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.
The study of nonlocal properties and their consequences on the dynamics in addition to the violation of the area law for the entanglement entropy are certainly, at the present date, a very challenging field of research. The concept of locality plays a crucial role in physical theories, with far reaching consequences, a fundamental one being the cluster decomposition property hastings (); nachtergaele (). This property implies that two-point connected correlation functions go to zero when the separation of the points goes to infinity. This is the reason why two systems very far apart, separated by a large distance, behave independently.
Another aspect related to correlations is the quantum entanglement. In bipartite systems the von Neumann or entanglement entropy quantifies how the two parts of the whole system are entangled. This quantity measures non-local quantum correlations and has universal properties, like the fact that, for gapped systems, it scales with the area of the boundary of the two subsystems eisert (). This property is called area law and is valid for systems with short-range interactions. In other words, if the interactions are short-ranged the information among the constituents of the system propagates with a finite speed involving a surface surrounding the source of the signal, like an electromagnetic impulse propagating with the speed of light. For critical 1-dimentional short-range systems the area law is violated logarithmically pasquale (). Quantum spin chains are promising tools for universal quantum computation thompson () and the efficiency may be related to the amount of quantum entanglement. Spin systems with entanglement entropy larger than that dictated by the area law can be used for quantum computing even more efficiently, and breaking down the speed of the propagation of the excitations can represent a breakthrough for quantum information processing.
Recently, novel quantum spin models have been introduced, with integer bravyi12 (); ramis16 (); ramis16+ () (Motzkin model) and half-integer luca16 (); olof () spins (Fredkin model), which, in spite of being described by local Hamiltonians, exhibit violation of the cluster decomposition property and of the area law for the entanglement entropy, with the presence of anomalous and extremely fast propagation of the excitations after driving the system out-of-equilibrium luca16 (). These models seem, therefore, extremely promising for applications in quantum information and communication processes. Very recently, also deformed versions of Motzkin klich () and of Fredkin olof2 () chains have been introduced and studied ramis17 (); sugino (), which can exhibit a quantum phase transition separating a phase with an extensively entangled phase klich (); olof2 () from a topological one lucaB ().
Quite recently, it has been given also a continuum description for the ground-state wavefunctions of those models, as originally formulated and in the colorless cases, which can reproduce well some quantities like the local magnetization and the entanglement entropy, and whose scaling Hamiltonian is not conformally invariant fradkin (). Some results on the Renyi entropy for these models have been also reported sugino18 ().
In this work we focus on the study of quantum entanglement after the discovery that cluster decomposition property in such systems can be violated luca16 (). This behavior occurs for colorful 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 lewenstein (); vidal () and the mutual information groisman (); wolf () shared by two disjoint segments of the chains in the ground state. We show that such systems exhibit long-distance entanglement campos (), namely given a measure of entanglement, e.g. the mutual information, for the system made by two disjoint subsystems and , the quantity does not vanish when the distance between and 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 luca16 (), being the the mutual information an upper bound for normalized connected correlators wolf (); chen ().
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 fradkin (), 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 pasquale11 (). 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 for conformal field theories of two adjacent intervals pasquale12 ().
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.
ii.1 Fredkin model
composed by a bulk, , and a boundary, , Hamiltonians, where is a projector operator acting on quantum states made by local spin-states, located at site with half-integer spin along -quantization axis and with local half-integer spin , with and . The maximum value of the index , namely , is called the number of colors of the model. The quantum states appearing in Eq. (1) are defined as follows
For colorless case, (spin ), 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 gates (controlled-swap operators), , where acts on three -spins (three qbits), swapping the -th and -th if the -th is in the state while does nothing if it is in the state .
ii.2 Motzkin model
where now acts on quantum states made by local integer spin-states, located at site with integer spin and with integer spin , with, again, . Also in this case, 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, (spin ), we have that the third and the last term are absent.
Iii 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, , 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 - plan connecting the origin to the point with steps , , , where is an integer number. Any point of the path is such that and are not negative.
