Long-distance entanglement in Motzkin and Fredkin spin chains

We derive some entanglement properties of the ground states of two classes of quantum spin chains described by the Fredkin model for half-integer spins and the Motzkin model for integer ones. Since the ground states of the two models are known analytically, we can calculate the entanglement entropy, the negativity and the quantum mutual information exactly. We show, in particular, that these systems exhibit long-distance entanglement, namely two disjoint regions of the chains remain entangled even when the separation is sent to infinity, i.e. these systems are not affected by decoherence. This strongly entangled behavior is consistent with the violation of the cluster decomposition property occurring in the case of colorful versions of the models (with spin larger than 1/2 or 1, respectively), but is also verified for colorless cases (spin 1/2 and 1). Moreover we show that this behavior involves disjoint segments located both at the edges and in the bulk of the chains.

cases, when also the area law for the entanglement entropy is violated more than logarithmically, and is more pronounced for correlation functions measured close to the edges of the chains. What is presented here is the calculation of other entanglement measures, the quantum negativity [18,19] and the mutual information [20,21] shared by two disjoint segments of the chains in the ground state. We show that such systems exhibit long-distance entanglement [22], namely given a measure of entanglement, e.g. the mutual information, I AB for the system A ∪ B made by two disjoint subsystems A and B, the quantity I AB does not vanish when the distance between A and B goes to infinity. For colorful cases this result is consistent with the fact that also the connected correlation functions do not vanish in the thermodynamic limit [9], being the the mutual information an upper bound for normalized connected correlators [21,23]. Moreover we show that this non-vanishing mutual information, persisting for infinite distances, is verified not only when the cluster decomposition is violated (for colorful cases, where the entanglement entropy scales as a square-root law) but also when the connected spin-correlators go to zero (for colorless cases, where the entanglement entropy scales logarithmically). Finally, contrary to what found in the continuum limit [16], this behavior occurs, and is even more pronounced, also when the subsystems are located deep inside the bulk, showing a stronger entanglement as compared to the one obtained in conformal field theories, where the mutual information vanishes upon increasing the distance with a power law behavior [24]. On the other hand, quite surprisingly, we show that the mutual information for two disjoint subsystems inside the bulk has the same form of the logarithmic negativity and of the mutual information for conformal field theories of two adjacent intervals [25].

Models
In this section we report the Hamiltonians for recently introduced half-integer and integer spin models, called Fredkin and Motzkin models whose ground state is known exactly and has the peculiarity of being related to some random lattice walks.

Fredkin model
The Fredkin model [9,10] composed by a bulk, H 0 , and a boundary, H ∂ , Hamiltonians, where P(|. ) = |. .| is a projector operator acting on quantum states made by local spin-states, ↑ c j located at site j with half-integer spin along z-quantization axis s z = c − 1 2 and ↓ c j with local half-integer spin s z = 1 2 − c , with c ∈ N and c = 1, . . . , q. The maximum value of the index c, namely q, is called the number of colors of the model. The quantum states appearing in Eq. (1) are defined as follows For colorless case, q = 1 (spin 1/2), we have that the third and the last term, so-called crossing term, are not present. In such a case the bulk term terms of Pauli matrices In terms of Fredkin gatesF ijk (controlled-swap operators), H 0 = L−2 j (2 −F j,j+1,j+2 − σ x j+2F j+2,j+1,j σ x j+2 ), whereF i,j,k acts on three 1 2 -spins (three qbits), swapping the j-th and k-th if the i-th is in the state |↑ while does nothing if it is in the state |↓ .

Motzkin model
The Motzkin model [6,7] is described by the following integer spin Hamiltonian H = H 0 + H ∂ with where now P(|. ) = |. .| acts on quantum states made by local integer spin-states, ⇑ c j located at site j with integer spin s z = c and ⇓ c j with integer spin s z = −c, with, again, c = 1, . . . , q. Also in this case, q is called the number of colors of the model and, in the Motzkin case, it correspond to the maximum value of the spins. . The quantum states appearing in Eq. (1) are defined as follows Also in this case, for colorless case, q = 1 (spin 1), we have that the third and the last term are absent.

