Tensor chain and constraints in tensor networks
Yi Ling Yuxuan Liu Zhuo-Yu Xian Yikang Xiao








 




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            This paper accompanies with our recent work on quantum error
                correction (QEC) and entanglement spectrum (ES) in tensor networks
                (arXiv:1806.05007). We propose a general framework for planar
                tensor network state with tensor constraints as a model for
                AdS_3/CFT_2 correspondence, which could be viewed as a
                generalization of hyperinvariant tensor networks recently proposed
                by Evenbly. We elaborate our proposal on tensor chains in a tensor
                network by tiling H^2 space and provide a diagrammatical
                description for general multi-tensor constraints in terms of
                tensor chains, which forms a generalized greedy algorithm. The
                behavior of tensor chains under the action of greedy algorithm is
                investigated in detail. In particular, for a given set of tensor
                constraints, a critically protected (CP) tensor chain can be
                figured out and evaluated by its average reduced interior angle.
                We classify tensor networks according to their ability of QEC and
                the flatness of ES. The corresponding geometric description of
                critical protection over the hyperbolic space is also given.
        
        Tensor chain and constraints in tensor networks
            Yikang Xiao
            
        ===============================================
        
        
        
        § INTRODUCTION
        
        Tensor network as a powerful tool for building the ground state of
        a many-body system has been greatly investigated in recent
            years <cit.>. One remarkable feature of tensor
            network states is the intuitive description of quantum
            entanglement among local degrees of freedom. For a subsystem
            composed of some uncontracted edges in a tensor network, its entanglement entropy is vividly bounded by the minimal cuts disconnecting this subsystem and its complementarity. This scenario can be viewed as
            the discretized description of Ryu-Takayanagi (RT) formula in
            holographic approach <cit.>. Inspired by this, people find that a
            holographic space can emerge from entanglement renormalization of
            a many-body system <cit.>. It has
            further been conjectured in <cit.> and <cit.> that the
            classical connectivity of spacetime arises by entangling the
            degrees of freedom in two components. As a bridge between quantum
            entanglement and the structure of spacetime, tensor networks have
            been providing a practical framework for exploring the emergence
            of spacetime in the context of gauge/gravity duality
            <cit.>.
        
        Another property of entanglement enjoyed by holographic duality is
        quantum error correction (QEC) <cit.>. Based on
        sub system duality, operators in the bulk can be reconstructed by
        the operators supported on a sub system of the boundary
        <cit.>.
        In other words, there are subspaces of the Hilbert space in the
        bulk which can still be reconstructed even if an amount of
        information on the boundary is erased
        <cit.>.
        Great progress has also been made in the realization of QEC by
        virtue of tensor networks
        <cit.>.
        In this framework, sub system duality is reflected by the
        isometry between two sub Hilbert spaces associated with sub tensor
        networks.
        
        The above properties of entanglement have been addressed in
        various tensor networks, including the multiscale entanglement
        renormalization ansatz (MERA) <cit.>,
        perfect tensor networks <cit.>,
        random tensor networks
        <cit.>,
         hyperinvariant tensor
        networks <cit.>, as well as spin networks
        <cit.>.
        
        
        Currently it is still a key issue whether tensor networks, or what
        kind of tensor networks could produce all the aspects of
        holography in the context of AdS/CFT correspondence. Taking
        AdS_3/CFT_2 as an example, we pick up some important
        properties that a tensor network is desired to possess.
        
          *  Such a tensor network is a discretization of
            2-dimensional hyperbolic space (H^2 space), which is a time
            slice of an AdS_3 spacetime in global coordinate system.
            Correspondingly, the tensor network is endowed with a symmetry
            described by a discrete subgroup of SL(2,R), which is the
            isometry of H^2 space.
            
          *  Such a tensor network respects RT formula and the entanglement entropy is characterized by a logarithmic
            law. Moreover, the entanglement spectrum (ES) of the ground state
            should be non-flat such that one can reproduce the Cardy-Calabrese
            formula of Renyi entropy for a CFT_2 with large central charge
            c, namely <cit.>
            S_n(A)=1+1/nc/6ln l_A,
             where A is a spatial interval on the boundary and l_A is
            its length with the unit of UV cutoff.
            
          *  Such a tensor network has the function of QEC as AdS spacetime enjoys.
            
          *  Such a tensor network can reproduce the behavior of Green's function in
            AdS_3/CFT_2.
        
        Of course, all these properties may not be independent of one
        another.
        
        One candidate for capturing above holographic features of AdS
            is hyperinvariant tensor networks, recently proposed by Evenbly in
            <cit.>. It is composed of identical polygons by
            uniformly tiling hyperbolic space. The key idea is to impose
            constraints on the product of multiple tensors to form isometric
            mappings. It turns out that this sort of networks may combine the
            advantages of multiscale entanglement renormalization ansatz
            (MERA)
            <cit.>
            which is characterized by non-flat ES and the network composed of
            perfect
            tensors<cit.>
            which is usually endowed with the function of QEC.
        
        But one key issue arises in this approach. That is, what kind
            of multi-tensor constraints could endow such features to a given
            tensor networks? or more quantitatively, is there any criteria to
            justify the ability of QEC and the non-flatness of ES for a given
            tensor networks with multi-tensor constraints? In
            <cit.> we have provided affirmative answers to these
            issues with the proposal for critical protection on tensor chains.
            In this paper we intend to elaborate our proposal and present the
            detailed analysis on tensor chains and constraints in tensor
            networks and prove the statements on the classification of tensor
            networks in <cit.>.
        
        We organize the paper as follows. In next section we will propose
        a generalized framework for the tensor networks with multi-tensor
        constraints in the tiling of H^2 space. To classify different
        types of multi-tensor constraints efficiently and describe the
        behavior of tensor contractions during the evaluation of ES, we
        introduce the notion of tensor chain to describe the contraction
        of tensor products. Moreover, we will introduce a quantity, called
        the average reduced interior angle, to characterize the geometric
        structure of CP chain. Based on this structure we will introduce
        the concept of critical protection in Section III, which should be
        viewed as the core concept in our paper, because it plays an
        essential role in measuring the quality of QEC as well as the
        non-flatness of ES in a quantitative manner. As the first
        consequence, we will immediately see that once the ES becomes
        non-flat under the multi-tensor constraints as proposed in
        <cit.>, then the ability of QEC from bulk to
        boundary has to be weakened. Among this sort of networks, we find
        that most of the perfect tensor networks as the limit case have
        the strongest ability of QEC, while they are always accompanied by
        a flat ES. Therefore, in order to construct tensor networks with a
        non-flat ES as AdS spacetime, one has to pay the price of
        sacrificing the ability of QEC. All above investigation is based
        on a tensor networks embedded into H^2 space which can be
        viewed as a discretization of the hyperbolic geometry.
        Correspondingly, we may also describe QEC and ES over the geometry
        of H^2 directly, which involves the notion of geodesics and the
        curves of constant curvature, etc. We present the description
        based on H^2 geometry in Section IV and the relevant backgrounds
        are given in Appendix <ref>. Keep going on, to
        intuitively understand the role of critical protection in the
        evaluation of QEC and ES, in Section V we present some specific
        examples of tensor networks and demonstrate how the realization of
        QEC could be reflected by the structure of CP tensor chain, and
        how the flatness of ES can be reflected by the region of critical
        protection. Moreover, we develop a generalized description of
        greedy algorithm by imposing multi-tensor constraints on tensor
        chains. After that we classify tensor networks with constraints by
        their properties of QEC and ES. We firstly study the relation
        between CP and QEC in Section VI, presenting a criteria for the
        existence of QEC, and then focus on the relation between CP and ES
        in Section VII, with detailed proofs of the propositions on
        various bounds for the flatness of ES. Section VIII is the
        conclusion and outlook.
        
        
        
        
        
        § TENSOR CHAINS IN A TENSOR NETWORK
        
        
        In this section we will present a general framework for tensor
        networks based on the tiling of hyperbolic space. We define a
        notion of tensor chain whose skeleton forms a polyline in a
        network. Associated with each tensor chain, the reduced interior
        angle can be defined, which in some sense could be viewed as the
        discrete description of the curvature of the ployline.
        
        
        
         §.§ Tiling of H^2 space
        
        In the global coordinate system of AdS_3 spacetime, the
        isochronous surface is a H^2 space 
        ds^2=L^2(dρ^2+cosh^2ρ dτ^2),  where L is the radius
        of H^2 geometry, the unique dimensional quantity introduced
        in this paper. So we are free to set L=1.
        
        Firstly, we intend to discretize H^2 space in a uniform
        version, which can be realized by the tiling of H^2 space with
        identical polygons. Consider many identical polygons composed of
        b edges in a 2 dimensional surface, then put them together by
        gluing their edges such that a edges share the same node. We
        call such discretization as the {b,a} tiling of H^2 space.
        In a space with
        negative curvature, because the sum of interior angles of a triangle is less than 2π, one can realize a {b,a} tiling of
        H^2 space only if
        1/a+1/b<1/2.
        
        Obviously, a≥3,b≥3.
        
        When a tiling of H^2 is specified by {b,a}, the geometry is
        determined (up to the radius L). We call the polygon with b
        edges as the elementary polygon, while the union of several
        elementary polygons as a composite polygon. The length of each
        edge of the elementary polygon is  P=2
        arccosh(cos(π/b)/sin(π/a)). 
        
         §.§ Tensor networks with {b,a} tiling
        
        Now we construct a tensor network based on a {b,a} tiling.
        Associated with each node, we assign a rank-a tensor T, each
        index of which is specified to an edge jointed at the node
        respectively. The elements of the tensor T are denoted as
        T^i_1i_2⋯ i_a, where all indexes have the same dimension
        d and d>1. Associated with each edge, we also assign a rank-2
        tensor E, whose elements are E_i_1i_2. We call the above
        indexes associated with tensors T and E as basic indexes,
        which are labelled by lowercase letters. As examples, two tensor
        networks with 7,3 tiling and 4,5 tiling are
        illustrated in Fig.<ref>, respectively.
        
        Because of the rotational invariance of H^2 space, we further
        demand that the indexes of tensor T and E have cyclic
        symmetry [Notice that the perfect tensor originally
            defined in <cit.> has no rotation symmetry as
            required. Here we further require it and will show the existence
            of such states in Appendix <ref>.] T^i_1i_2⋯ i_a=T^i_2i_3⋯ i_ai_1,  
        E_i_1i_2=E_i_2i_1.  In this paper, we adopt the convention
        that the index of tensor T can be lowered by contracting it
        with a tensor E T_j_1^i_2⋯ i_a≡∑_i_1E_j_1i_1T^i_1i_2⋯ i_a.  Correspondingly, the
        edge connecting two nodes represents the index contraction of two
        tensors T by a tensor E, namely, ∑_i_1j_1T^i_1i_2⋯ i_aE_i_1j_1T^j_1j_2⋯
        j_a=∑_i_1T^i_1i_2⋯ i_aT_i_1^j_2⋯
            j_a.
         Therefore, given a tensor network with b,a tiling, we
        can define a quantum state Ψ consisting of two sorts of
        tensors T and E by tensor products and contractions. For later
        convenience, we require that for a tensor network state Ψ,
        all the indexes of tensors T (with full upper indexes) should be
        contracted and those uncontracted indexes should only belong to
        tensors E, as shown in Fig.<ref>.
        
        A tensor network Ψ can define a state |Ψ⟩ in the
        Hilbert space on those uncontracted edges. In this paper, we will
        investigate the algorithms of QEC and the entanglement of Ψ
        by manipulating tensor networks.
        
        
        
         §.§ Tensor chain
        
        Given a tensor network by the {b,a} tiling, we intend to
        introduce a notion of tensor chain to depict the product structure
        of multi-tensors with index contractions, which will be convenient
        for us to impose a tensor constraint and quantitatively describe
        its geometric properties. Firstly, in order to define a tensor
        chain in an efficient way, we adopt a compact form to denote a
        single tensor T of rank-a which is subject to rotational
        symmetry. We divide all its indexes into four groups  in
        order and label each group with an abstract index, which is
        called collected index and labelled by a capital letter. Then
        the component of T can be generally written as T^ABCD. For
        instance, for a tensor T^i_1i_2⋯ i_5 with 5 indexes, we
        may collect them as i_1i_2=A, i_3=B, i_4=C, i_5=D or
        i_1i_2=A, i_3i_4=B, i_5=C, ∅=D, while the
        collection i_1i_2=A, i_3=C, i_4=B, i_5=D is prohibited.
        Furthermore, those collected indexes may also be lowered by tensor
        E, such as  T_A'^BCD≡
        T_i_1'i_2'^i_3i_4i_5≡∑_i_1i_2E_i_1'i_1E_i_2'i_2T^i_1i_2⋯ i_5≡∑_AE_A'AT^ABCD,  where
        i_1'i_2'=A', i_1i_2=A, i_3=B, i_4=C, i_5=D and
        E_A'A≡ E_i_1'i_1E_i_2'i_2. We further define #(A)
        as the number of basic indexes in a collected index A. For
        example, in the above collection, #(A)=2 and #(B)=1.
        
