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We identify patterns of ground state entanglement, or quantum discord, in qubit clusters with three and four qubits, that are induced by varying the couplings between next-nearest neighbors in the clusters. We show that these entanglement patterns can be associated with continuous multiply connected regions in parameter space, on which entanglement quantifiers, such as the pairwise concurrence, exhibit a particular type of behavior as a function of the couplings between next-nearest neighbors in the cluster. We present the distinct patterns in diagrams in parameter space with continuous boundary lines and we associate each pattern to a specific type of pure quantum correlation. We propose this procedure as a simple method to identify useful classes of pure quantum correlations in qubit networks.

Since the beginning of quantum mechanics, entanglement has been studied as a fundamental concept setting the main difference between classical and quantum mechanics [1,2]. Later, its study was intensified with the discovery that it is a valuable resource in quantum information protocols [

For the purposes of implementing quantum computation, the physical system can be treated as a qubit network in which the couplings between the qubits can be controlled externally. The concept of qubit networks, on the other hand, can be related to that of quantum graphs, in which a quantum particle is bound to move through connectors and nodes in a network. This relation was analyzed in detail in [

The distinction between classical and pure quantum correlations in general quantum states is a central question in quantum information theory. The subject has attracted much attention since the discovery that bipartite mixed separable states can have non-classical correlations that can be captured by a measure, denoted quantum discord [

In a recent paper [

In this paper, we identify ground-state entanglement, or quantum discord, patterns in qubit clusters by analyzing the behavior of pairwise concurrences as a function of a control parameter that changes the coupling between certain pairs of qubits in the cluster, which ultimately leads to a change in the cluster topology. In Section 2, we classify entanglement patterns generated by gradually introducing an XY coupling between extremal qubits in an XY trimer, leading to a triangle configuration. This additional interaction monotonically increases the pairwise concurrence between the extremal qubits, but it also leads to a non-trivial behavior for the nearest-neighbor concurrence,. Our results show the existence of multiply connected regions in parameter space, whose boundary lines separate different types of entanglement patterns, which are characterized by different classes of behavior of the pairwise concurrences as a function of the variable coupling. We observe in particular an entanglement pattern, associated with a non-monotonic behavior of the nearest-neighbor concurrence, C(1), with a maximum at an intermediate value of the coupling between the extremal qubits. In addition, we study for each entanglement pattern, the behavior of global entanglement, Q and its part stored in pairwise entanglement, Q_{p}, as a function of the same control parameter. We find that although Q always increases with the additional interaction, it also reduces Q_{p} for weak magnetic fields, thus showing that the new coupling not only induces more global entanglement, but also modifies its nature. In the Section 3, we consider clusters with four qubits. The extremal limits, which are interpolated by additional coupling constants, are the nearest-neighbor XY square and the LMG tetrahedron, where all qubits are homogeneously coupled. In this case, two pairs of next-nearest neighbor interactions are introduced via the transition from XY to LMG model, which leads to a very rich diagram of entanglement patterns. We find that the nextnearest neighbor concurrence, , does not necessarily increase with the transition from XY to LMG model. Consequently, the behavior of also defines a set of regions in parameter space that can be associated with different entanglement patterns. For both types of pairwise concurrence, and, we have regimes of monotonic increase, monotonic decrease and nonmono-tonic behavior, with a maximum at an intermediate value of the additional coupling parameter, m. A remarkable result in this case is the existence of a region between two limiting curves where for all values of m. This means that in spite of the fact that the coupling constants between pairs of qubits are not all equal, the pairwise entanglement is equally distributed among all pairs of qubits. A summary and conclusions are presented in Section 5.

We analyze the effect of introducing a new coupling in a three qubit cluster with XY nearest-neighbor interactions, performing a change of topology: from the trimer topology, where the two extremal qubits are not coupled, to triangle topology, where all pairs of qubits interact equally. The

The goal this work is analyzing the modification on the amount and nature of the ground-state entanglement which results of the changing of topology. In order to perform this analyzes, we use an interpolating model where the coupling between extremal qubits is adjustable with a simple control parameter. The XY Hamiltonians which interpolate the topologies is:

where is the anisotropy parameter, is the magnetic field. The operators are the Pauli’s operators corresponding to the direction. The parameter controls the next-nearest neighbor coupling.