Ground states
The most important property shared by these frustration-free Hamiltonians is that their ground states are unique, made by uniform superpositions of all states corresponding to Motzkin paths, for the integer case (for the Motzkin model) and all states corresponding to Dyck paths for the half-integer one (Fredkin model). This states are such that, denoting the spins up, ⇑, by /, the spins down, ⇓, by \ and spins zero, 0, by −, one can construct a Motzkin path, while by using only / for ↑ and \ for ↓ one can construct a Dick path. A Motzkin path is any path on a x-y plan connecting the origin (0, 0) to the point (0, L) with steps (1, 0), (1, 1), (1, −1), where L is an integer number. Any point (x, y) of the path is such that x and y are not negative. Analogously, a Dyck path is any path from the point (0, 0) to (0, L) (L now should be an even integer number) with steps (1, 1), (1, −1). As for the Motzkin path, any point (x, y) of the Dyck path is such that x and y are not negative. The corresponding colored path are such that the steps can be drawn with more than one color. The color attached to a path move is taken freely only for upward steps (up-spins) while any downward steps (down-spin) should have the same color of the nearest up-spin on the left-hand-side at the same level. This color matching is induced by the cost energy contribution described by the last term both in Eq. (1) and Eq. (7), which, in spite of being short-ranged, it produces non local effect in the ground state. As a result, a colorless Motzkin path |m can be defined as a string of L spins (or steps) such that, starting from the left by convention, the sum of the spins contained in any initial segment of the string is nonnegative, or alternatively, any initial segment contains at least as many up-spins (upward steps) as down-spins (downward steps), while the sum of all the L spins is zero (the total number of upward steps is equal to the number of downward steps). The colorful Motzkin or Dyck paths are the paths where, in addition, the upward steps can be colored at will while the colors of the downward steps are determined uniquely by the matching condition (any spin down has the same color of the adjacent upward spin on the left-hand-side at the same height). Examples of colored Motzkin and Dyck states are shown in Fig. 1. The ground state of the Fredkin Hamiltonian is then obtained by a uniform superposition of all possible Dyck paths for a given length L and a given number of colors q, where D (L) is the number of all possible colored Dick paths with q colors with C(n) = 2n! n!(n+1)! the Catalan numbers and p n = (1 − mod(n, 2)) selects even integers. Analogously, the ground state of the Motzkin Hamiltonian is a uniform superpositions of all possible Motzkin paths where, in the normalization factor, is the colored Motzkin number, i.e. the number of all the possible colored Motzkin paths. Because of this mapping between the ground states and the lattice paths several ground state properties can be studied exactly resorting to combinatorics.

Decomposition in two parts
The ground state for both the models can be written in terms of states defined on two subsystems, A and B, as follows where, h m = min( A , L − A ) and A h are some Schmidt coefficients depending of the number of paths, whose expressions will be given in the next section for the two cases, in Eq.

Decomposition in three parts
Let us now divide our spin chains in three parts, a left and a right part, A and B, and a central part C, see Fig 2. The ground state can decomposed in terms of states defined in these three regions as follows is the an orthonormal state composed uniformly by all the paths with (L − A − B ) steps starting at height h and ending at height h , with (h − z) unmatched down-spins and (h − z) unmatched up-spins, namely those paths which touch at most ones the horizontal line defined by z. In our notation the indices in Eq. (18) for unmatched spins are useful also for colorless case to classify the paths by the level z. Actually the minimum of the values of z which contribute to the sum appearing in Eq. (17) is . This horizontal quantity can be seen, therefore, as a quantum number classifying all the state in the central region C, since for any z the states |P are orthogonal to each other simply because composed by local spin states express in the canonical orthogonal basis. An example of this classification is shown in Fig. 3 for the Fredkin and the Motzkin case.
23 c 3 , the dashed red lines are all the paths which touch z = 1 that contribute to |P (7) 23 c 2 ,c 2 ,c 3 , the dotted green lines are all those which touch z = 0 that contribute to |P