        Now we can construct a tensor chain M by contracting k
        tensors T with tensors E
        M^A_B=M^A_1A_2⋯ A_k_B_1B_2⋯ B_k
        =∑_C_1C_2⋯ C_k,C_1=C_k+1∏_i=1^k T^A_iC_i_B_iC_i+1 where A=A_1A_2⋯ A_k and B=B_1B_2⋯ B_k are
        uncontracted indexes, while C_1,C_2,⋯ C_k are specified as
        the indexes which are contracted in the chain. Moreover, since
        each tensor T occupies a node in the tiling, we call k the
        number of nodes in tensor chain M as well. The sum index i in
        (<ref>) runs from 1 to k, counting the number of nodes in M. Alternatively, a tensor chain M can be viewed as a mapping from the Hilbert space on uncontracted indexes A to the Hilbert space on uncontracted indexes B.
        
        Obviously, in a {b,a} tiling any two nodes are only possibly connected by single edge, otherwise they are not connected directly. Thus, here we only consider tensor chain with #(C_i)=1 for i=2,3,⋯,k.
        Furthermore, if #(C_1)=#(C_k+1)=1, then we call M as a
        closed tensor chain; if #(C_1)=#(C_k)=0, we call M as an
        open tensor chain. Two typical samples of tensor chain are
        illustrated in Fig.<ref>.
        
        Since a diagram of tensor chain can be specified by the number of
        uncontracted edges in the tensor product, we propose a notation
        m_1   m_2   ⋯   m_k
         n_1   n_2   ⋯   n_k to denote an
        open tensor chain M with elements M^A_1A_2⋯
            A_k_B_1B_2⋯ B_k, where m_i≡#(A_i) and
        n_i≡#(B_i). Obviously, we have
        
        m_i,n_i≥0,    m_i+n_i=a-2+δ_i1+δ_i,k,   i=1,2,⋯,k.
         Similarly, we use m_1   m_2   ⋯   m_k
        
        n_1   n_2   ⋯   n_k to denote a closed tensor chain with
         m_i,n_i≥0,    m_i+n_i=a-2,  
        i=1,2,⋯,k.
        
        Since one can reconstruct m_i from n_i
        or vice versa according to (<ref>)(<ref>),
        some time for convenience we abbreviate
        either m_i or n_i to *. For instance, *   *   ⋯   *
        
        n_1   n_2   ⋯   n_k ≡ m_1   m_2   ⋯   m_k
           *   ⋯   * ≡ m_1   m_2   ⋯   m_k
        
        n_1   n_2   ⋯   n_k with (<ref>) and *   *   ⋯   *
         n_1   n_2   ⋯   n_k ≡
        m_1   m_2   ⋯   m_k
           *   ⋯   * ≡
        m_1   m_2   ⋯   m_k
         n_1   n_2   ⋯   n_k with
        (<ref>).
        
        We provide the following definition to describe some
        relations among tensor chains.
        
            Given a tensor network with {b,a} tiling and an open tensor chain M= m_1   m_2   ⋯   m_k
           *   ⋯   *, we define its sub tensor chain M_p,q as m_p   m_p+1   ⋯   m_q
           *   ⋯   * for 1≤ p≤ q≤ k. We say M'⊑ M if M' is a sub tensor chain of M. Furthermore, we say an open tensor chain m_1'   m_2'   ⋯   m_l'
           *   ⋯   *≼  M where 1≤ l≤ k, if ∃ p∈1,2,⋯,k-l+1 s.t.m_i' ≤ m_p+i-1 for i∈1,l and m_i'= m_p+i-1 for 1<i<l.
            
            Given a closed tensor chain M= m_1   m_2   ⋯   m_k
           *   ⋯   *, we define its sub tensor chain M_p,q for 1≤ p≤ k, 1≤ q≤ k as m_1”   m_2”   ⋯   m_l”
           *   ⋯   * where m_i”=m_p+i-1  mod  k for 1≤ i≤ l and 1≤ l=q-p+1<k. We say M'⊑ M if M' is a sub tensor chain of M. Furthermore, we say an open tensor chain  m_1'   m_2'   ⋯   m_l'
           *   ⋯   *  ≼ M where 1≤ l<k if ∃ p s.t.m_i'≤ m_p+i-1  mod  k for i=1,l and m_i'=m_p+i-1  mod  k for 1<i<l.
        
        For example, in 4,5 tiling, 2   1
        2   3 ⊑ 2   1   2
        2   2   2 and 2   1   1
        2   2   3 ≼ 2   1   2
        2   2   2.
        
        We always require that the subscripts p,q in M_p,q belong to
        integers and satisfy 1≤ p≤ q≤ k, where k is the
        number of nodes in M.
        
        
        
         §.§ The reduced interior angle
        
        
        When a tensor chain is embedded into a tensor network in H^2
        space, its skeleton can be marked by a directed polyline
        concisely, as shown in Fig.<ref>. Along the direction
        of the polyline, we require that the sequence number of nodes
        increases and the edges on the left (right) hand side of the
        polyline are always associated with the upper (lower) indexes
        of the tensor chain. For a closed tensor chain, conventionally
        the direction of the closed polyline is specified to be
        anticlockwise, so the inward or left-handed (outward or
        right-handed) edges of the polyline are associated with the upper
        (lower) indexes of the tensor chain. We remark that a closed
        polyline with clockwise direction can be analyzed in parallel,
        with the requirement that its inward (outward) edges are
        associated with the lower (upper) indexes.
        
        The curvature of a polyline at the ith node can be captured by
        its interior angle θ_i, which is defined as the angle on
        the left hand side of the polyline and is a multiple of 2π/a.
        We further define the reduced interior angle as s_i=θ_i/2π/a, which is an integer. Obviously,
        the reduced interior angle is related to the number of upper
        edges at each node and we intend to give the following definition.
        
            For a closed tensor chain M=
            m_1   m_2   ⋯   m_k
           *   ⋯   *, the reduced interior
            angle of the i-th tensor is s_i=m_i+1. For an open tensor chain
            M= m_1   m_2   ⋯   m_k
           *   ⋯   *, the reduced
            interior angles are s_i=m_i+1-δ_i1.
        
        For later convenience, we further introduce several quantities
        based on reduced interior angles to evaluate the curvature of
        a tensor chain M. Specially, we define the prime tensor chain,
        which is the core notion for the construction of the algebra of
        tensor constraints in next subsection.
        
            Given a tensor chain M with k nodes, the average reduced interior angle κ(M) is defined as
            κ(M)=1/k∑_i=1^k s_i.
            
            The sub reduced interior angle κ_p,q(M) from the
            p-th tensor to the q-th tensor is defined as
            κ_p,q(M)=κ(M_p,q),    1≤ p≤ q≤ k,
            
            and the maximal reduced interior angle κ_p,q(M)
            is defined as
            κ_max(M)=max_p,qκ_p,q(M).
            
            Given an open tensor chain M with finite k nodes, when
            that κ_p,q(M)=κ_max(M) if and only if p=1 and
            q=k, then we say tensor chain M is prime.
        
        
        Based on above definitions, we have following theorems for
        tensor chains.
        
        
            Given two tensor chains M and M', if M≼ M', then
            κ_max(M)≤κ_max(M').
        
            Given a tensor chains M and a prime tensor chain M', if M≼ M' but M≠ M', then
            κ_max(M)<κ(M').
         The uniqueness theorem of prime tensor chain. Given a
            {b,a} tiling and a rational number  κ∈1,a/2,
             the prime tensor chain M with κ(M)=κ exists and
            is unique. Furthermore, if κ=u/v where u and v are
            coprime integers, then M=
            m_1   m_2   ⋯   m_v
        n_1   n_2   ⋯   n_v with
            
            m_i=[iκ]-[(i-1)κ]-1+δ_i1,
            
            where [x] is the floor function. So, M has reversal
            symmetry, namely 
            m_i=m_v-i+1.
             The reduced interior angles of M are given by
            
            s_i=[iκ]-[(i-1)κ].
            
            Set a prime tensor chain M= m_1   m_2   ⋯   m_k
        n_1   n_2   ⋯   n_k satisfying κ(M)=κ.
            Then, for l=1,2,⋯,k-1,
            1/k∑_i=1^k m_i+k-1   =   κ,  
        1/l∑_i=1^l m_i+l-1   <   κ,  
        1/k-l∑_i=1^k-l m_i+k-l-1   <   κ.  
            On one hand, from (<ref>)(<ref>)(<ref>),
            one has
            ∑_i=1^l m_i-1<l(κ-1)<∑_i=1^l m_i.
            
            Then
            
            l(κ-1)∉ℤ⇒ l/v∉ℤ,  
            l=1,2,⋯,k-1.
            
            Thus
            
            v≥ k.
            
            On the other hand, from (<ref>), one has
            
            kκ∈ℤ⇒ k/v∈ℤ.
            
            Combining above two statements, we have
            
            k=v.
            
            From (<ref>)(<ref>), one has
            ∑_i=1^l m_i=[l(κ-1)]+1,     l=1,2,⋯,v.
            
            So
            
            m_l=∑_i=1^l m_i-∑_i=1^l-1 m_i=[lκ]-[(l-1)κ]-1,   
            l=2,3,⋯,v.
             Specially, when l=1, 
            m_1=[1*κ]-[0*κ]-1+1.
             In summary, we have proved (<ref>). Moreover,
            using identities
            
                [-x]   =-[x]-1+δ_x,[x],    
        
                [x+n]   =[x]+n,   n∈ℤ,
            
            we can derive (<ref>). Then according to
            Definition <ref>, we have (<ref>). At
            last, from (<ref>) we find 
            m_1=[κ]  or  [κ]-1.
            
            It leads to the conclusion that conditions in
            (<ref>) can always be satisfied if we require
            (<ref>). So the prime tensor chain exists and is
            unique.
        
            For a tensor chain M, ∃ unique prime tensor chain M' satisfying M'⊑ M and κ(M')=κ_max(M).
        ∃ p,q s.t.κ_p,q(M)=κ_max(M) and κ_p',q'(M)<κ_max(M), ∀ p',q' satisfying p< p'≤ q'≤ q or p≤ p'≤ q'< q. Then M'=M_p,q is prime, whose κ(M')=κ_p,q(M)=κ_max(M). Because of Theorem <ref>, M' is unique.
        
        
        § TENSOR CONSTRAINTS AND CRITICALLY PROTECTED (CP) TENSOR CHAINS
        
        
        In this section we propose a notion of critical protection to
        describe the behavior of tensor networks under the
        contractions of tensor product which are subject to tensor
        constraints.
        
        
        
         §.§ Tensor constraint
        
        The notion of tensor chain provides us a convenient way to
        describe a general constraint on the product of tensors T and
        E, which plays an essential role in pushing operators through
        nodes or edges in network in the context of QEC. Usually we impose
        the constraint requiring that some contraction of tensors
        should be proportional to an isometry. Of course the
        contraction of tensors may or may not form a tensor chain. Here
        for simplicity, we only consider imposing tensor constraints on
        tensor chains which can be concisely written as ∑_B  M^A_B (M_B^A')^*   ∝δ^AA',
         where M= m_1   m_2   ⋯   m_k
        n_1   n_2   ⋯   n_k is
        an open tensor chain with n_i to be the number of edges that are
        contracted with its conjugate tensor at the ith node, as
        illustrated in Fig.<ref>. Notice that the contraction on
        two lower indexes B in (<ref>) implies that it involves in
        #(B) contractions of ∑_j E_ij(E_jk)^*. For
        convenience, in the remainder of this paper, when we say a tensor
        constraint M, we actually refer to the constraint in terms of
        tensor chain M which is subject to (<ref>). Obviously, a
        non-trivial constraint M requires ∑_i=1^k m_i≥1.
        Moreover, an isometry can be realized only if the number of
        degrees of freedom in A is less than or equal to that in B,
        namely ∑_i=1^k m_i=#(A)≤#(B)=∑_i=1^k n_i.
        
        Thus, any non-trivial tensor constraint M should satisfy
        κ(M)∈1,a/2.
        
        
        One may immediately find that for a set of tensor constraints some
        of them may not be logically independent. In general, there are
        four fundamental operations to derive new constraints from given
        tensor constraints, which can be listed as follows.
        
          Reversal If m_1   m_2   ⋯   m_k
        n_1   n_2   ⋯   n_k is a tensor constraint, then m_k   m_k-1   ⋯   m_1
        n_k   n_k-1   ⋯   n_1 is a tensor constraint as well.
          
          Contraction If m_1   m_2   ⋯   m_k
        n_1   n_2   ⋯   n_k is a tensor constraint, then m_1-1   m_2   ⋯   m_k
        n_1+1   n_2   ⋯   n_k is a tensor constraint as well.
          
          Reduction If m_1   m_2   ⋯   m_k
        n_1   n_2   ⋯   n_k and *   *   ⋯   *
        n_p   n_p+1   ⋯   n_q are tensor constraints, where 2≤ p≤ q≤ k, then m_1   m_2   ⋯   m_p-1
           *   ⋯   * is a tensor constraint as well.
          
          Combination If *   *   ⋯   *
        n_1   n_2   ⋯   n_k and m_1'   m_2'   ⋯   m_l'
           *   ⋯   * are tensor constraints, then 
        *   *   ⋯   *   m_1'   m_2'   ⋯   m_l'
        n_1   n_2   ⋯   n_k   *   *   ⋯   * is a tensor constraint as well.
        
        As an example, we demonstrate the derivation of new constraints
        by contraction and reduction in Fig.<ref>.
        
