy of Ψ⟩ and Φ⟩ with respect to M n . First, the relative Tomita operator S n Ψ Φ is defined to map O Ψ⟩ to O † Φ⟩ for all O ∈ M n . The Tomita operator should be viewed as a map between two different Hilbert spaces, M n Ψ⟩ and M n Φ⟩. Since M n is finite-dimensional, S n Ψ Φ is a bounded operator on M n Ψ⟩. The relative modular operator ∆ n Ψ Φ = S n † Ψ Φ S n Ψ Φ is a self-adjoint operator on M n Ψ⟩, and it may be defined to act as the identity operator on the orthogonal complement M n Ψ⟩ ⊥ . Then, the relative entropy is defined as S n = − ⟨Ψ log ∆ n Ψ Φ Ψ⟩ . (11.1) Equation (11.1) will appear again when we consider the limit of larger subalgebras. We now relate S n to the more familiar finite-dimensional relative entropy formula. Because M n is a finite-dimensional von Neumann algebra that acts on the finite-dimensional Hilbert space M n Ψ⟩, we note that M n Ψ⟩ may be written as [7] M n Ψ⟩ = ⊕ α (H Aα ⊗ HĀ α ) , (11.2) while M n may be written as Restricting our attention to M n Ψ⟩, the vector Ψ⟩ is cyclic and separating with respect to M n . This implies that for each α, dim H Aα = dim HĀ α [18].
We now explain how to obtain a density matrix on M n Ψ⟩ from Ψ⟩. Intuitively, one simply needs to perform a partial trace on Ψ⟩ ⟨Ψ , since Ψ⟩ ∈ M n Ψ⟩. However, we follow a different procedure that will also allow us to obtain a density matrix on M n Ψ⟩ from Φ⟩, even though we might have that Φ⟩ ∉ M n Ψ⟩. Let us define a linear map T Ψ ∶ M n → C such that T Ψ (O) = ⟨Ψ O Ψ⟩ ∀O ∈ M n . The map T Ψ is positive. Assuming that Ψ⟩ is normalized, T Ψ (I) = 1. The map T Ψ is also faithful because Ψ⟩ is separating with respect to M n . If we restrict the domain of T Ψ to the set of operators in M n that annihilate H Aα ⊗ HĀ α for all α ≠ 1, then we can naturally define a hermitian, positive operator on H A 1 as follows. Let i⟩ , i ∈ {1, 2, ⋯, dim H A 1 } denote an orthonormal basis of H A 1 . Any operator in B(H A 1 ) may be written as a linear combination of the operators i⟩ ⟨j ∀i, j ∈ {1, 2, ⋯, dim H A 1 }. To treat i⟩ ⟨j as an operator in M n that acts on all of M n Ψ⟩, we define i⟩ ⟨j to act as the identity on HĀ 1 and to annihilate the subspaces H Aα ⊗ HĀ α ∀α ≠ 1. Then, we define the operator ρ Ψ to an operator on H A 1 ⊗ HĀ 1 by defining ρ (1) Ψ to act as the identity on HĀ 1 . In this way, we can define an operator ρ (α) Ψ acting on each H Aα ⊗ HĀ α . Then, we define the density matrix ρ Ψ ∈ M n to be the direct sum of all the ρ (α) Ψ for all values of α. That is, Note that ∑ α Tr Aα ρ (α) Ψ = 1 by construction and that ρ Ψ only depends on Ψ⟩ through the linear map T Ψ . Also, Ψ⟩ must be a purification of ρ Ψ on M n Ψ⟩.