Entanglement properties of the Fredkin chain
We will study the entanglement properties of the ground state for the Fredkin model, reviewing the entanglement entropy after a bipartition, and then calculating the negativity and the mutual information shared by the two spins at the edges resorting to the decomposition we obtained Eq. (17). We will show that these quantities, particularly the mutual information, revel an unconventional long-distance behavior. Before to proceed ne need to know the coefficient in Eq. (16) for the Fredkin ground state decomposed in two parts , which is and the coefficient in Eq. (17) after its decomposition into three parts, which reads where D (n) is the number of colored Dyck-like paths (q the number of colors) between two points at positive heights h and h with n steps. We assume D (n) hh to be zero for negative h or h by definition. In particular we have D (n) 00 ≡ D (n) = q n 2 C n 2 p n . Moreover we notice that D h0 .

Entanglement entropy
In this section we will briefly review the calculation for the von Neumann entanglement entropy. The reduced density matrix after a bipartition of the whole systems into two subsystems A and B, after tracing out one of them, is obtained from Eq. (16) where A h given by Eq. (19) therefore, since there are q h eigenvalues equal to A h , the entanglement entropy is simply Since is a normalized probability, h q h A h = 1, the first term of Eq. (23) is log q times the average height of the paths at a given position located at distance A from the edge which, for large L and A , when the binomial factors can be approximated by gaussian factors and the sum by an integral, scales as a square root, The second term, instead scales as 1 2 log A (L− A ) L , therefore, for large systems and for a sizable bipartition one gets Notice that this approximation is very good when the bipartition occurs in the bulk while Eq. (23) is exact for any A and L. For instance, if A = 1, the entanglement entropy, from Eq. (23), is exactly S A = log q, for any L, while Eq. (25) deviates from it.

Reduced density matrix for the edges
Let us consider the system A ∪ B made by the two spins located at the edges of our spin chains, as shown in Fig. 4. We will study the entanglement properties between these two spins at the edges for the Fredkin spin chain by tracing out all the spins between the first and the last one described by so that Eq. (17), dropping the site indices to simplify notation, reads The joint reduced density matrix of the subsystem A ∪ B, after tracing out all the degrees of freedom of the central part, and keeping only the two spins at the edges, is where the coefficients, from Eq. (20), are The normalization condition is fulfilled since the trace of ρ AB is .. t we can write the q 2 × q 2 reduced density matrix as follows where J is a matrix of all ones, 0 is a matrix of all zeros and 1 the identity matrix.

Negativity
We calculate now the quantum negativity which detects the entanglement between two disjoint regions and can be defined as follows where λ α are the eigenvalues of the partial transpose of the reduced density matrix with respect to a region, say B, namely obtained when the indices related to the degrees of freedom of one part, B, are transposed, which is Taking the same basis as for Eq. (33) the partial transpose of the reduced density matrix reads where the last block is a Kronecker product of an identity matrix and a 2 × 2 matrix which is verified for q ≥ 3, for any finite L. Therefore N = 0 for q ≤ 2 (and q = 3 in the limit which, in the large L limit goes to

Mutual Information
The eigenvalues of the reduced density matrix ρ AB , from Eq. (33), are (A 110 + qA 111 ) and A 110 , the latter with multiplicity (q 2 − 1), so that the entanglement entropy is with A 111 and A 110 given by Eqs. (30) and (31). On the other hand, from Eq. (19), since A 1 = q −1 which is the eigenvalue of ρ A (and ρ B ) with multiplicity q, we have We can, therefore calculate and study another entanglement measure which is the mutual information as a function of the size L, being L − 2 the distance between the two disjoint spins in A and B, and as a function the color number q.
Colorless case : For q = 1, we have S AB = 0 as well as S A = S B = 0, therefore I AB = 0 exactly, for any size of the chain L. This is due to the fact that the first and the last spins of the colorless Fredkin model are uncorrelated in the ground state. For that reason one has to increase the size of the subsystems A and B including further spins, as done in the next Sec. 4.5, where we will consider two spins at each edge, revealing in this way that there is a long-distance entanglement even for colorless case.
Colorful case : For colorful cases (q > 1) instead I AB turns to be finite also for large distances, namely for large L 1, as shown in Fig. (5). Actually we can calculate the limit of L → ∞, since I can be written in terms of D (L−2) D (L) only and where lim means the limit of a sequence, therefore A 111 → 1 4q and A 110 → 3 4q 2 , so that We show therefore that the two spins located at the edges of the chain, even when the distance is infinite, are strongly entangled for any q > 1. This behavior is consistent with the violation of the cluster decomposition occurring in such colorful cases.