        
        
        We remark that the strength of a tensor constraint M can be
        quantified by its maximal reduced interior angle
        κ_max(M). By comparing κ_max of new derived
        constraints with those of original constraints, we have the
        following theorem.
        
        
        Any tensor constraint M” derived from M and M' through
        above ways satisfies
            κ_max(M”)≤maxκ_maxM,κ_maxM'.
            
        where M= m_1   m_2   ⋯   m_k
        n_1   n_2   ⋯   n_k ,M'= m_1'   m_2'   ⋯   m_l'
        n_1'   n_2'   ⋯   n_l'.
        
          For reversal, κ_max(M”)=κ_max(M), then we have (<ref>).    
        
          For contraction and reduction, M”≼ M. Thanks to Theorem <ref>, we have (<ref>). 
        
          For combination, ∃ p,q s.t.κ_p,q(M”)=κ_max(M”). In the case of q≤ k or k<p, similarly, we have (<ref>).
          In the case of p≤ k<q, we observe that
          
            1+∑_i=p^k m_i-1/k-p+1   ≤   κ_max(M),  
        
            1+∑_i=1^k-q m_i'-1/q-k   ≤   κ_max(M'),
          
        which leads to
          ∑_i=p^k m_i+∑_i=1^q-k m_i'≤ 2+(κ_max(M)-1)(k-p+1)+(κ_max(M')-1)(q-k).
          
          So,
          κ_p,q(M”)
               =1+∑_i=p^k m_i+∑_i=1^q-k m_i'-2/q-p+1
           ≤k-p+1/q-p+1κ_max(M)+q-k/q-p+1κ_max(M')    
           ≤maxκ_maxM,κ_maxM'.
          
        In summary, we have (<ref>).
        
        
        Next we intend to study a set of tensor constraints with the form
         S={M_s, M_1,M_2,⋯}, 
        where M_s= 1
        a-1 is called step tensor chain and other
        tensor constraints are not specified. Step tensor chain has the
        minimal average reduce interior angle κ(M_s)=1. Generally,
        those tensor constraints in S may not be mutually independent.
        With the help of step tensor chain, we have the following theorem
        for the relation of tensor constraints.
        
            Any tensor constraint M satisfying M≼ M' can be derived
            from S={M_s, M'}.
        
        
        For any general set of tensor constraints, we can find a unique
        set of tensor constraints S_c which is logically equivalent to
        S and only contains two elements 
            S_c={M_s, M_t}.
        S_c is called central set. M_t is called top tensor chain,
        which should be prime and satisfy the condition
        (<ref>). The equivalence between S and S_c require
        κ(M_t)=max_M∈ Sκ_max(M), as proved in
        Theorem (<ref>). Without loss of generality, we will
        only consider central set S_c hereinafter.
        
        Thanks to Theorem <ref>, given a b,a tiling,
        we have a one-to-one mapping between all the possible top tensor
        chains M_t and the rational numbers in 1,a/2. Thus,
        we have classified all the general sets of tensor constraints with
        the form in (<ref>) by the rational number
        κ(M_t)∈ [1,a/2]. Given a κ(M_t), with the use
        of (<ref>), we can directly construct top tensor chain
        M_t as well as those tensor chains M satisfying M≼
        M_t.
        
        
        
            Given a central set S_c={M_s,
            M_t}, we define a set S_D=M|κ_max(M)≤κ(M_t).
         A tensor constraint M can be derived from S_c if and only if M∈ S_D.
        
            Let M_t= m_1   m_2   ⋯   m_k
        n_1   n_2   ⋯   n_k ,
            M= m_1'   m_2'   ⋯   m_k''
        n_1'   n_2'   ⋯   n_k''.
            We denote the proposition that M can be derived from S_c as P1.
        
            If  P1 is true, thanks to Theorem <ref> and Theorem
            <ref>, we have κ_max(M)≤κ(M_t), thus M∈ S_D.
        
        Next we will apply the method of induction on k' to prove that
        if κ_max(M)≤κ(M_t) then  P1 is true.
        
        Firstly, for the simplest case with k'=1, κ_max(M)=m_1'
        is an integer. Because of Theorem <ref>,
        m_1=[κ(M_t)]. Then κ_max(M)≤κ(M_t)⇒ m_1'≤ m_1⇒ M≼ M_t.
        Because of Theorem <ref>, P1 is true.
        
        Now assume that when 1≤ k'<lP1 is true, we are going to
        prove that for k'=l, P1 is also true.
        
        At current stage, the length of tensor chain M, namely l,
        could be either longer or shorter than the length of M_t, namely
        k. In either case, we will compare the number of upper basic
        indexes at each node within the parts with the same length,
        min(k,l). To describe the difference of this part in two tensor
        chains, it is convenient to define the proposition that ∃
        j∈1,2,⋯,min(k,l) s.t.m_j'≠ m_j as
         P2. Then for k'=l, We split the situation into the
        following three cases.
            
          * P2 is false and l≤ k. It means that the tensor chain
        M is shorter or has equal length, and has the same number of
        upper basic indexes as M_t at each node. Then obviously one has
        M⊑ M_t, so  P1 is true. 
          * P2 is false and
        l>k. It means the tensor chain M has the same number of upper
        basic indexes as M_t at first k nodes but it is longer. One
        can pick out the extra part of M
        by setting M'= m_k+1'+1   m_k+2'   ⋯   m_l'
           *   ⋯   *. For p>1, one can easily derive that
        κ_p,q(M')=κ_p+k,q+k(M)≤κ_max(M)≤κ(M_t). While for p=1, one has
        κ_1,q(M')=1/q(k+q)κ_1,k+q(M)-kκ(M_t)≤1/q(k+q)κ(M_t)-kκ(M_t)=κ(M_t). So
        κ_max(M')≤κ(M_t). Because of the assumption of
        induction and l-k<l, M' can be derived from S_c. Now since
        M can be derived from M' and M_t by reversal and
        combination,  P1 must be true. 
          * P2 is true. It means the
        number of upper basic indexes at some nodes are different in two
        tensor chains. Set the minimal j satisfying m_j'≠ m_j as
        r. If m_r'>m_r, then m_r'≥ m_r+1. So κ_1,r(M)
        ≥1/rm_1+⋯+m_r+1+r-1 =1/r∑_i=1^r([iκ(M_t)]-[(i-1)κ(M_t)]-1+δ_i1)+r
        =1/r[rκ(M_t)]+1 >κ(M_t), where
        (<ref>) is used. It is contradictory to the condition
        κ_max(M)≤κ(M_t). So only m_r'<m_r is possible.
        Now we set M'= m_1'   m_2'   ⋯   m_r-1'   m_r'+1
           *   ⋯   *   * and M”= m_r+1'   m_r+2'   ⋯   m_l'
           *   ⋯   * (If r=1, set M'= m_1'+1), then
        M'≼ M_t and M”≼ M. So M' can be derived from
        S_c and κ_max(M”)≤κ(M). Because of the
        assumption of induction and l-r<l, M” can be derived from
        S_c as well. Since M can be derived from M' and M” by
        combination, P1 is true.
            
            In conclusion, P1 is true when κ_max(M)≤κ(M_t).
        
        Any general set of tensor constraints S=M_s,M_1,M_2,⋯
        is logically equivalent to a unique central set of tensor
        constraints S_c=M_s,M_t, where M_t is specified by
        κ(M_t)=max_M∈ Sκ_max(M).
        ∃ M∈ S s.t.κ_max(M)=max_M'∈ Sκ_max(M'). Because of Theorem <ref>, ∃ unique prime tensor chain M_t satisfying M_t⊑ M and κ(M_t)=κ_max(M). Because of Theorem <ref>, S⇒M_s,M⇒ S_c and κ(M_t)=max_M∈ Sκ_max(M).
        
            On the other hand, because of Theorem <ref>, S_c⇒M_s,M, ∀ M∈ S. Thus, S_c⇒ S.
        
            Thus, ∃ unique prime M_t s.t.S ⇔ S_c and κ(M_t)=max_M∈ Sκ_max(M).
        
        
        
        One may ask whether there always exist tensor T and tensor
        E satisfying the tensor constraints S_c. We do not have a
        general proof about the existence here. Nevertheless, for some
        specific tensor constraints, we can actually solve them by
        constructing specific tensors indeed. Some examples are
        demonstrated in Appendix <ref> and see more in <cit.>.
        
        
        
         §.§ Protection
        
        
        In this subsection we describe the behavior of tensor chains
        under the action of tensor constraints. For this purpose we first
        give the following definitions.
        
        The transpose of an open tensor chain M= m_1   m_2   ⋯   m_k
        
        n_1   n_2   ⋯   n_k is M^T=
        n_1   n_2   ⋯   n_k
        m_1   m_2   ⋯   m_k. The transpose of a
        closed tensor chain M= m_1   m_2   ⋯   m_k
        
        n_1   n_2   ⋯   n_k is M^T=
        n_1   n_2   ⋯   n_k
        m_1   m_2   ⋯   m_k.
        
        Given a set of tensor constraints S_c={M_s, M_t},
        say a tensor chain M is unprotected, if ∃ M'∈ S_D such
        that M'^T⊑ M^T. Otherwise, we say that M is
        protected.
        
        The notion of protection can be intuitively understood as
        following. If we find such M' in S_D satisfying
        M'^T⊑ M^T, then the contraction ∑_B_iB_i+1⋯ B_j M^A_1A_2⋯ A_k_B_1B_2⋯ B_k(M'_B_iB_i+1⋯ B_j^D_iD_i+1⋯ D_j)^*,   1≤ i≤ j≤ k
         can be simplified under the constraint
        S_c={M_s,M_t}. Diagrammatically, the tensor chain
        M becomes disconnected under the contraction with the tensor
        chain M'^*, as illustrated in Fig.<ref>. In other
        words, when we say a tensor chain is protected, it means that one
        can not factorize it by contracting its lower indexes with any
        M∈ S_D derived from S_c. Actually, the condition in
        Definition <ref>, namely “∃ M'∈
        S_D”, can be simplified as “∃ M'≼ M_t”.
        
        
        
         §.§ CP tensor chains
        
        In this subsection we point out that given a tiling and S_c,
        there exists a tensor chain which is critically
        protected. We notice that whether a tensor chain M is protected
        or not can  be reflected by the value of interior angles, which
        roughly speaking measures the curvature of the skeleton of the
        tensor chain. Specifically, the larger is κ_max(M^T),
        the easier it is for M to become unprotected. Therefore, there
        is a critical value for κ at which tensor chain is
        critically protected.
        
        Given an open tensor chain M= m_1   m_2   ⋯   m_k-1   m_k
        
        n_1   n_2   ⋯   n_k-1   n_k, we can define a periodic tensor chain by joining infinitely many Ms as
        
        M_period=[     ⋯ m_1-1   m_2     ⋯ m_k-1   m_k m_1-1   m_2     ⋯ m_k-1   m_k     ⋯;     ⋯   n_1   n_2     ⋯ n_k-1 n_k-1   n_1   n_2     ⋯ n_k-1 n_k-1     ⋯ ]
        with a loop body m_1-1   m_2   m_3   ⋯   m_k
        
        n_1   n_2   n_3   ⋯   n_k-1[Here we have exceptionally
            used the notation of closed tensor chain to denote a periodic tensor chain and a loop body, because both of them satisfy (<ref>).].
        k is called the period of M_period.
        
        Obviously, κ(M_period)=κ(M).
        
        Given a tiling and the set S_c, we define the critically
        protected (CP) tensor chain M_c as the periodic tensor chain
        generated by M_t. We further define the CP reduced interior
        angle as κ_c=κ(M_c)=κ(M_t).
        
        
        We demonstrate the construction of M_c with an example in
        Fig.<ref>.
        
        The exact meaning of critical protection is characterized by the following theorem.
        
        With a given {b,a} tiling and a given S_c, an open tensor chain M= *   *   ⋯   *
        n_1   n_2   ⋯   n_k is unprotected if and only if ∃ p,q satisfying 1≤ p≤ q≤ k such that
        ∑_i=p^q (a-1-n_i)≤(q-p+1)κ_c-1.
        
        A closed tensor chain M= *   *   ⋯   *
        n_1   n_2   ⋯   n_k
        is unprotected if and only if ∃ p,h satisfying h≤ k such that
        ∑_i=1^h (a-1-n_(p+i-1) k)≤ hκ_c-1.
        
        We will present the proof for the case of open tensor chain in
        detail and claim that it can be applied to closed tensor chain in
        parallel. The main difference will be mentioned in the end of
        proof.
        
        We first prove the proposition: if ∃ p,q satisfying 1≤ p≤ q≤ k such that (<ref>) is true, then M is unprotected.
        Without lose of generality, we start with the assumption that ∄p',q' with p< p'≤ q'≤ q or p≤ p'≤ q'< q satisfying
        ∑_i=p'^q' (a-1-n_i)≤(q'-p'+1)κ_c-1. Otherwise, we
        just replace p,q by p',q'.
        
        Let h=q-p+1 and x_i=a-1-n_i, ∀ i.
        There are two cases:
        
        1. If h=1, then a-n_p≤κ_c⇒ a-n_p≤ [κ_c]. Because of
        (<ref>), a-n_p
        n_p ≼ [κ_c] 
         a-[κ_c]≼
        M_t. We know a-n_p
        n_p ^T ⊑ M^T, then M is unprotected.
        