Even though Φ⟩ is not necessarily in M n Ψ⟩, we can still define a density matrix ρ Φ on M n Ψ⟩ with the linear map T Φ , which is defined analogously to T Ψ . Let Φ ⟩ ∈ M n Ψ⟩ be a purification of ρ Φ . We want to ask how Φ⟩ is related to Φ ⟩. Note that ⟨Φ O Φ⟩ = ⟨Φ O Φ ⟩ ∀O ∈ M n . Define the linear map U ′ ∶ M n Φ⟩ → M n Ψ⟩ such that U ′ O Φ⟩ = O Φ ⟩ ∀O ∈ M n . Because M n is finite-dimensional, U ′ is a bounded operator, and U ′ has trivial kernel because Ψ⟩ is separating with respect to M n . Because O Φ⟩ = O Φ ⟩ ∀O ∈ M n , U ′ is an isometry. Because U ′ is invertible, U ′ satisfies U ′ † U ′ = I, and from its definition we can see that U ′ commutes with all operators in M n . Because Φ ⟩ = U ′ Φ⟩, we see that the relative modular operator ∆ n Ψ Φ defined at the beginning of this section equals the relative modular operator ∆ n Ψ Φ . Then, the relative entropy of Ψ⟩ and Φ⟩ computed with respect to M n is given by Since Ψ⟩ and Φ ⟩ are both vectors in the same finite-dimensional Hilbert space M n Ψ⟩, it is straightforward to see [18] that S n , defined in equation (11.1), is given by equation (A.21) of [7] for ρ = ρ Ψ , σ = ρ Φ , M = M n , which is the finite-dimensional relative entropy formula.
The relative entropy defined with respect to M n of the vectors Ψ⟩ and Φ⟩ only depends on Ψ⟩ and Φ⟩ through the linear maps T Ψ and T Φ . As long as we can represent M n on a finite-dimensional Hilbert space with a cyclic and separating vector, we can decompose the Hilbert space as in (11.2) (see [7] for the details) and compute the relative entropies using ρ Ψ and ρ Φ , which are defined from T Ψ and T Φ .
Applying the above discussion to our tensor network model, we let M n ⊂ M code be a finite-dimensional subalgebra of M code that consists of operators that act on the black qutrits labeled i (see Figure 4.1) in the first n collections. Let Ψ⟩ , Φ⟩ ∈ H code be cyclic and separating with respect to M code . To compute the relative entropy with respect to M n of Ψ⟩ and Φ⟩, we consider the action of M n on the Hilbert space associated with the first n qutrit pairs. The relative entropy may be computed from the density matrices ρ Ψ and ρ Φ , which are constructed using the linear maps T Ψ and T Φ . This intuitively amounts to performing a partial trace on Ψ⟩ ⟨Ψ and Φ⟩ ⟨Φ over all of H code except the Hilbert space of the first n qutrits. In this subsection, we have shown that the result is equivalent to equation (11.1). In the remainder of this section we will show that the infinite n limit of equation (11.1) yields the relative entropy of Ψ⟩ and Φ⟩ with respect to M code .

Monotonicity of Relative Entropy
To show that the limit of finite-dimensional relative entropies equals the infinite-dimensional relative entropy, we use the monotonicity of relative entropy, which is nicely explained using a graph argument in [18,32]. However, our proof of the monotonicity of relative entropy is slightly different, as we do not assume that cyclic states remain cyclic after restricting the von Neumann algebra to a subalgebra. In the remainder of section 11, we make use of definitions and theorems given in [17], such as the spectral theorem.