Entanglement between the two couples of spins at the edges
As we know, for the colorless case, the first and the last spins are completely uncorrelated in the ground state, since for all the configurations of the spins in the bulk which contribute to the ground state, the first spin is always up and the last is always down. For colorful case instead they are correlated because of the color matching condition. For that reason we will consider more than one spin at the edges, studying the entanglement properties of two couples of spins at the borders, namely A and B made by two spins instead of one, as shown in Fig. 6. In this case, for any q the states at the edges are given by so that, dropping the site indices to simplify the notation, Eq. (17) reads where the coefficients are Let us now consider the colorless case (q = 1) for simplicity. The reduced density matrix of A ∪ B, after tracing out over the states of the central region, is . On the basis (|↑↑ |↑↓ , |↑↓ |↑↓ , |↑↑ |↓↓ , |↑↓ |↓↓ ) t the reduced density matrix can be written as . Its partial transpose matrix with respect to B is which, on the same basis of Eq. (58), reads , and since D (2n) 20 ≥ D (2n) , ∀n ≥ 1, the negativity is zero, N = 0. Tracing out the degrees of freedom of one of the two parts we get ρ A = Tr B ρ AB , or ρ B = Tr A ρ AB which can be written as One can notice that D which are the matrices one can get after a bipartition of the system, from Eqs. (19), (22), with Now one can easily calculate S AB , S A and S B , finding as a result, the large L limit of the mutual information which is which is I AB ≈ 0.01745 using the natural logarithm. We have shown, therefore, that, even for the colorless case, with the lowest value for the entanglement entropy, the mutual information, shared by two couples of spins at the edges, does not go to zero but remain finite also for infinite distance.

Entanglement properties of the Motzkin chain
As done for the Fredkin model, we will study the entanglement properties of the Motzkin model in the ground state, briefly reviewing the entanglement entropy after a bipartition, then calculating the negativity and the mutual information shared by the two spins at the edges. We will show that also in this case the latter quantity reveals long-distance entanglement. The coefficients in Eq. (16) for the ground state of the Motzkin chain, after a bipartition, are while the coefficients in Eq. (17) after decomposing the state into three parts are where is the number of colored Motzkin-like paths between two points at heights h and h . Moreover

Entanglement entropy
In this section we briefly review the calculation for the von Neumann entanglement entropy for the Motzkin chain. The reduced density matrix after a bipartition of the system into two subsystems A and B, after tracing out one of them, is obtained from Eq. (16) and has the same form reported in Eq. (22) where now A h is given by Eq. (66). Since ρ A have q h eigenvalues equal to A h , the entanglement entropy reads as follows Also in this case, since is a normalized probability, h q h A h = 1, the first term of Eq. (69) is log q times the average height of the paths at a given position located at distance A from the edge which, for large L and A , scales as a square root, The second term, instead, scales as 1 2 log A (L− A ) L , therefore, for large systems and for a sizable bipartition one gets As for the Fredkin case, this approximation is very good when the bipartition occurs in the bulk while Eq. (69) is exact for any A and L.