        2. If h≥ 2, then ∀ l∈{1,2,⋯,h-1}, we have
        ∑_i=p^p+l-1 x_i > l κ_c - 1,   
        ∑_i=p+l^q x_i > (h-l) κ_c - 1 .
        From (<ref>)(<ref>)(<ref>), we have
        
            lκ_c-1 < ∑_i=p^p+l-1 x_i < l κ_c    ⇒   ∑_i=p^p+l-1 x_i=[lκ_c] ,
        
            hκ_c-2 < ∑_i=p^q x_i ≤ h κ_c-1    ⇒   ∑_i=p^q x_i = [h κ_c]-1 . Let the rational number κ_c=u/v, where u,v∈ℕ^+ and u,v are coprime. From
        (<ref>), we know lκ_c ∉ℤ,
        then l/v∉ℤ,∀ l∈{ 1,2,...,h-1}. Thus
        v≥ h. From (<ref>)(<ref>),
        we have x_l+p-1=[lκ_c]-[(l-1)κ_c]-δ_lh. We define M'=
        m_1'   m_2'   ⋯   m_h'
        n_1'   n_2'   ⋯   n_h' where
        n_i'=n_i+p-1. So M'^T⊑ M^T and m_i'=[iκ_c]-[(i-1)κ_c]-δ_i1-1. Comparing M' with (<ref>), we have M'≼ M_t such that M is unprotected.
        
        
        
        Now we prove the converse proposition: If M is unprotected, then
        ∃ p,q satisfying 1≤ p≤ q≤ k such that
        (<ref>) is satisfied. Suppose that the top tensor chain
        M_t= m_1'   m_2'   ⋯   m_v'
        n_1'   n_2'   ⋯   n_v',
        and M is unprotected when its tensors from the pth to the
        qth are acted on by a tensor constraint M'. Since
        M'^T⊑ M^T and M'≼ M_t, ∃ c,d
        satisfying 1≤ c≤ d≤ v and d-c+1=q-p+1=h such that
        x_p+l-1≤ m_c+l-1'+1 - δ_l1 -δ_lh, ∀
        l∈{1,2,⋯,h}. By using (<ref>), we further
        have ∑_i=p^q x_i     ≤ h-2+ ∑_i=c^d m_i' 
           = h-2+∑_i=c^d  [iκ_c]-[(i-1)κ_c]-1+δ_i1
           = [dκ_c]-[(c-1)κ_c]-2+δ_c1
           =[d κ_c]+[(1-c)κ_c]-1  
           ≤ [hκ_c]-1   
           ≤ hκ_c-1,
        
        where (<ref>) is used.
        So (<ref>) is satisfied.
        
        As far as closed tensor chain is concerned, the only difference is
        that it has cyclic symmetry with modular k, namely *   *   ⋯   *   *
        n_1   n_2   ⋯   n_k-1   n_k  = 
        *   *   ⋯   *   *
        n_2   n_3   ⋯   n_k    n_1, then above x_i is
        nothing but reduced interior angles, namely s_i=a-1-n_i=x_i. The
        proposition can be proved with the same algebra.
        
        
        From above theorem, we claim that any tensor chain M
        satisfying M^T ≼ M_c^T is protected; while any tensor
        chain M satisfying M≼ M_c and M≠ M_c is unprotected.
        
        Similarly, thanks to Theorem <ref>, we have a
        one-to-one mapping between CP tensor chain M_c and CP
        reduced interior angle κ_c.
        
        Critical protection characterizes the limit of mapping the
        information from one side (with upper indexes) to another side
        (with lower indexes) with full fidelity. So the physical
        correspondence of CP tensor chain is the maximal boundary of the
        region where the interior information can be mapped to the
        boundary without loss.
        
        
        
        § GEOMETRIC DESCRIPTION
        
        
        In this section we elaborate some geometric properties of the
        tensor network with {b,a} tiling in H^2 space, which will
        be essential for us to provide a quantitative description for the
        QEC and ES in tensor networks. The isometry group of H^2 space
        is SL(2,R). In H^2 space, the curves of constant curvature
        (CCC) include circles, hypercircles and horocircles, depending
        on their values of geodesic curvature. Geodesic is a kind of
        hypercircle [In many literature, `geodesic' in tensor
            network often refers to the polyline with minimal cuts. However,
            it may not always coincide with the geometrical geodesic in H^2
            space. We will not adopt `geodesic' to describe the polyline
            with minimal cuts through this paper.].
        A brief review on SL(2,R) and CCC is given in Appendix <ref>.
        
        
        
         §.§ The curve of constant curvature corresponding to a periodic polyline
        
        The b,a tiling
        breaks the isometry group SL(2,R) into a discrete subgroup
        G_tiling, which is the set of all transformations
        preserving the tiling. We are interested in two specific
        generators V,S of G_tiling, where V is the
        anticlockwise rotation around a node by an angle 2π/a and S
        is the clockwise rotation around the midpoint of an edge linked to
        this node by π. V,S should satisfy the following equations
        
        S^2=V^a=(VS)^b=-1,
        (S)=0,
        (V)=2cos(π/a),
        (VS)=2cos(π/b).
        
        The solutions up to SL(2,R) are
        
        S=(
        [  0 -1;  1  0;    ]),  
        V=(
        [        cos(π/a)   e^P/2sin(π/a); -e^-P/2sin(π/a)        cos(π/a);                 ]).
        
        where length P is given in (<ref>).
        
        Recall that a tensor chain can be embedded into a tensor network
        in H^2 space, as discussed at the beginning of Subsection
        <ref>. Similarly, a periodic tensor
        chain can also be embedded, whose skeleton forms an endless
        and periodic polyline.
        When the scale of a chain is much greater than the period of a
        polyline, the roughness of the skeleton can be zoomed out such
        that it looks like a CCC in H^2 space, whose geodesic curvature
        λ is a constant. In general, given the embedding of a
        periodic tensor chain, we can define a unique CCC corresponding to
        this chain by specific operation. Next we will firstly present the
        procedures to locate such a CCC and finally discuss some
        exceptional cases that such CCC could not be defined.
        
        The periodic tensor chain M can be constructed from an open
        tensor chain according to Definition <ref>. We
        choose a node of M and number it by i. The loop body beginning
        at the (i+1)th node is m_i+1   m_i+2   ⋯   m_i+k
        n_i+1   n_i+2   ⋯   n_i+k, where k is the period of M. Now we choose the center of
        the rotation generated by V as the ith node and the center of
        the rotation generated by S as the midpoint of the edge between
        the ith node and the (i+1)th node. Then we further define a
        transformation preserving the structure of periodic polyline as
         W= V^s_i+kS V^s_i+k-1S ⋯ V^s_i+2S
        V^s_i+1S,    s_j=m_j+1,   ∀ j,  which maps
        each period in the polyline to the next period along the direction
        of the polyline. Starting from a point q in H^2 space, the set
        of all the points generated by W^n, namely Q_W={W^nq|n∈ℤ}, will be located on a CCC. When |Q_W|≥3, the
        CCC can be uniquely determined.
        
        We are interested in the case that point q is the midpoint of
        one edge with lower index in M. With some algebra we finally
        derive that the geodesic curvature λ of this kind of CCC
        can be calculated by λ^2=[ (π_q
        W)]^2/( W)^2+[(π_qW)]^2-4,  where π_q is the
        matrix of clockwise rotation by angle π around point q,
        which belongs to SL(2,R). π_q can be generated by the
        generators V and S, according to the relative position between
        point q and the ith node.
        
        Obviously, different choices of point q generate different CCCs
        and λs. To determine the unique CCC corresponding to M,
        we remark that one just need to choose the point q which
        minimizes |λ-1| in (<ref>). The process of
        generating CCC from a periodic tensor chain is illustrated in
        Fig.<ref>.
        
        
        
        Once the CCC corresponding to a periodic tensor chain can be
        uniquely determined, the classification of CCC in
        (<ref>) can be reformulated by the trace of W,
        [    |(W)|<2,      circle;    |(W)|=2,  horocircle;    |(W)|>2, hypercircle;             ]
        
        From Theorem <ref> and Definition
        <ref>, for a given rational number
        κ∈[1,a/2], one can construct a unique prime tensor
        chain and its periodic tensor chain M whose κ(M)=κ.
        As a result, one can further figure out the corresponding CCC as
        well as its geodesic curvature λ. So one can define a
        mapping from κ to λ, as illustrated in Fig.<ref>.
        
        It may be noticed that not all the λ can be inversely
        mapped to κ, because λ is a positive number while
        κ is a rational number. Nevertheless, horocircle, whose
        curvature λ=1, is a special kind of CCC in H^2 space. It
        corresponds to a closed tensor chain (closed polyline) in large
        radius limit. Furthermore, given a {b,a} tiling, we can prove
        that the average reduced interior angle κ_h of such a
        closed tensor chain is κ_h=a/2 -
        a-2/2√(ab-2a-2b/ab-2a-2b+4).  Since this
        quantity plays a crucial role in classifying the tensor networks,
        we provide the detailed proof as follows.
        
            Horocircle is the limit of a circle with
            infinitely long radius in H^2 space. To construct the polyline corresponding to
            horocircle in a tensor network, we may consider the process of
            increasing the scale of a closed polyline. In a network with
            {b,a} tiling, we consider a closed polyline M with k nodes
            and its total reduced interior angle is l. So the average
            reduced interior angle of M is κ=l/k. Now it is
            helpful for us to plot the Poincare dual of the network with
            {b,a} tiling, which is a network with {a,b} tiling and all
            the nodes located at centers of dual polygons, as illustrated in
            Fig.<ref>.
        
            Given a polyline M, its outer adjoint polyline M' in Poincare
            dual network can be constructed step by step: (1) find out
            those elementary polygons in Poincare dual network whose center
            is located at the node of M; (2) pick out the edges of
            those polygons outside of M; (3) link those edges in order
            and then obtain a closed polyline which is just M'.
        
            The relation between M and M' is illustrated in Fig.<ref>. One can find that M' has (a-1)k-l nodes and its total reduced interior
            angle is ak-l. Keep going on, one can find the outer adjoint
            polyline M” of polyline M' will fall back into the
            network with {b,a} tiling. Similarly, polyline M” has
            (b-1)((a-1)k-l)-ak+l nodes and its total reduced interior angle
            is b((a-1)k-l)-ak+l. So its average reduced interior angle is
            κ”=b((a-1)k-l)-ak+l/(b-1)((a-1)k-l)-ak+l
            =(1-b)κ+(b-2)c_1(a-c_1)/(2-b)κ+ab-2a-b+1≡ f(κ),
             where  c_1≡a/2 -
            a-2/2√(ab-2a-2b/ab-2a-2b+4).  The number of
            nodes enclosed by M” is always more than that enclosed by
            M. Thus, if
            we begin with an elementary closed polyline with b nodes and
            total reduced interior angle b and continuously find the outer
            adjoint polyline, we will approach the polyline corresponding to a
            horocircle. Thus we have κ_h=lim_n→∞f^n(1),
            
            where f^n represents f applied n times. To evaluate
            above limit, we observe that
            κ”-c_1/κ”+c_1-a=c_2κ-c_1/κ+c_1-a,
            
            where
            
           c_2≡1-b-2c_1+bc_1/1-b+(2-b)(c_1-a),   0<c_2<1.
            
            Thus
            lim_n→∞f^n(1)-c_1/f^n(1)+c_1-a=0.
            
            Since f^n(1) must be finite, we finally have
            κ_h=lim_n→∞f^n(1)=c_1.
            
        
        Now we discuss some exceptional cases. The first case is
        |Q_W|≤ 2, namely the number of generated points less than
        two, so CCC can not be uniquely defined through the above process.
         The second case is that M_period bends in an irregular way such
        that the embedding of periodic tensor chain may lead to a
        self-crossed polyline and no CCC could form, such as the
        M_period with loop body 1   1   1
         5   4   5 in the
        tensor network with 3,7 tiling. Such two cases only happen for some κ∈ [1,κ_h).
        
        
        
         §.§ CP curves
        
        The CCC corresponding to
        a CP tensor chain is called CP curve, whose geodesic curvature
        is called CP curvature λ_c. CP curve is a generalization of
        the greedy geodesic in <cit.>.
        
        Given a tiling, CP curvature λ_c and CP reduced interior
        angle κ_c are inversely related to each other, as shown in
        Fig.<ref>. If the CP tensor chain form a closed
        polyline, the CP curve is a circle, κ_c<κ_h and
        λ_c>1. If the CP tensor chain form an open polyline
        which extends to the boundary, the CP curve is a hypercircle,
        κ_c>κ_h and λ_c<1.
        
        Roughly speaking, a periodic tensor chain is unprotected if its
        corresponding CCC has geodesic curvature λ>λ_c;
        while it is protected if the corresponding CCC has
        λ<λ_c .
        
        
        
        § TENSOR NETWORKS AND GREEDY ALGORITHM
        
        
        So far we have established a framework to describe tensor
        chains and tensor constraints. We will impose the central set S_c to a tensor network in the sense that those constraints M∈ S_D derived from S_c are valid, while those M ∉ S_D are not valid.
        
        
        
         §.§ Tensor networks with 7,3 tiling or 4,5 tiling
        
        
        Before analysing the role of critical protection in QEC and ES for general tensor
        networks, we construct some specific examples of tensor networks
        and provide an intuitive understanding on the notion of critical
        protection.
        