Let S be a closed, densely defined, antilinear operator on H. Define X ∶= KS. Note that X † X = S † S and that X is a closed, densely defined, linear operator on H. The graph Γ X is thus a closed linear subspace of the Hilbert space H ⊕ H. We define Π X to be the projection operator onto Γ X , which satisfies Π 2 X = Π † X = Π X . Since any vector in H ⊕ H can be represented as a column vector ψ⟩ φ⟩ for ψ⟩ , φ⟩ ∈ H, (11.8) we may represent Π X as a two by two matrix: The condition Π X = Π † X implies that p † ij = p ji ∀i, j ∈ {1, 2}, and the condition Π 2 X = Π X implies that ∑ 2 k=1 p ik p kj = p ij ∀i, j ∈ {1, 2}. With these relations, one may show that p i1 (X † X + 1)p 1j = p ij ∀i, j ∈ {1, 2}, (11.11) which implies that p 11 (X † X + 1)(p 11 ψ⟩ + p 12 χ⟩) = (p 11 ψ⟩ + p 12 χ⟩). (11.12) Note that the domain of X is given by D(X) = {p 11 ψ⟩ + p 12 χ⟩ ∶ ψ⟩ , χ⟩ ∈ H}. (11.13) Because D(X) is a dense subset of H, it follows that (11.14) Then, we see that In the following theorem, we study modular operators as opposed to relative modular operators. We will make an explicit connection to monotonicity of relative entropy later. Theorem 11.3. Let M be a von Neumann algebra that acts on a Hilbert space H. Let Ψ⟩ ∈ H be cyclic and separating with respect to M . Let S M Ψ be the Tomita operator defined with respect to M and Ψ⟩. Let N be a von Neumann subalgebra of M (we do not assume that Ψ⟩ is cyclic with respect to N ). On the closed subspace N Ψ⟩ ⊂ H, let S N Ψ be the Tomita operator defined with respect to N and Ψ⟩. On the orthogonal complement N Ψ⟩ where K is given in equation (11.7). Then for all Φ⟩ ∈ N Ψ⟩ and all s > 0, Proof. Let X M = KS M Ψ and X N = KS N Ψ . Let Γ X M ⊂ H ⊕ H and Γ X N ⊂ H ⊕ H be the graphs of X M and X N respectively, with projections Π X M and Π X N . Let Π N Ψ denote the projection onto the closed subspace Π N Ψ (H ⊕ H), which is defined by Note that the closed subspace Γ X N ∩ Π N Ψ (H ⊕ H) is completely determined by the Tomita operator defined with respect to Ψ⟩ and N on the subspace N Ψ⟩. Because N is a subalgebra of M , it follows that The projection onto the closed subspace If we evaluate the expectation value of the above equation in the state Φ⟩ 0 for Φ⟩ ∈ N Ψ⟩, we find that which implies Theorem 11.4. Let ∆ 1 , ∆ 2 be operators on H that are densely defined, closed, self-adjoint and positive. Assume that, for some Φ⟩ ∈ D(∆ 1 ) ∩ D(∆ 2 ) and all s > 0, Also assume that ⟨Φ log ∆ 1 Φ⟩ and ⟨Φ log ∆ 2 Φ⟩ are finite. Then Proof. Let P 1 Ω , P 2 Ω denote the projection-valued measures associated with ∆ 1 , ∆ 2 . We use the spectral theorem 11 to write By Fubini's Theorem ( [21], page 26), we may interchange the order of integration above if the following integral converges: (11.23) We may switch the order of integration above for the same reason as in equation (11.22). Thus, − ⟨Φ log ∆ 1 Φ⟩ ≥ − ⟨Φ log ∆ 2 Φ⟩ . (11.28) In this section, we use the above theorems to show how one could compute the relative entropy of two cyclic and separating vectors of a hyperfinite von Neumann algebra as a limit of finite-dimensional relative entropies. Theorem 11.5. Let M be a von Neumann algebra acting on a Hilbert space H such that M is generated by ∪ ∞ n=1 M n , where {M n } is a sequence of finite-dimensional von Neumann subalgebras of M satisfying M n ⊂ M n+1 ∀n ∈ N. Let Ψ⟩ , Φ⟩ ∈ H both be cyclic and separating with respect to M . Let S n denote the relative entropy of Ψ⟩ and Φ⟩ defined with respect to M n (see equation (11.1) for details). Let S denote the relative entropy defined with respect to M . Then lim n→∞ S n = S.
In particular, if the limit does not converge, then S is infinity.