Reduced density matrix of the edges
Let us consider the system A ∪ B made by the two spins located at the edges of our spin chains, as shown in Fig. 4, and study the entanglement properties between these two spins at the edges of a Motzkin spin chain by tracing out all the spins between the first and the last one described by so that Eq. (17), dropping the site indices to simplify the notation, reads The joint reduced density matrix of A ∪ B, after tracing out all the degrees of freedom of the central part C (see Fig. 4) and keeping only the two spins at the edges, is where the Schmidt coefficients are given by One can verify that the trace of ρ AB is one, Writing ρ AB on the basis |0 |0 , |⇑ 1 |⇓ 1 , ... , |⇑ q |⇓ q , |0 |⇓ 1 , ... , |0 |⇓ q , |⇑ 1 |0 , ... ,

Negativity
We can calculate the negativity defined as in Eq. (34) where now λ α are the eigenvalues of the partial transpose of the reduced density matrix with respect to B Expressing this matrix on the same basis of Eq. (83) we can write (85) where the last block on the bottom-right corner is a Kronecker product of an identity matrix and a 2 × 2 matrix where, according to Eqs. (78-80), A 111 = A 000 and A 100 = A 010 . The eigenvalues of ρ t B AB are A 000 with multiplicity one, (A 000 + A 110 ) with multiplicity q, (A 100 + A 000 ) with multiplicity q, (A 100 − A 000 ) with multiplicity q, (A 110 + A 000 ) with multiplicity q(q − 1)/2 and (A 110 − A 000 ) with multiplicity q(q − 1)/2. Since A 100 ≥ A 000 , namely M n 10 ≥ M n , ∀n ≥ 1, the only possibility for having a negativity greater than zero is when A 000 > A 110 , which occurs for ) their ratio, in the limit L → ∞, goes to 4 from below for any q. As a result, for any finite L, we have and N = 0 otherwise (q = 1, 2), as in the case of the Fredkin model. For L 1, we have that M (L−2) 11 → 4M (L−2) for any q so that Eq. (88) becomes

Mutual Information
The eigenvalues of the reduced density matrix, Eq. (83), are together with A 000 with multiplicity (q − 1), A 100 with multiplicity 2q and A 110 with multiplicity (q 2 − q). The entanglement entropy S AB is, therefore, with the coefficients fulfilling A 0 + qA 1 = 1. The mutual information that we report here for convenience can be calculated for any value of L and q through Eqs.
whose eigenvalues are twice 2/9 and 5 ± √ 13 /18. Summing over the right or left spin degrees of freedom we get so that the mutual information is given by which is I AB ≈ 0.05361, in natural logarithm, as shown in Fig. 7.
Colorful case : Here we derive an explicit expression for the asymptotic value of I AB as a function of q. For a generic q we can simplify the expression for I AB , in the limit L → ∞, writing it in terms of only one colored Motzkin ratio since A 1 = (1 − A 0 )/q and, from Eqs. (78)-(81), where we recognize that M (L) in Eq. (15) can be seen as 2 F 1 ( −L 2 , 1−L 2 , 2, 4q), an hypergeometric serie reduced to a polynomial since at least one on the first two arguments is a nonpositive integer number. Substituting these values in Eq. (90) we get and, from Eqs. (91), (92) we get, for the mutual information, the following asymptotic exact expression The result is, therefore, that the mutual information shared by the two disjoint spins at the edges is always finite, even when the separation is sent to infinity, also for the colorless case which, from the point of view of the entanglement entropy resembles a critical system.

Entanglement properties in the bulk
Let us generalize what seen so far considering two disjoint subsystems also in the bulk. This requires to divide our system in five subsystems with different lengths such that L = A + B + C + D + E , as shown in Fig. 8, and the ground state has to be decomposed in five parts where the states located in each region is defined as before, see Eq. (18), and the coefficients depend on the model.