        The structure of a tensor network with 7,3 tiling is shown
        in Fig.<ref>. According to (<ref>), one has
        κ_h=1.28. We impose the central set S_c= 1
         2 , M_t for some typical M_t and discuss
        the entanglement property of the tensor network. The structure
        of M_t, M_c, and the values of κ_c and λ_c are
        listed in Table.<ref>. The corresponding diagrams of
        tensor constraints and CP tensor chains in the tiling are
        illustrated in Fig.<ref>, <ref>, <ref> and
        <ref> respectively.
        
        
        
        In parallel, a tensor network with 4,5 tiling is shown in
        Fig.<ref>. In this case, one has κ_h=1.63. The
        entanglement properties of this tensor
        network with different top tensor chains
        are collected in Table.<ref>. The corresponding diagrams
        of tensor constraints and the embedded CP tensor chains are
        plotted in Fig.<ref>, <ref>, <ref> and
        <ref> respectively.
        
        
        
        In above figures, we divide the boundary of the tensor network
        into two intervals A and A̅. The shaded region with
        different colors presents the effect of the greedy algorithm
        starting from A and from A̅ respectively, which will be
        further discussed in next subsection in detail. CP tensor chains
        are marked in each figure and its significance in greedy algorithm
        will be stressed as well. In next two sections, we will further
        take these figures as examples to disclose the relation
        between greedy algorithm and quantum error correction as well
        as entanglement spectrum.
        
        
        
         §.§ Greedy algorithm on tensor chains
        
        
        For a tensor network Ψ, we generalize the greedy algorithm in
        <cit.>, based on the set S_D derived from a
        central set S_c. After choosing an interval A on the boundary,
        we consider a sequence of cuts C_n and a sequence of sub
        tensor network Φ_n, where each C_n is bounded by
        ∂ A and each Φ_n consists of those tensors enclosed
        by C_n and A, shaded with strips. So each Φ_n is a
        mapping from the Hilbert space on C_n to the Hilbert space on
        A. Let C_1=A, then Φ_1 is an identity. Next one figures
        out a tensor chain in the tiling which belongs to the set of S_D
        and all of its lower indexes can be contracted with Φ_n.
        Then Φ_n+1 is constructed by absorbing such M_n
        into Φ_n. The greedy algorithm stops when no such tensor
        chain can be found. The way of iteration guarantees that each
        Φ_n is proportional to an isometry. As explained in
        <cit.>, above greedy algorithm for a tensor network
        Ψ is equivalent to the procedure of simplifying the
        contraction of tensor chains in (<ref>), where M is any
        tensor chain embedded in the tensor network Ψ.
        
        To describe the process of greedy algorithm precisely, which is
        essential in the proofs for the properties of ES, we intend to
        extend the notion of protection to a directed cut in greedy
        algorithm.
        
        One may notice that process of a greedy algorithm is not unique.
        Actually, one may have a lot of ways to arrange the sequence of
        absorbing tensor chains into the shaded region Φ_n such that
        during the course of greedy algorithm, C_n need not to be
        connected. In the greedy algorithm starting from an interval A,
        each cut C_n is specified a direction such that its
        corresponding Φ_n is on its right hand side. A cut may
        consist of one or more connected components, as illustrated in
        Fig.<ref>. A connected component can be an open curve or a
        closed curve. The open curve is bounded by ∂ A, while the
        closed curve need not. We define the sequence of nodes
        corresponding to a connected component as follows.
        
        Given a connected component of a directed cut, we find out those
        nodes on its left hand side and define the sequence of them along
        the direction of the connected component.  The sequence of nodes
        given by an open curve is denoted as [N_1,N_2,⋯,N_l], where
        N_1 and N_l are located at the boundary. The sequence of nodes
        given by a closed curve is denoted as (N_1,N_2,⋯,N_l),
        where N_1 and N_l are neighbor.
        
        Just for convenience, one may allow the sequence number of nodes
        to start from any integer, such as [N_-1,N_0,N_1,N_2]. But the
        sequence number must be monotonically increasing with unit
        step.
        
         Say a tensor chain M is connected to a directed cut C, if M
        lies on the left hand side of C and all the edges associated with its lower indexes are cut by C while all the edges
        associated with its upper indexes are not cut by C.
        
        Say a directed cut C is unprotected, if there exists
        unprotected tensor chain which is connected to C. Otherwise, say
        C is protected.
        
        So a greedy algorithm progresses (stops) when the cut is
        unprotected (protected).
        
        
        
        
        A greedy algorithm can start from the interval A̅ as well.
        In Fig.<ref><ref><ref><ref><ref><ref>, we show the final results of greedy
        algorithm on several tensor networks with specific intervals A
        and A̅, which are shaded with different colors respectively.
        In Fig.<ref> and <ref>, all the tensor chains are
        protected under the action of greedy algorithm such that no shaded
        region presents in those networks.
        
        In above plots one may notice that some CP tensor chains are
        absorbed by greedy algorithm, which apparently conflicts with the
        fact that CP tensor chain should be protected. We point out that
        this phenomenon ascribes to the fact that the endpoints of CP
        tensor chains belong to the interval A or A̅ on the
        boundary as well. For instance, consider the greedy algorithm
        starting from A̅, as shown in Fig.<ref>.
        We define Ȧ̅̇ to be the interval between one endpoint
        of a CP tensor chain, which is an uncontracted edge on the
        boundary within A̅, and the most neighboring endpoint of
        A̅. Generally, the width of Ȧ̅̇ is equal to the
        geodesic distance between the CP curve and its axis,  i.e.d_c=arctanh(λ_c). We firstly consider the greedy
        algorithm starting from a sub-interval A̅ - Ȧ̅̇. At
        this stage greedy algorithm stops before it touches the CP tensor
        chain indeed. However, for practice when we calculate the reduced
        density matrix ρ_A, the contraction on the endpoints of CP
        tensor chain, namely uncontracted edges within Ȧ̅̇,
        must be taken into account by definition. At this stage the CP
        tensor chain may fail to be protected under the action of greedy
        algorithm, as shown in Fig.<ref>. We refer it as the
        boundary effect of greedy algorithm. Nevertheless, we remark that
        this boundary effect is weak in the sense that it just absorbs
        finite layers (most possibly, only one layer) of tensors, and we
        will elaborate it when we study the ES of tensor networks in
        Section <ref>.
        
        
        
        § QUANTUM ERROR CORRECTION (QEC)
        
        In this section we will concentrate on how to justify the
        ability of QEC for a tensor network based on the properties of CP
        tensor chain.
        
        
        
         §.§ Greedy algorithm and QEC
        
        The whole story of QEC on tensor networks is based on the Hilbert
        space associated with uncontracted edges which introduce extra
        degrees of freedom in the bulk and the corresponding code subspace
        in the Hilbert space on the boundary. The correction to the code
        subspace after erasing an interval A on the boundary is
        equivalent to pushing a bulk operator in the wedge of the interval
        A̅ to the interval A̅ on the boundary
        <cit.>. Technically, following
        <cit.>, the procedure of QEC involves in three
        steps: (1) acting on an uncontracted index in the bulk with an
        operator; (2) pushing the operator from this index in the bulk to
        the indexes in the network; (3) pushing it to the boundary
        further.
        
        In our current work we will ignore the Hilbert space in the bulk
        since our main purpose is to realize the algorithm of QEC on
        tensor networks. We will skip step 1 and begin at step 2, by
        directly inserting an operator into the contracted edges in the
        interior. In the language of tensor chain, we can insert an
        operator O between tensor chain M and M'∑_B M^A_B M'^B_C →∑_BD M^A_B O^B_D M'^D_C.  No
        matter which way one adopts to insert an operator into the
        network, the latter processes of QEC are the same. Thanks to
        tensor constraint, we can push an operator O through a tensor
        chain M∈ S_D and the output operator can be rewritten as O',
        namely ∑_B O^A_BM^B_C ∝∑_B M^A_B
        O'^B_C,    O'^B_C= ∑_AD (M^B_A)^* O^A_D M^D_C. 
        
        Specifically, without uncontracted indexes in the bulk, equation
        (<ref>) is just the reflection of step 2 above and equation
        (<ref>) depicts step 3. After all, the terminology `QEC' in
        this paper refers to the above interpretation.
        
        
        All above operations can be demonstrated by diagrams. Taking
        the tensor network with 7,3 tiling as an example. The
        insertion of an operator is illustrated as
        
            < g r a p h i c s >
        .
        
        While employing tensor constraints (<ref>), the
        process of pushing an operator through tensor chains can be
        illustrated as
        
            < g r a p h i c s >
        ,    
            < g r a p h i c s >
        
        
            < g r a p h i c s >
        ,    
            < g r a p h i c s >
        .
        
        
        One can successively push operators through tensor chains in
        S_D. Operators may be finally pushed to an interval on the
        boundary or not, depending on the structure of tilings and tensor
        constraints. Actually, tensor pushing is the reverse procedure of
        greedy algorithm, where pushing an operator through tensor chain
        M∈ S_D is reverse to the procedure of absorbing a tensor chain
        M into the shaded region of a tensor network.
        
        
        We say that a tensor network enjoys QEC if any operator inserted
        into the bulk of the network can be pushed to an interval on
        the boundary.
        
        
        
        
        In Fig.<ref> and <ref>, an inserted operator O
        is successfully pushed to A̅. We remark that if the
        operator is inserted into the region enclosed by the CP tensor
        chain and interval A̅, as illustrated in
        Fig.<ref>, then it can be pushed to A̅. In other
        word, after erasing an interval A, most of those points in the
        wedge of A̅ can be recovered by QEC.
        
        While if the operator is inserted into the region enclosed by the
        CP tensor chain and the geodesic bounded by ∂A̅, the
        situation becomes subtle and it is not guaranteed that the
        operator can always be pushed to A̅. On one hand, if the
        inserted operator is close to CP tensor chain, as illustrated
        in Fig.<ref>, then it may still be pushed to a
        subinterval in A̅. While, now the bound of such
        subinterval is approaching to ∂A̅. In this
        figure we notice that a lot of arrows, which denote the
        trajectory of pushing the operator through, go across the geodesic
        and then radiate out in a wide region, in contrast to the
        process in Fig.<ref>. Such phenomenon indicates that the
        information of an operator can only be recovered in a wide range
        of the boundary, implying the function of QEC in
        Fig.<ref> is weaker than that in the tensor network in
        Fig.<ref>. It may be related to the approximate QEC
        <cit.>. On the other hand, if
        the operator is rather close to the geodesic bounded by ∂A̅, it may not be pushed to A̅ any more.
        
        
        
         §.§ CP curves and QEC
        
        
        The geometric description of CP tensor chain in Section
        <ref> provides us a way to describe QEC over H^2
        space as well. Given a subsystem A̅ on the boundary, one may
        ask whether an operator acting on point x which locates inside
        the wedge of A̅ can be pushed to A̅. For a simply
        connected interval A̅, we denote its two endpoints as u
        and v, respectively. Then these two points together the point
        x can uniquely determine a hypercircle H in H^2 space. The
        sub network in region Ω enclosed by H and A̅
        defines a mapping Φ from the Hilbert space associated with
        the edges on H to the Hilbert space associated with the edges on
        A̅. If and only if Φ is proportional to an isometry,
        then the operator can be pushed to the boundary, thus implementing
        QEC in an operator scenario. Otherwise the operator can not be
        pushed into the region specified by A̅, and the recovery of
        such an operator will be prevented by erasing A such that QEC
        fails.
        
        To check whether Φ is isometric or not, one needs to evaluate
        the inner product ΦΦ^†, which is directly
        determined by the imposed tensor constraints S_c. During the
        evaluation process, the most difficult step is to simplify the
        contraction M M^†, where M is the boundary tensor chain
        of Φ on H. So to figure out whether Φ is isometric or
        not, our final task is to justify whether M is protected or not
        under tensor contractions which are subject to S_c.
        
        Fortunately, our discussion in the section of critical
        protection has provided an answer to this question. One can
        justify this by comparing the geodesic curvature λ of the
        hypercircle H with the the curvature of CP curve λ_c. If
        λ>λ_c, then M is unprotected; if
        λ<λ_c, then M is protected.
        
        As a result, given a subsystem A on the boundary, we find a geodesic connecting two end points of the subsystem A and a CP curve between the geodesic and A̅. Whether an
        operator at x can be pushed into A̅ depends on the geodesic
        curvature of the hypercircle passing through x. For those points
        inside the region enclosed by boundary A̅ and the CP curve, an operator can be recovered by QEC since the geodesic curvature of hypercircles is greater than λ_c; while for those points inside the region enclosed by the CP curve and the geodesic an operator can not be recovered by QEC since the
        geodesic curvature is less than λ_c.
        
        When a tiling of H^2 is specified, κ_c is inversely
        related to λ_c. Because κ is more easily calculated
        than λ, one can alternatively compare the average reduced
        interior angle κ of a hypercircle with the average reduced
        interior angle of CP tensor chain κ_c.  For a given tiling,
        recall that the tensor chain corresponding to a horocircle with
        λ_h=1 has average reduced interior angle κ_h in
        (<ref>). Moreover, once S_c is specified, then
        λ_c and κ_c are determined as well. Whether a
        tensor network enjoys QEC or not can be justified by comparing the
        value of λ_c with λ_h or κ_c with
        κ_h, as described below.
        