Proof. We mostly follow the logic of the proof of Lemma 3 of [12]. We consider the tensor product Hilbert space H ⊗ K, where K is a four-dimensional Hilbert space spanned by orthonormal basis vectors e ij ⟩ (i, j ∈ {1, 2}). Let M 2×2 be a four-dimensional von Neumann algebra spanned by the operators u ij (i, j ∈ {1, 2}), which act on the basis vectors of K as u ij e k ⟩ = δ jk e i ⟩. It follows that Note that Φ ⟩ is cyclic and separating with respect toM . Let∆ denote the modular operator defined with respect toM and Φ ⟩. Let the operator∆ n act onM n Φ ⟩ as the modular operator defined with respect to Φ ⟩ andM n , and let∆ n act as the identity onM n Φ ⟩ ⊥ . Note that where ∆ Ψ Φ is the relative modular operator defined with respect to M , Ψ⟩, and Φ⟩. We also have that∆ n Θ⟩ ⊗ e 12 ⟩ = (∆ n Ψ Φ Θ⟩) ⊗ e 12 ⟩ , Θ⟩ ∈ M n Ψ⟩, (log∆ n )( Θ⟩ ⊗ e 12 ⟩) = ((log ∆ n Ψ Φ ) Θ⟩) ⊗ e 12 ⟩ , Θ⟩ ∈ M n Ψ⟩, (11.30) where ∆ n Ψ Φ is the relative modular operator defined with respect to the finite-dimensional algebra M n (see the paragraph before equation (11.1) for details). Thus, (11.31) Thus, we need to show that lim n→∞ ⟨u 12Φ log∆ n u 12Φ ⟩ = ⟨u 12Φ log∆ u 12Φ ⟩ . (11.32) Note that Theorems 11.3 and 11.4 imply relations between the finite-dimensional relative entropies S n . That is, S n ≤ S n+1 ∀n ∈ N because M n ⊂ M n+1 . Also S n ≤ S ∀n ∈ N. Thus, if lim n→∞ S n does not converge, then S must be infinity. For the remainder of the proof, we will thus assume that lim n→∞ S n converges to a quantity that is less than or equal to S.
Given the definitions of∆ and∆ n , it follows that (see [12] and references therein) where g N (λ) is a continuous, bounded function on R, defined for any N ≥ 1, such that (11.34) Let P Ω denote the spectral projections of∆, and let P n Ω denote the spectral projections of ∆ n . By definition, (11.35) Note that (11.36) (11.37) We have used the inequality λ −1 log λ N ≤ (N e) −1 . Note that ⟨u 12Φ ∆ u 12Φ ⟩ = ⟨Ψ ∆ Ψ Φ Ψ⟩ = ⟨Φ Φ⟩ is a finite quantity [18]. Likewise, we have that From equation (11.33), we have that   Using equations (11.37) and (11.38), we can take the large N limit of equation (11.41) to obtain lim n→∞ ⟨u 12Φ log∆ n u 12Φ ⟩ ≤ ⟨u 12Φ log∆ u 12Φ ⟩ , (11.43) which implies that lim n→∞ S n ≥ S, (11.44) which implies that S is finite. Using the monotonicity properties proved earlier, it follows that lim n→∞ S n = S. (11.45)

Conclusion and outlook
It is widely believed that entanglement in a holographic field theory encodes properties of the bulk spacetime. In particular, boundary states with semiclassical bulk duals must be highly entangled so that local operators in the bulk may be reconstructed from different subregions of the boundary [1,6]. The Reeh-Schlieder theorem implies that generic boundary states are highly entangled. The implications of boundary entanglement for bulk reconstruction have been explicitly studied using tensor networks with a finite number of tensors [15,25], which necessarily involve finite-dimensional Hilbert spaces. However, various existing toy models are not well-suited to study the implications for AdS/CFT of the Reeh-Schlieder theorem, which is formulated in the continuum limit of quantum field theory with an infinite-dimensional Hilbert space. Our primary motivation is to construct a model of bulk reconstruction where the Reeh-Schlieder theorem is manifestly true. More precisely, we want to associate von Neumann algebras with boundary subregions so that cyclic and separating states with respect to these algebras are dense in the boundary Hilbert space.
Since tensor networks and quantum error correction have proven to be useful tools in understanding AdS/CFT [7,27,28], it is natural to generalize existing tensor network models to models with an infinite number of tensors. Our strategy for constructing an infinitedimensional QECC is to first construct a QECC that relates a code pre-Hilbert space to a physical pre-Hilbert space. Tensor networks with a repeating pattern provide a natural way to do this. The HaPPY Code [15] is a tensor network constructed from a pentagonal tiling of hyperbolic space with a natural AdS/CFT interpretation. We plan to apply our strategy to the HaPPY code, as the HaPPY tensor network can be naturally extended to an arbitrarily large size. The explicit example described in this paper uses multiple disconnected tensor networks, but it would be more satisfying to use a connected tensor network such as the HaPPY code. If we can generalize the HaPPY code to a QECC with infinite-dimensional Hilbert spaces, we will be able to construct a more accurate toy model of entanglement wedge reconstruction.