Analogously, a Dyck path is any path from the point to ( now should be an even integer number) with steps , . As for the Motzkin path, any point of the Dyck path is such that and 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 or Dyck path can be defined as a string of 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 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 and a given number of colors ,
where is the number of all possible colored Dick paths with colors
with the Catalan numbers and 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.
iii.1 Decomposition in two parts
The ground state for both the models can be written in terms of states defined on two subsystems, and , as follows
where, and 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. (19) for the Fredkin model and Eq. (66) for the Motzkin one.
is an orthonormal state defined on the subsystem made by a uniform superposition of lattice paths (of the Motzkin or Dyck type) which start from the origin and reaching the height after steps, with therefore unmatched up-spins with indices . Analogously, is an orthonormal state defined on the subsystem made by a uniform superposition of lattice paths (of the Motzkin or Dyck type) which start from the point and reaching the ending point after steps, with unmatched down-spins with indices .
iii.2 Decomposition in three parts
Let us now divide our spin chains in three parts, a left and a right part, and , and a central part , see Fig 2.
The ground state can decomposed in terms of states defined in these three regions as follows
with , and where an orthonormal state defined on the region as before, namely as a uniform superposition of lattice paths starting from the origin and ending at with unmatched colored up-spins and a uniform superposition of lattice paths defined on , starting from the point and ending at with unmatched colored down-spins. Moreover
is the an orthonormal state composed uniformly by all the paths with steps starting at height and ending at height ,
with unmatched down-spins and unmatched up-spins, namely those paths which touch at most ones the horizontal line defined by .
In our notation the indices in Eq. (18) for unmatched spins are useful also for colorless case to classify the paths by the level .
Actually the minimum of the values of 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 , since for any the states 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.
Iv 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
is the number of colored Dyck-like paths ( the number of colors) between two points at positive heights and with steps. We assume to be zero for negative or by definition. In particular we have . Moreover we notice that .
iv.1 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 and , after tracing out one of them, is obtained from Eq. (16)
where given by Eq. (19) therefore, since there are eigenvalues equal to , the entanglement entropy is simply
Since is a normalized probability, , the first term of Eq. (23) is times the average height of the paths at a given position located at distance from the edge
which, for large and , when the binomial factors can be approximated by gaussian factors and the sum by an integral, scales as square a root, . The second term, instead scales as , 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 and . For instance, if , the entanglement entropy, from Eq. (23), is exactly , for any , while Eq. (71) deviates from it.
iv.2 Reduced density matrix for the edges
Let us consider the system 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 , 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 is
On the basis we can write the reduced density matrix as follows
where is a matrix of all ones, is a matrix of all zeros and the identity matrix.
We calculate now the quantum negativity which detects the entanglement between two disjoint regions and can be defined as follows
where are the eigenvalues of the partial transpose of the reduced density matrix with respect to a region, say , namely obtained when the indices related to the degrees of freedom of one part, , 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 matrix
The eigenvalues of are with multiplicity and with multiplicity , therefore the negativity is greater than zero if , namely, if
which is verified for , for any finite . Therefore for (and in the limit ) while
which, in the large limit goes to
iv.4 Mutual Information
The eigenvalues of the reduced density matrix , from Eq. (33), are and , the latter with multiplicity , so that the entanglement entropy is
We can, therefore calculate and study another entanglement measure which is the mutual information
as a function of the size , being the distance between the two disjoint spins in and , and as a function the color number .
: For , we have as well as , therefore exactly, for any size of the chain . 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 and including further spins, as done in the next Sec. IV.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.
: For colorful cases () instead turns to be finite also for large distances, namely for large , as shown in Fig. (5).
Actually we can calculate the limit of , since can be written in terms of only and
where means the limit of a sequence, therefore and , 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 . This behavior is consistent with the violation of the cluster decomposition occurring in such colorful cases.
iv.5 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 and made by two spins instead of one, as shown in Fig. 6.
In this case, for any 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 () for simplicity. The reduced density matrix of , after tracing out over the states of the central region, is
where and . On the basis , the reduced density matrix can be written as
with . Its partial transpose matrix with respect to is
which, on the same basis of Eq. (58), reads
whose eigenvalues are , and , and since , , the negativity is zero, .
Tracing out the degrees of freedom of one of the two parts we get , or