        If λ_c≥1 or κ_c≤κ_h, the CP curve is a
        circle or a horocircle. The geodesic curvature of all hypercircles
        must be less than λ_c, so no QEC can be implemented by
        inserting an operator into any point in the bulk and such a tensor
        network do not enjoy QEC. For instance, those tensor networks in
        Fig. <ref> and <ref> belong to this class. It
        matches the fact that greedy algorithm does not iterate in these
        tensor networks.
        
        Similarly, if λ_c<1 or κ_c>κ_h, the CP curve
        is a hypercircle. An operator inserted into the region
        enclosed by the CP tensor chain and A̅ can be recovered by
        QEC and such a tensor network enjoy QEC. For instance, all the
        other tensor networks except Fig.<ref> and <ref> in
        this paper belong to this class. Nevertheless, given an interval
        A̅, the region that can be recovered is different for
        different constraints.
        
        We summarize the above results about the function of QEC in a
        network in Fig.<ref>.
        
        
        
        § ENTANGLEMENT SPECTRUM (ES)
        
        
        Next we focus on the evaluation of entanglement spectrum for a
        given tensor network, and argue that the flatness of ES can be
        justified with the power of critical protection in general
        cases.
        
        
        
         §.§ Reduced density matrix
        
        
        A tensor network Ψ gives a state |Ψ⟩ in the
        Hilbert space defined on its uncontracted edges on the
        boundary. Given an interval A on the boundary, one can obtain
        the reduced density matrix of A by tracing out the
        complementary region A̅, namely
        ρ_A=_A̅|Ψ⟩⟨Ψ|.
        
        
        We are concerned with the issue whether the reduced density
        matrix ρ_A has a flat spectrum, which means that all the
        non-zero eigenvalues of ρ_A are identical. This statement
        can be alternatively rephrased as the following propositions:
        
          * 
        All the orders of Renyi entropy
        
        S_A,n=-1/n-1lnρ_A^n/ρ_A^n
        are identical, namely independent of n.
            
          * 
        Reduced density matrix satisfies the relation
        ρ_A^2∝ρ_A.
        
        
        As indicated at the beginning of this paper, ρ_A of the
        ground state of CFT_2 satisfies (<ref>) and
        exhibits a non-flat ES. The gravitational dual result of AdS_3
        vacuum coincides with the above result as well. Now we would like
        to check whether the ES of a tensor network state is flat or not.
        For this purpose it is convenient to check the relation in
        (<ref>) by manipulating tensor networks.
        
        First we disclose the key role of CP tensor chain in identifying
        the protected region in a tensor network. Recall the boundary
        effect in greedy algorithm, we intend to separate the procedure of
        taking trace on A̅ into following two steps
         ρ_A=_A̅|Ψ⟩⟨Ψ|=_Ȧ̅̇_A̅-Ȧ̅̇|Ψ⟩⟨Ψ|. 
        
        In the following, we will take tensor networks with 7,3
        tiling as examples to demonstrate the evaluation of ES by
        manipulating tensor networks. The results have previously been
        collected in Table <ref>.
        
        
        
          §.§.§ Non-flat ES
        
        
        First of all, we point out the evaluation of ES depends on the
        choice of the interval A on the boundary. We will see that, for
        constraints S_c= 1
        2 ,  1   0   1
        1   1   1,
        the ES of ρ_A for any relatively large interval A is
        non-flat. We call the tensor network generally has a non-flat ES. The word “generally” means that the ES is always non-flat unless fine-tuning tensor T and E. Throughout this paper when we say that a tensor network has a non-flat ES, we refer to the above statement.
        
        Firstly, we trace out the degrees of freedom in A̅-Ȧ̅̇ to obtain the reduced density matrix. During this procedure
        the structure of tensor network is simplified due to the tensor
        constraints generated by S_c, see Fig.<ref> (a-c).
        Specifically, those tensors in the wedge of A̅ are
        contracted into identity matrices, which is just the process of
        the greedy algorithm starting from A̅ - Ȧ̅̇ in the
        previous section. One can repeatedly consider this process until
        it reaches a final stage that the network can not be simplified
        any more, as shown in Fig.<ref>(c). The terminal
        boundary forms a polyline in red as marked in
        Fig.<ref>. As a matter of fact, such a polyline is
        nothing but a CP tensor chain as we have defined in previous
        section. From this figure we perceive that, before the trace of
        uncontracted edges in Ȧ̅̇ is taken into account, the
        operation induced by the greedy algorithm can not enter the region
        enclosed by CP tensor chain and A, which is exactly the reason
        why we call it critically protected tensor chain.
        
        Now the next step is to evaluate _Ȧ̅̇, namely
        tracing the degrees of freedom associated with uncontracted edges
        on the boundary which are mostly neighboring to A. The process
        is illustrated in Fig.<ref>(d)(e). We notice that the
        network structure can be further simplified such that CP tensor
        chains are absorbed into the shaded region at this step, which
        is the boundary effect of greedy algorithm as we described in
        previous section.
        
        The boundary effect of greedy algorithm results from the
        discretization of H^2 space, which may not appear in a
        continuous geometry. In the context of tensor networks, however,
        according to (<ref>) the uncontracted edges in Ȧ̅̇
        should be contracted. Sometime the contribution of this effect to
        reduced density matrix becomes subtle, and we should cautiously
        handle this effect. In other words, whether the ES is flat or not
        can only be justified after the boundary effect is taken into
        account.
        
        Now with the reduced density matrix ρ_A at hand, we can
        compute ρ_A^2 by further contracting those uncontracted edges
        in A. One can simplify ρ_A^2 by virtue of tensor
        constraints, which is parallel to the above process on A̅.
        Boundary effect of greedy algorithm appears as well. Before the
        boundary effect is taken into account, the greedy algorithm stops
        at a CP tensor chain, which is the reflection of the CP tensor
        chain appearing in the contraction on A̅ about the geodesic
        bounded by ∂ A. Thus, the simplification of ρ_A^2 is
        equivalent to applying the greedy algorithm to A and A̅
        successively, as shown in Fig.<ref>.
        
        The calculation of ρ_A^2 is demonstrated in
        Fig.<ref>. Obviously from this diagram we find that
        ρ_A^2 can not be simplified to be proportional to ρ_A
        such that equation (<ref>) is not satisfied. Equivalently,
        from Fig.<ref>, we notice that some tensors are not
        absorbed by the greedy algorithm starting from A and A̅,
        thus (<ref>) is not satisfied.
        
        Given the above constraints, we point out that as long as A
        is large enough, ρ_A always gives rise to a non-flat ES,
        independent of the choice of A. This assertion will be proved in
        Subsection <ref>. Right now we just conclude that
        such a tensor network has a non-flat ES, in agreement with what is
        found by explicitly computing the eigenvalues of the reduced
        density matrix in <cit.>.
        
        We remark that for the above constraints the corresponding CP
        curve is a hypercircle. When the CP curve is a horocircle, it
        approaches the boundary with single intersecting point. Or when
        the CP curve is a circle, it does not reach the boundary. For both
        cases one need not consider the boundary effect separately, and
        the ES is usually non-flat for CP circles since the region
        enclosed by the circle is protected.
        
        
        
          §.§.§ Flat ES
        
        
        We have pointed out that one equivalent way to check the relation
        in (<ref>) is to consider the greedy algorithm starting
        from A and from A̅ successively. Let us take Fig.<ref>
        as an example, where S_c= 1
        2 ,  1   1
        1   1. We observe that the union of these two shaded regions
        covers the whole tensor network, implying that all the tensors are
        absorbed by the greedy algorithm. Therefore, (<ref>) is
        satisfied and ES has to be flat. We call the tensor network has a
        flat ES.
        
        
        
          §.§.§ Mixed ES
        
        
        From Fig.<ref>, we know that, for S_c= 1
        2
        ,  1   0   1   0   1
        1   1   0   1   1, the ES of ρ_A can
        be flat or non-flat, depending on the choice of A. We call
        the tensor network has a mixed ES.
        
        
        
        
         §.§ Geometric point of view on ES
        
        
        In the tensor network realization of AdS/CFT, a tensor network is
        usually treated as the wavefunction Ψ of the ground state.
        Alternatively, when an interval A on the boundary is given, we
        notice that Ψ can be understood as a mapping from the Hilbert
        space on A to the Hilbert space on A̅. So Ψ can be
        regarded as a matrix Ψ^A_A̅, where two indexes A and
        A̅ represent the degrees of freedom on two subsystems A
        and A̅, respectively.
        
        The notion of critical protection provides us an efficient way to
        visualize the simplification of tensor networks under the tensor
        contractions which are subject to tensor constraints. To make this
        process more transparent, we firstly intend to decompose a network
        into some sub networks. As seen in previous subsections, when the
        indexes on A or A̅ are contracted, the greedy algorithm
        will stop at some nodes. Let us firstly neglect the boundary
        effect, then the skeletons of connecting those nodes will form two
        CP tensor chains, which are neighboring to the geodesic bounded
        by ∂ A.
        
        First of all, we point out that when λ_c≥1 or
        κ_c≤κ_h, all the hypercircles are protected since
        their geodesic curvatures are less than λ_c. So a non-flat
        ES is guaranteed. In the following, we will focus on the
        non-trivial case, λ_c<1 or κ_c>κ_h, where CP
        curves are hypercircles.
        
        We denote the CP curve close to A or A̅ as H_A or
        H_A̅, respectively. The region enclosed by two CP curves
        is called CP region Ω_c. Those tensors in the CP region
        form a sub tensor network Ψ_c, which is a mapping from H_A
        to H_A̅ and is denoted as (Ψ_c)^H_A_H_A̅.
        Similarly, H_A and A enclose a sub tensor network Φ_A,
        which defines a mapping (Φ_A)^A_H_A; H_A̅ and
        A̅ enclose a sub tensor network Φ_A̅, which
        defines a mapping (Φ_A̅)^A̅_H_A̅. Since
        the tensors outside H_A are not protected under the contraction
        of A, the mapping (Φ_A)^A_H_A from H_A to A should
        be proportional to an isometry. Similarly, the mapping
        (Φ_A̅)^A̅_H_A̅ from H_A̅ to
        A̅ is proportional to an isometry as well. It is denoted as
        Φ_A^†Φ_A∝ I,   Φ_A̅^†Φ_A̅∝ I',
        
        where the indexes are abbreviated and I (I') is identity matrix on A (A̅).
        
        Finally, the full matrix Ψ^A_A̅ can be represented as the product of matrices
        Ψ=Φ_AΨ_cΦ_A̅^†.
        
        Then it is easy to see
        ρ_A   =   ΨΨ^†= Φ_AΨ_cΦ_A̅^†Φ_A̅Ψ_c^†Φ_A^†∝Φ_AΨ_cΨ_c^†Φ_A^† , 
        ρ_A^2   =   ΨΨ^†ΨΨ^†∝Φ_AΨ_cΨ_c^†Ψ_cΨ_c^†Φ_A^† ,
        
        where (<ref>) is used. A flat ES in (<ref>) means that
        Ψ_cΨ_c^†Ψ_cΨ_c^†∝Ψ_cΨ_c^† .
        
        We present a schematic diagram to demonstrate the decomposition of tensor network state as well as the calculation of ρ_A and ρ_A^2 in Fig.<ref>. The condition for flat ES (<ref>) is illustrated in Fig.<ref>.
        This figure reveals that whether the ES is flat or not depends on the thickness of the CP region where the thickness of the CP region is defined by the distance between the two CP curves.
        
        Equivalently, from above derivation we notice that the flatness of
        ES may be checked by observing the result of the greedy algorithm
        starting from A and from A̅ successively, which
        figures out the region of isometry between tensor chains in
        (<ref>). If all the tensors are absorbed by the greedy
        algorithm, then (<ref>) is valid and the ES is flat, and
        vice versa.
        
        Once the boundary effect is considered, as we showed in previous
        section, CP tensor chains on the boundary of the CP region
        Ω_c are not protected any more under the greedy algorithm.
        Nevertheless, only a finite thickness of the CP region will be
        absorbed. In Fig.<ref>, since those tensors close to
        the geodesic are not absorbed, the tensor network has a non-flat
        ES.
        
        
        
        The experience one has gained from this picture is that the
        thickness of CP region determines whether the ES is flat or not.
        Without the boundary effect, the boundary of CP region is composed
        of two CP curves, so its thickness is 2d_c, where
        d_c=arctanh(λ_c) is the geodesic distance between
        the CP curve (hypercircle) and its axis. Due to the boundary
        effect, the CP tensor chain will not be protected any more and the
        outer layer of the original CP region will be absorbed by greedy
        algorithm. The thickness of such layers is proximately given by
        P, which is the length of an edge (<ref>). So the
        thickness of CP region decrease to 2d_c-P. Since Ψ_c is
        protected, (<ref>) is true only if the thickness of the
        CP region vanishes.
        
        The evaluation of the geodesic curvature λ in a
        general tensor network is difficult, which prevents us from
        justifying the flatness of ES with CP curvature λ_c.
        Alternatively, this job can be done by calculating CP reduced
        interior angle κ_c, as described in the next subsection.
        