An important aspect of AdS/CFT that our toy model captures is that subregions in the boundary theory are associated with von Neumann algebras. In our example, we study type II 1 factors acting on both the bulk and boundary Hilbert spaces. However, the local operator algebras that arise in quantum field theory are generically of type III 1 . [14,18,20]. Thus, it would be satisfying to have a toy model of entanglement wedge reconstruction where the von Neumann algebras are of type III 1 .
Our infinite-dimensional QECC satisfies both statements in Theorem 1.1 [17]. The assumptions and statements in Theorem 1.1 are physically motivated by the Reeh-Schlieder Theorem [26] and previous work on error correction and AdS/CFT [1,7,8,23]. Toy models with infinite-dimensional Hilbert spaces should allow us to better understand the physics of entanglement wedge reconstruction and holographic relative entropy, including the role that the Reeh-Schlieder theorem plays.
In light of the fact that the equivalence between bulk and boundary relative entropies is only approximately correct at large N , approximate entanglement wedge reconstruction has been studied in [24] using finite-dimensional von Neumann algebras and universal recovery channels. It would be interesting to see if an appropriate generalization of the explicit formulas given for finite-dimensional entanglement wedge reconstruction can be checked in an infinite-dimensional toy example. In the future, we want to apply the study of infinitedimensional von Neumann algebras to entanglement wedge reconstruction beyond the planar/semiclassical limit.
While our primary motivation has been to understand the bulk reconstruction in AdS/CFT, we note that infinite tensor networks may also be useful in studying two-dimensional conformal field theories. In the algebraic approach to 2d conformal field theory, every interval I on the circle is assigned a von Neumann algebra A, and if I 1 ⊂ I 2 for two intervals I 1 and I 2 , the associated algebras A 1 and A 2 satisfy A 1 ⊂ A 2 . In the case of 2d chiral conformal field theory studied in [20], each algebra is isomorphic to the unique hyperfinite type III 1 factor. Furthermore, note that there is also a unique hyperfinite type II 1 factor [14]. In our setup, we use an infinite tensor network to characterize the type II 1 factor M code as a particular subalgebra of M phys on H phys . If infinite tensor networks can relate the algebra associated with an interval to a subalgebra associated with a subinterval, they could be used to probe aspects of 2d conformal field theory. It would be interesting to see how infinite tensor networks could be related to quantities such as primary operator dimensions or three-point function coefficients.
Another way to write Y k is where x k and y k are real numbers satisfying x k ≤ 1, y k ≤ 1. One may show that 1 − 2 1 − cos θ k ≤ r k ≤ 1 + (1 − cos θ k ) 2 + 2 1 − cos θ k + sin 2 θ k (A. 12) and that, if θ k < π 2 , φ k ≤ arctan sin θ k 1 − 1 − cos θ k . (A.13) Up until now we did not specify the choice of angles θ k . Now, we make a choice. First, we choose an arbitrary η ∈ R such that 0 < η < 1. We choose θ k such that each r k satisfies e −η k < r k < e η k (A. 14) and such that each φ k satisfies − η k < φ k < η k . We can therefore determine that the real part of ∏ m k=n+1 Y k is arbitrarily close to 1 for n sufficiently large. This means that ψ m ⟩ − ψ n ⟩ is arbitrarily close to 0 for n sufficiently large and m > n. This is enough to show that the sequence { ψ n ⟩} is Cauchy, meaning that lim n→∞ a n M, {p, q}⟩ converges for every basis vector M, {p, q}⟩. Hence, lim n→∞ a n ψ⟩ converges for every ψ⟩ ∈ pH code . Since the sequence of norms { a n } is bounded from above (in particular, a n = 1 ∀n ∈ N), then the sequence of operators {a n } converges strongly.