Fredkin model
For the half-integer spin model the coefficients appearing in Eq. (104) are the following Let us consider the case where both in A and B there is only a single spin ( A = B = 1).
In this case, using the definition given in Eq. (18) and remembering that non-zero contributions |P and that the difference of the heights is limited by , namely |h i − h j | ≤ , we have only the following possibilities for the states defined in A and B Eq. (109) is null for h 1 = 0 and Eq. (111) if h 4 = 0, however the coefficients in Eq. (104) take into account these possibilities when some heights in the argument of the Dyck numbers become negative. Calling h = h 1 , h = h 4 , z = z 2 and the residual spin indicesc =c (h+1) andc =c (h +1) to simplify the notation, the ground state can be written as and where the sums over h is limited by min( D , L − D ), the sums over h is limited by min( E , L − E ) and the sum over z is defined differently for the four terms according to the different heights of the borders of the central region as given by Eq. (107), where z i = z, = C , h i = h ± 1 and h j = h ± 1. More importantly, we notice that in Eq. (112), for any z, the central states in the first and in the last term are orthogonal to any other while the central states in the second and third term can overlap, more explicitly any (z + 2)-th state of the second term coincides with the z-th state of the third term. This overlaps cause the coherent terms in the reduced density matrix ρ AB . In order to calculate the mutual information between A and B we have to calculate also ρ A and ρ B . This can be done using the tripartition, Eq. (17) and Eq. (20) where, using the same notation of Eq. (20), Colorless case. Let us consider for simplicity the colorless case (q = 1). From Eq. (112) we can derive the reduced density matrix for the joint system A ∪ B after tracing out the rest of the chain +V X |↓ |↑ ↑| ↓| + |↑ |↓ ↓| ↑| which, on the basis (|↑ |↑ , |↑ |↓ , |↓ |↑ , |↓ |↓ ) t , can be written as where the coefficients, from Eqs. (113)-(116), are and the crossing term which actually causes coherence and long-distance entanglement between the spins One can verify through Eqs. (113)-(116) that Eq. (123) is actually the generalization of Eq. (57) and reduces to it for D = E = 1 since the first and he last spins are fixed to be up and down respectively. The eigenvalues of ρ AB are V ↑↑ , V ↓↓ , and Together with the following reduced density matrices for the regions A and B where the coefficients, from Eqs. (119)-(122), are (132) Explicit calculation shows that also for very large system size L the mutual information between two spins also in the bulk does not vanishes, as depicted in the left panel of Fig. 9. Actually, its asymptotic value increases when considering two spins more deeply in the bulk (see right panel of Fig. 9. Actually we expected and verified that increasing D and E , so going more deeply in the bulk, the probabilities of getting ↑ and ↑ (V ↑↑ ) or ↑ and ↓ (V ↑↓ ) or ↓ and ↑ (V ↓↑ ) or finally ↓ and ↓ (V ↓↓ ) should become the same and equal to 1/4, since also the probabilities of having one spin ↑ or ↓ both in A and B should be 1/2. This means that and consequently, for large system size L and for D , E 1, completely in the bulk, which is also the same value, I AB ≈ 0.35, obtained for C = 0 and D , E 1, as shown by the first points in the left panel of Fig. 9). Eq. (134) is actually the asymptotic value of the curve in the right panel of Fig. 9.

Motzkin model
For the integer spin model the coefficients appearing in Eq. (104) are the following As done for the Fredkin chain, let us consider the case where in A and B there are only a single spin ( A = B = 1). Proceeding analogously as done for the half-integer case, using Eq. (104) we get, for the ground state of the Motzkin model, the following decomposition On the other hand the reduced density matrix of a single spin in A and B can be determined by using the tripartition, Eq. (17), and Eq. (67) where, using the definition reported in Eq. (67), the coefficients are Colorless case. As done for Fredkin chain, let us consider for simplicity the colorless Motzkin model (q = 1). From Eq. (138) we can derive the reduced density matrix for the joint system A ∪ B after tracing out the rest of the chain where, denoting σ, σ = ⇑, 0, ⇓, the coefficients are defined by Choosing an opportune basis the reduced density matrix can be written as a block diagonal matrix as follows We verified that Tr(ρ AB ) = 1. Now, together with the reduced density matrices for the single spins on A and B where the coefficients, from Eqs. (149)-(154), are we can calculate the entanglement entropies and finally the mutual information, I AB = S A + S B − S AB , exactly. Examples for the exact mutual information shared by two spins in the bulk have been given in Fig. 10 where it is shown that it does not vanishes when the distance of the spins in the bulk goes to zero but saturates at some values which increases going more deeply into the bulk. Increasing the distances from the edges, D and E , we expected and verified that the and upon further increasing D and E , very deeply in the bulk, they become homogeneous, V σσ → 1 9 and A σ → 1 3 , B σ → 1 3 . The eigenvalues of ρ AB , in this limit, become 1/3, 2/9, 2/9, 1/9, 1/9, 0, 0, 0, 0, getting for the entanglement entropy S AB → 5 3 log 3 − 4 9 log 2. As a result the asymptotic upper bound limit for the mutual information between two spins in the bulk is