        
        
         §.§ The reduced interior angle of CP tensor chain and ES
        
        
        
        In previous subsections we have shown the relation between the
        flatness of ES and the structure of CP tensor chains under the
        action of greedy algorithm. In this subsection we show that the
        flatness of ES can be justified based on the value of κ_c.
        Specifically, we find that the bigger is κ_c, the stronger
        is the ability of QEC while ES more easily becomes flat, as shown
        in Fig.<ref>. In this figure we further introduce
        three quantities which are determined by the {b,a} tiling: κ_h=a/2 - a-2/2√(ab-2a-2b/ab-2a-2b+4),
          κ_1=b/b-2,  κ_0=a/2.  Because
        of (<ref>), the relation κ_h<κ_1<κ_0
        always holds. If κ_c ∈ (1, κ_h), it turns out the
        network is not able to implement QEC but has non-flat ES, as
        indicated in Fig.<ref> and <ref>. If κ_c ∈
        (κ_h,κ_1), then the network can implement QEC and has
        non-flat ES, as shown in Fig.<ref> and <ref>. If
        κ_c ∈ [κ_1,κ_0), the ability of QEC will become
        stronger but the ES will become “mixed"
        , as shown in Fig.<ref> and <ref>. Finally,
        if κ_c = κ_0, the quality of QEC
        becomes better but the ES has to be flat, which is exactly the
        property of perfect tensors, as shown in Fig.<ref> and <ref>.
        
        Correspondingly, we may propose a geometric quantity in H^2
        space which plays a similar role as κ_c in tensor network.
        This quantity is the geodesic curvature λ_c of CP curve.
        Given a tiling, λ_c can be calculated by using κ_c.
        A schematic relation between λ_c and QEC and ES is also
        illustrated in Fig.<ref>. While, we do not have
        general expressions for the bounds λ_0 and λ_1 so
        far, which corresponds to κ_0 and κ_1, respectively.
        [The main difficulty probably results from the
        specification of an unique CP curve corresponding to a CP tensor
        chain. Tensor chains are discrete, while curves are continuous. To
        assign an unique curve, we have to impose more conditions such as
        requiring that the CP curve has the maximal value of geodesic
        curvature, which is difficult to handle in practice for a general
        tiling.]
        
        Until now, we have constructed a general framework for tensor
        networks with tensor constraints, and developed a generalized
        greedy algorithm to describe the property of critical protection.
        In the remainder of this paper, we will provide detailed
        proofs for the quantitative relation between CP tensor chain and
        QEC as well as ES, and finally complete the classification of
        tensor networks as illustrated in Fig.<ref>.
        
        
        
        Those statements can be rephrased into following propositions:
        
          *  If κ_c=a/2, then ρ_A has flat ES for any choice of A;
            
          *  If κ_c < a/2, then ρ_A may have non-flat ES for some choices of A;
            
          *  If κ_c≥b/b-2, then ρ_A may have flat ES for some choices of A;
            
          *  If κ_c<b/b-2, then ρ_A has non-flat ES for any choice of  largeA.
        
        Now we intend to prove these propositions separately.
        
        
        
          §.§.§ κ_c=a/2⇒ flat ES
        
        
        Based on the discussion in Subsection <ref>, we
        will prove the flatness of ES by showing that any directed cut
        appearing in the process of greedy algorithm is unprotected such
        that the greedy algorithm will not stop until all the tensors are
        absorbed.
        
        To prove a directed cut in the process of greedy algorithm is
        unprotected, one need to find out an unprotected tensor chain
        connected to the cut. Recall that those directed cuts in greedy
        algorithm may have many disconnected components. We firstly
        prove a lemma for a cut containing the structure of twigs or
        loops, which will greatly simplify the rest of proofs.
        
        
        Given a tensor network with b,a tiling and κ_c≥b/b-2, if a directed cut C contains a connected
        component whose corresponding sequence of nodes has a form as
        [⋯,N_0,N_1,⋯,N_k,⋯] with N_0=N_k, or
        (⋯,N_0,N_1,⋯,N_k,⋯) with N_0=N_k, or
        (N_1,N_2,⋯,N_k), then the cut C is unprotected.
        
        Denote the sequence of nodes [N_1,⋯,N_k] as L_N. We
        assume that all these nodes in L_N are distinct,
        otherwise we just replace N_1 and N_k by any two nodes which
        are identical and the following proof is still valid.
        
        When k=1, it is only possible that the connected component is a
        single node N_1, then the tensor chain 0
        a on
        N_1 is connected to C. Since 0
        a ∈ S_D, C is
        unprotected.
        
        When k=2, the shape of L_N is a twig and N_1 is the endpoint
        of the twig. Step tensor chain 1
        a-1 on N_1 is
        connected to C, so C is unprotected. For instance, in
        Fig.<ref>, N_27=N_29 and the sequence
        L_N=[N_28,N_29] forms a twig, N_28 is the endpoint and
        M_s at N_28 is connected to the cut.
        
        When k≥3, those edges between N_i and N_i+1, and the
        edge between N_1 and N_k in L_N form a closed polyline,
         e.g., see the sequence [N_10,N_11,⋯,N_16] in
        Fig.<ref>. We define the region enclosed by the polyline as
        Y, which consists of F elementary polygons, E edges
         and V nodes (vertices)
        which satisfy Euler's formula
            
            F-E+V=1.
            
            Let the reduced interior angle of Y at N_i be x_i for i∈1,2,⋯,k. We have
            ∑_i=1^k x_i +k +a(V-k)=2E,
        
            k≤ V, 
        
            bF=2E-k.
            
            From above four formulas, we have
            ∑_i=1^k x_i ≤b/b-2 (k-2).
            
            Because x_k≥1, we further have
            ∑_i=1^k-1 x_i ≤b/b-2 (k-2) - 1.
            
        Those nodes [N_1,N_2,⋯,N_k-1] form a tensor chain M=
        *   *   ⋯   *
        n_1   n_2   ⋯   n_k-1 connected to C, where
        n_i=a-1-x_i for i∈1,2,⋯,k-1. Recall that
        κ_c≥b/b-2. Finally,
            ∑_i=1^k-1 (a-1-n_i) ≤κ_c (k-2) - 1<κ_c(k-1)-1.
            
        From Theorem <ref>, M is unprotected and thus
        C is unprotected.
        
        In conclusion, when κ_c≥b/b-2, any cut C
        containing twigs or loops must be unprotected.
        
        
        Obviously, κ_c=a/2> b/b-2. Lemma
        <ref> is applicable to this case and those branches
        forming twigs or loops in a cut will be absorbed by the greedy
        algorithm. Taking the cut in Fig.<ref> as an example,
        we claim that those nodes in N_8,N_9,⋯,N_17,
        N_27,N_28,N_29, and N_1',N_2',⋯,N_7' will
        be absorbed.
        
        As a result, now we can focus on the case that the cut C is
        single connected and bounded by ∂ A, which is denoted
        as C=[N_1,N_2,⋯,N_l]. Furthermore, these nodes in C are
        distinct.
        
        With any choice of single interval A on the boundary of a tensor
        network Ψ, we apply the greedy algorithm starting from A
        and from A̅ simultaneously. So two cuts, C_A and C_A̅, appear in Ψ at the same time. We will prove with
        mathematical induction that when κ_c=a/2, either of
        these two cuts is unprotected until all the tensors are absorbed.
        
        
        
        Now we consider the configuration of C_A and C_A̅. Both
        of them are connected to ∂ A. Besides, they may overlap
        at some place, where their directions are opposite, as illustrated
        in Fig.<ref>. Then, we define the “directed sum”
        of C_A and C_A̅ as the union of them but excluding their
        overlapped parts. The directed sum consists of one or more closed
        curves, as shown in Fig.<ref>. Select one of them
        and denote it as C_I, which is a directed cut as well. Set the
        sequence of nodes corresponding to C_I to be
        (N_1,N_2,⋯,N_k). We connect these nodes
        (N_1,N_2,⋯,N_k) with edges in order and enclose a region
        I, which is a union of elementary polygons and edges. At node
        N_i, let the reduced outer angles of I be y_i and let the
        number of edges cut by C_I be n_i. Obviously, y_i=n_i+1.
        Gauss-Bonnet theorem tells that ∑_i=1^k
        (2π/ay_i-π)-2π= Area(H)≥0,  then
        ∑_i=1^k n_i≥a-2/2 k+a. 
        
        Obviously, those edges cut by C are divided into two parts, one
        part is cut by C_A and the other is cut by C_A̅.
        Without loss of generality, we suppose that C_A runs from N_1
        to N_u+1. Moreover, l_1 edges of N_1 are cut by C_A and
        l̅_k+1 edges cut by C_A̅. While for N_u+1,
        l_u+1 edges are cut by C_A and l̅_u+1 edges are cut
        by C_A̅. Obviously, l_1+l̅_k+1=n_1 and
        l_u+1+l̅_u+1=n_u+1. We further define that l_i=n_i
        for i=2,3,⋯,u and l̅_i=n_i for i=u+2,u+3,⋯,k.
        Then we know that tensor chain M_A= *   *   ⋯   *
        
        l_1   l_2   ⋯   l_u+1 is connected to C_A and tensor
        chain M_A̅= *   *   ⋯   *
        l̅_u+1   l̅_u+2   ⋯   l̅_k is connected to C_A̅. From
        (<ref>), ∑_i=1^u+1 l_i+∑_i=u+1^k+1l̅_i=∑_i=1^k n_i ≥a-2/2 k+a = (u+1)a-2/2+1 +
        (k-u+1)a-2/2+1.  So, ∑_i=1^u+1 l_i≥
        (u+1)a-2/2+1   or  ∑_i=u+1^k+1l̅_i≥ (k-u+1)a-2/2+1,  i.e.∑_i=1^u+1 (a-1-l_i)≤ (u+1)κ_c - 1
          or  ∑_i=u+1^k+1 (a-1-l̅_i) ≤
        (k-u+1)κ_c - 1.  From Theorem <ref>, either
        of M_A or M_A̅ is unprotected, so either of C_A or
        C_A̅ is unprotected. Thus the greedy algorithm will keep
        going on until Area(H)=0 at least, which means  two cuts
        C_A and C_A̅ are overlapped such that all tensors are
        absorbed. Then the ES is flat.
        
        
        
          §.§.§ κ_c < a/2⇒∃ non-flat ES
        
        
        Next we intend to prove when κ_c < a/2, there
        exists non-flat ES for some choices of single interval A on the
        boundary.
        
        Thanks to Theorem <ref>, when a is odd, we can
        specifically choose A on the boundary such that the structure as
        shown in Fig.<ref> is protected under the action of the
        greedy algorithm starting from either side. A tensor network
        with a special choice for A is shown in Fig.<ref>,
        where the structures enclosed by dashed red circles are
        protected and prevent the ES from being flat.
        
        When a is even, similarly one can choose A appropriately
        such that the structure as shown in Fig.<ref> is protected.
        Then the ES is non-flat.
        
        We remark that such kind of protected structures is common in
            tensor networks, especially when the network is large enough. So
        we intend to argue that when κ_c < a/2, most
            choices of interval A will lead to non-flat ES.
        
        
        
          §.§.§ κ_c≥b/b-2⇒∃ flat ES
        
        
        In previous subsection we have learned that when κ_c <
        a/2, ES being non-flat is a common phenomenon.
        Nevertheless, we point out that when κ_c≥b/b-2,
        it is possible to construct single interval whose ES is flat.
        
        Next we just prove the existence of flat ES by constructing a
        specific interval A with “minimal secant geodesic”, which is
        obtained by following steps (Fig.<ref>). We start from the
        midpoint of an edge between two uncontracted edges on the
        boundary, then connect this point with the midpoint of another
        edge in the polygon which has the farthest distance to this point.
        Next we choose the neighboring polygon of this new midpoint in the
        bulk and connect the midpoint with the other farthest midpoint in
        this polygon. Repeat above steps until it reaches the boundary of
        the network. The trajectory forms a geodesic called minimal secant
        geodesic, denoted by G_m. It should be noticed that for a
        polygon with odd edges, there are two middle points which are the
        farthest from the specified midpoint, one to the left and the
        other to the right, as shown in Fig.<ref>. We need choose
        these two midpoints by turn in above steps, as shown in
        Fig.<ref>.
        
        A minimal secant geodesic G_m divides the boundary of network into two
        parts A and A̅, which almost have the same size.
        
        We will show that for such a division, the corresponding ES is
        flat by proving that the greedy algorithm starting from either
        A or A̅ does not stop until the sequence of cuts reaches
        G_m. The proofs for A and A̅ are parallel. So we only
        prove the case for A.
        
        Similarly, thanks to Lemma <ref>, we focus on the
        case that the cut C is single connected and connected to
        ∂ A, which is denoted as C=[N_1,N_2,⋯,N_l],
        with distinct nodes.
        
        We give G_m a direction such that Φ is on its right
        hand side. Then G_m becomes a directed cut which is denoted as
        [N_1',N_2',⋯,N_m']. By definition, these nodes are
        distinct.
        
        When C and G_m are not overlapped, the edges connecting
        those nodes in C and G_m at least form a polygon. In general,
        they may enclose one or more polygons, as illustrated in
        Fig.<ref>.
        
        We pick out any one of them and label it as Y. Let the set
        of those nodes on the boundary of Y to be the union of
        [N_p+1,N_p+2,⋯,N_p+u] in C and
        [N_q+1',N_q+2',⋯,N_q+v'] in G_m. N_p+1 and
        N_q+1' are neighboring to each other. N_p+u and
        N_q+v' are neighboring to each other. We naturally have
        u≥2 after excluding the cases in Lemma <ref>.
        Let the reduced interior angle of Y at N_p+i as x_i for
        i∈1,2,⋯,u and the reduced interior angle of Y at
        N_q+j as x_j' for j∈1,2,⋯,v. Similar to the
        relation in (<ref>) in the proof of Lemma
        <ref>, for Y, we have
        ∑_i=1^u x_i + ∑_j=1^v x_j' ≤b/b-2(u+v-2).
         Suppose that the part [N_q+1',N_q+2',⋯,N_q+v']
        crosses w elementary polygons. Due to the special construction
        of G_m, we have the relation 
            v < w/2 (b-2)+2.
        
        So, we have
        ∑_j=1^v x_j'
            =v+w-1
            > b/b-2(v-2)+1.
        
        Plugging it into (<ref>), we obtain
        ∑_i=1^u x_i
            < b/b-2u-1
            ≤κ_c u - 1.
        
        On [N_p+1,N_p+2,⋯,N_p+u], tensor chain M=
        *   *   ⋯   *
        n_1   n_2   ⋯   n_u  is connected to C, where
        n_i=a-1-x_i for i∈1,2,⋯,u. From (<ref>) and
        Theorem <ref>, we know M is unprotected, thus C
        is unprotected.
        
        In conclusion, once C≠ G_m, C is unprotected and the greedy
        algorithm progresses. So those tensors between A and G_m
        will be absorbed. In parallel, those tensors between A̅
        and G_m will be absorbed under the greedy algorithm starting
        from A̅. Finally, the sequence of cuts reaches G_m,
        leading to a flat ES.
        
        
        
          §.§.§ κ_c<b/b-2⇒ non-flat ES
        
        
        Here we prove that when κ_c<b/b-2, the ES of a
        single and large interval A is non-flat. Perhaps this argument
        is the most important part in this section because it supplies us
        a quantitative criteria to justify if a tensor network has a
        not-flat ES.
        
        Consider a single interval A and its complement A̅ on the
        boundary of a given tensor network. There exists a continuous
        line, called G, connecting two ending points of A with a
        minimal cuttings on the edges of the network. The line G divides
        the whole network into two sub tensor networks (see Fig.<ref>).
        
        It is noticed that the nearest neighboring tensors of line G
        form two tensor chains. We call these two tensor chains as M_A
        and M_A̅, respectively. As an example, the skeletons of
        these two tensor chains are marked in Fig.<ref>. We set
        all the indexes associated with the edges cut by line G as upper
        indexes, while the other indexes are lower indexes.
        
        Assume that M_A has k_A nodes, and M_A̅ has k_A̅ nodes. Set the number of elementary polygons crossed by
        line G to be F. Then we have two equations
           κ(M_A)k_A + κ(M_A̅)k_A̅  =  b F,
           k_A + k_A̅  =  (b-2) F + 2.
        
        
        Now we provide a proof by contradiction. We assume that the ES
        would be flat, then M_A,M_A̅∈ S_D. According to Theorem
        <ref>, we have
        κ(M_A) ≤κ_c,  κ(M_A̅) ≤κ_c.
        
        We substitute
        (<ref>) into (<ref>) and get an inequality as
        κ _c≥b F/(b-2) F+2.
        
        
        To simulate real AdS spacetime, the number of layers in a network
        is expected to be large enough. Then for large interval A,
        F≫ 1. Since 2κ_c/b-(b-2)κ_c is a finite
        number,  F> 2κ_c/b-(b-2)κ_c. 
        From (<ref>) and (<ref>), we have
        κ_c≥b/b-2, contradictory to the initial
        assumption. Thus, when κ_c<b/b-2, the ES of a
        single interval A must be non-flat in a network with large
        layers.
        
        
        
        § CONCLUSIONS AND OUTLOOKS
        
        
        In this paper we have presented a general framework for tensor
        networks with tensor constraints based on the tiling of H^2
        space. A notion of critical protection based on the tensor chain
        has been proposed to describe the behavior of tensor networks
        under the action of greedy algorithm. In particular, a criteria
        has been developed with the help of the average reduced interior
        angle of CP chain such that for a given tensor network the ability
        of QEC and the flatness of ES can be justified in a quantitative
        manner. We have also demonstrated a lot of examples of tensor
        network and discussed their properties of QEC and ES. In general,
        once the ability of QEC of a tensor network becomes stronger, then
        its ES becomes flat more easily, and vice versa. By contrast, it
        is fascinating to notice that AdS spacetime is endowed with these
        two holographic features with perfect balance indeed. Currently it
        is still challenging to construct tensor networks which could
        capture all the holographic features of AdS spacetime. What we
        have found in this paper may shed light on this issue. Firstly, we
        have learned that the notion of critical protection provides a
        description on the limit of information transmission with full
        fidelity. In the case that the CP curve H_c is a circle, 
            i.e.λ_c L^2>1, the information in the interior of H_c
        can be transmitted to its surface without loss, where we have
        restored the AdS radius L. While, for a circle H which is
        larger than H_c, its interior information can not be transmitted
        to its surface without loss. So we can say that H_c is the
        maximal boundary which can holographically store the interior
        information <cit.>. Thus, for a
        tensor network which captures the feature of QEC as AdS space, it
        must not contain circular CP curves, which requires λ_c
        L^2≤1. Furthermore, if we intend to construct a single tensor
        network which exhibits both QEC and non-flat ES, it seems that the
        tensor networks with κ_c ∈ (κ_h, κ_1) might
        have more likelihood to approach this goal.
        
        Next we address some open issues that should be crucial for one to
        explore the role of tensor networks with constraints in
        holographic approach. Firstly, because of the chain structure of
        tensor constraint, in our present framework we have
            investigated QEC and ES only for a single interval on the
        boundary, which is just similar to the setup for hyperinvariant
        tensor network in <cit.>. It is an open question
        whether QEC can be realized for multi-intervals on the boundary,
        as investigated in network with perfect tensors or random tensors
        <cit.>. Actually,
        our preliminary investigation reveals that if the number of
        intervals is large enough, it would be very hard to realize QEC
        with non-flat ES for multi-intervals, because it involves in
        constructing tensor constraint with scales as large as the
        entanglement wedge of the multi-intervals, which is rather
        complicated. We would like to leave this issue for further
        investigation.
        
        Secondly, in order to simulate AdS space, it is desirable to send
        the number of layers of tensor network to infinity. Then the area
        of its boundary goes to infinity as well. Under this limit, the
        treatment on the boundary effect of tensor constraints is subtle.
        When the CP curve is a hypercircle with λ_c L^2<1, it has
        a constant distance to the geodesic which is d_c=L arctanh(λ_c L^2). The CP curve is unprotected once
        the boundary effect is considered, so the boundary effect scales
        as d_c, which is independent of the number of the layers. When
        d_c/L is small, the boundary effect becomes negligible in this
        limit comparing to the infinite area of the boundary. However,
        when d_c/L is very large, such as κ_c→κ_h+0, the
        boundary effect can not be neglected.
        
        Finally, we are concerned with the issue how to reproduce the
        Cardy-Calabrese formula of Renyi entropy (<ref>)
        in the framework of tensor networks. It is known that Renyi
        entropy depends not only on the tiling and tensor constraints, but
        also on the matrix elements of tensors, such as the elements of
        tensor U and Q in Appendix <ref>. In
        addition, we are interested in the possible relation between the
        CP curve and the gravity dual of Renyi entropy. In
        <cit.>, the nth-order holographic Renyi entropy can
        be calculated by the area of a cosmic brane_n with tension
        T_n, namely 
        n^2∂_nn-1/nS_n=Area(Cosmic
                Brane_n)/4G_N.
         The cosmic brane_n backreacts to the geometry at order
        T_nG_N where G_N is the Newton constant. However, if we simply
        set T_nG_N→0, all the cosmic branes become probe branes
        [ Notice that the tension is T_n=n-1/4nG_N and
            the product T_nG_N is fixed in AdS/CFT
            <cit.>. However, now the question we are asking is
            which tensor network can mimic AdS/CFT. So we loose T_nG_N. ]. Then, for a given subsystem on the
        boundary, those cosmic branes would have the same area and
        flat entanglement spectrum appears. According to Subsection
        <ref> in our paper, when d_c/L is large, the
        entanglement spectrum becomes non-flat, while when d_c/L is
        small, the entanglement spectrum becomes flat. It would be
        interesting to explore the possible relation between T_nG_N and
        d_c/L in the light of this observation.
        
        
        
        We are grateful to Long Cheng, Glen Evenbly, Wencong Gan, Muxin
        Han, Ling-Yan Hung, Hai Lin, Wei Li, Fuwen Shu, Yu Tian, Menghe
        Wu, Xiaoning Wu and Hongbao Zhang for helpful discussions and
        correspondence. This work is supported by the Natural Science
        Foundation of China under Grant No. 11575195. Y.L. also
        acknowledges the support from Jiangxi young scientists (JingGang
        Star) program and 555 talent project of Jiangxi Province. Z. Y.
        Xian is supported by National Postdoctoral Program for Innovative
        Talents BX20180318.
        
        
        
        
        § HYPERBOLIC GEOMETRY IN 2 DIMENSIONAL SPACE
        
        
         §.§ SL(2,R)
        
            In
            this section we present a brief review on the geometric property
            of H^2 space. Without loss of generality, we choose the radius
            of H^2 to be 1. Then the scalar curvature of H^2 geometry is
            -2. The metric in Poincare coordinate {x∈ R,0<z<∞} is
            
            ds^2=dx^2+dz^2/z^2.
            
            We define new coordinate ζ=x+i z to rewrite the
            metric as
            
            ds^2=-4 dζ dζ^*/(ζ-ζ^*)^2.
            
            The isometry of H^2 geometry is SL(2,R), which means
            the form of the metric is unchanged under the coordinate
            transformation
            ζ→αζ+β/γζ+δ,
            
            where real parameters α,β,γ,δ satisfy αδ-βγ=1.
        
            
        
         §.§ Curves of constant curvature
        
            One key notion that we have frequently used in this paper is the
            curve of constant curvature (CCC) in H^2 space.  The geodesic
            curvature of a curve with an affine parameter s is given by λ^μ=d^2 x^μ/ds^2+Γ^μ_νρdx^ν/dsdx^ρ/ds.
            
        
            The curves with λ^μ=0 are geodesics in H^2
            space. The geodesic distance of any two points with
            coordinates (x_1,z_1) and (x_2,z_2) can be derived as
            d=arccosh(x_1-x_2)^2+z_1^2+z_2^2/2z_1z_2.
        
            There are three kinds of CCC in H^2 space, namely, the
            circle, horocircle and hypercircle, as illustrated in
            Fig.<ref>.
        
            A circle is a curve whose geodesic distance to a given point
            (the center of the circle) is a constant r. The geodesic
            curvature of a circle with radius r is λ=(r).
        
            A horocircle (or horocycle) is a curve whose normal geodesics
            all converge asymptotically to its center in the same
            direction, so it is also called limit circle. The geodesic
            curvature of a horocircle is equal to 1.
        
            A hypercircle (or hypercycle) is a curve whose points have the
            same orthogonal distance d from a given geodesic, so it is also
            called equidistant curve. The corresponding geodesic is called
            its axis. The geodesic curvature of a hypercircle is
            λ=tanh(d).
             Of course, a geodesic is a hypercircle with d=0.
        
            As a summary, one can classify all the CCCs in H^2 space by their geodesic curvature
            [        λ>1,      circle;        λ=1,  horocircle;      0≤λ<1, hypercircle;             ]
        
        § SPECIFIC CONSTRUCTION OF TENSORS SUBJECT TO TENSOR CONSTRAINTS
        
        
        Analogous to the construction of tensors in <cit.>
        and <cit.>, we define tensors U, Q and R as
        the building blocks for tensors T and E, as shown in
        Fig.<ref>, <ref> and <ref>. The elements of tensor
        U are U_μν, which satisfy
            
                U_μν=U_νμ,   ∑_ν U_μνU_ρν^*∝δ_μρ.
            
            The elements of tensor Q are Q_μνρσ which satisfy
            
              Q_μνρσ=Q_ρσμν=Q_νμσρ,   ∑_ρσ Q_μνρσQ_μ'ν'ρσ^*∝δ_μμ'δ_νν'.
            
            where two indexes μν (ρσ) are grouped together.
            The elements of tensor R are R_μνρσ, which satisfy
            
                R_μνρσ=R_ρσμν=R_νμσρ,   ∑_ρσ R_μνρσR_μ'ν'ρσ^*∝δ_μμ'δ_νν',   ∑_νσ R_μνρσR_μ'νρ'σ^*∝δ_μμ'δ_ρρ'.
            
        
        
        For the tensor network with 7,3 tiling and S_c={
        1
        2 , 1   1
        1   1 }, and the tensor network with
        {4,5} tiling and S_c={ 1
        4 , 1   1   1
        3   2   3
        }, we construct tensors T and E in
        Fig.<ref> and <ref>, respectively.
        The specific structures of the top tensor chain in these two
        tensor networks are also shown in Fig.<ref>.
        
        
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