General results for any disjoint intervals
The physical explanation of what seen so far, at least for the colorless cases, is the following. Any state defined on a segment of the chain |P hh (z) can be defined by two quantum numbers: i) the magnetization m = (h − h) (the number of up-spins minus the number of down-spins), ii) the horizon z (the lowest level of the paths), so that we can write |P hh (z) = |m, z . Considering the decomposition as depicted in Fig. 8 we have that the ground state Eq. (104) can be written schematically as where the sum is such that m D + m A + m C + m B + m E = 0 and is restricted by the request that for any set of magnetizations one has to get a Fredkin or a Motzkin path (therefore, m D has to be non-negative, as well as any initial sums, for instance m D + m A + .., while m E has to be non-positive). The reduced density matrix of A ∪ C ∪ B, after tracing over D and E is where the central magnetizations are restricted by namely, the overlap is one if the total magnetizations in A and B are the same. This constraint implies that there are coherent terms in the reduced density matrix for A and B after integrating over C. As a result the reduced density matrix ρ AB can be written as a block diagonal matrix, where each block is defined by a magnetization sector, The blocks V S −i are square matrices defined by the maximum total magnetization where the sum runs over h with step 1 for the Motzkin and step 2 for the Fredkin, and Θ[n] = 1 for n ≥ 0 and Θ[n] = 0 otherwise. Defining where L hh z defined by Eq. (106), for the colorless Fredkin model and where L hh z defined by Eq.  with p n = (1 − mod(n, 2)) which selects even integers, as introduced before, while which are the numbers of all possible configurations of 1 2 -spins and 1-spins, respectively. Remarkably, we find that all off-diagonal terms in each block are written in terms of the diagonal probabilities, therefore these coherent terms persist for any set of distances. In particular, looking at Eqs. (180), (182), we notice that the blocks, identifying the magnetization sectors, deeply in the bulk, become singular matrices, since any two rows or columns are equal except for a factor. As a result only a single eigenvalue of a generic block V n is not zero, therefore, it is given by  We have shown that the entropy for A ∪ B behave as if the two regions compose a single unit subsystem of size A + B , no matter how far the two regions are. The effect of the off-diagonal coherent terms in the reduced density matrix is to glue together, through the central region, the two subsystems.
As a final result, for 1, approximating the binomial factors with a Gaussian distribution and sums with integrals, we get for the Fredkin and the Motzkin spin chains respectively. These approximations are in perfect agreement with the results reported in Fig. 11. Surprisingly, the latter results for the mutual information have the same form of the logarithmic negativity and of the mutual information for conformal field theories [25] of two adjacent intervals.

Conclusions
In this paper we have shown that the ground states of the novel quantum spin models under study exhibit a robust non-local behavior, the long-distance entanglement, in addition to violation of cluster decomposition property which occurs mainly on the boundaries of the chains in their colorful versions. On the contrary the strong entanglement shared by any segments of the spin chains survives at infinite distances either if the subsystems are located close to the edges or inside the bulk. This anomalous behavior has not been observed previously in the continuum version of the models [16] and, therefore, one should resort to an exact calculation in order to reveal it, taking afterwards the continuum limit. This peculiar non-local behavior takes origine from the presence of coherent terms in the reduced density matrix which do not vanish in the thermodynamic limit. Intriguingly, we show that the mutual information of two disjoint subsystems inside the bulk of these spin chains does not depend on their distance and has the same form of the logarithmic negativity and of the mutual information for conformal field theories of two adjacent subsystems in an infinite system. This finding strengthens the belief that these models, which in spite of being described by local short-range Hamiltonians show non-local behaviors, can be promising tools for quantum information technologies.