In matrix representation, we have a block diagonal matrix corresponding respectively to the canonical basis and. The matrix elements of the two blocks are transformed onto one another by the change. Thus, by determining the eigenvalues and eigenvectors of one of these matrices, we can determine the ground-state and thus the entanglement. We restrict our analysis to the anti-ferromagnetic case.

Beyond the degeneracy at, there is a special field value, which depends on the anisotropy parameter, where the ground state is degenerate. In

two configurations, trimer and triangle. For the triangle the special field value can be written as. These two degeneracy curves separate three regions in parameters space, which will be important in the analysis of the behavior of the pairwise concurrences as a function of the coupling between the extremal spins.

In order to calculate the pairwise concurrence, we need to obtain the reduced density matrix of two selected qubits. In the triangle configuration, due to translation invariance, the reduced density matrices and, consequently, the pairwise concurrences, of every pair of qubits are the same. However, in the trimer configuration, we have two different types of two-qubit reduced density matrix, which can be written in the following form:

where and are functions of and.

The concurrence can be calculated using Wooters’ Formula [

With this model we can analyze the behavior of the pairwise concurrences and as a function of the coupling parameter. In

ferent patterns of entanglement, which are characterized by the shape of as a function of. Varying in the interval, monotonically decreases in the region denoted (I) in

In this subsection we analyze the possibility to transform entanglement stored in pairwise concurrences in global entanglement which has not this nature. As a quantifier of global entanglement, we use the Q-measure [

to the average purity of the reduced density matrices of each qubit

where is the reduced density matrix of the i-th qubit and N is the number of qubits. In the three qubit case, the reduced matrices of extremal qubits are identical. Thus, the Q-measure reduces to

As we have seen, the pairwise concurrences can decrease or increase with the parameter. However, the the global entanglement should be an increasing monotonic function of, which is indeed what we observe in

The part of the global entanglement stored in pairwise concurrences can be calculated using the concept of distributed entanglement or monogamy of entanglement [35,36]. In this concept, the entanglement between a qubit and the others in the cluster is used as a bound for the amount of entanglement stored in pairwise concurrences. It is defined by the sum of squares of all pairwise concurrences involving that qubit. Thus, the part of global entanglement stored in pairwise concurrences can be estimated by the average of these sums over all qubits in the cluster, which we denote by. For a three-qubit cluster, we obtain. In

but also converts entanglement stored in pairs into genuine multipartite entanglement.

In this section we identify entanglement patterns in clusters with four qubits induced by the introduction of nextnearest neighbor interactions. The model systems are the Heisenberg $XY$ model with nearest neighbor interactions and the $LMG$ model with identical interactions between all pairs of qubits. The model Hamiltonian contains an interpolating parameter, used to control the next-nearest neighbor interaction, and is given by

In

fit, which is given by. For, the two degeneracy fields have also been obtained numerically, but we found that they adjust well to sectors of the ellipses and. The fact that there are two degeneracy curves as a function of the anisotropy parameter implies a much richer structure of entanglement patterns in comparison with the three qubits case.

Since we are not interested in the behavior of the pairwise concurrencies as functions of the magnetic field, we present in this section only results obtained by varying the interpolation parameter. Although it is hard to obtain analytically the exact ground-state of the interpolating model, it is possible to calculate it explicitly in the intermediate region. It has the form

and

The states and are known as W-states and have maximum multipartite entanglement and the pairwise concurrences in any of these states is invariant under pair exchange. The ground state in this region is a coherent superposition of these states and thus it preserves this invariance, which in turn explains the observed property for all values of.

In order to analyze in more detail the effect of the additional couplings in the four-qubit cluster, we study the behavior of the concurrence as a function of the interacttion parameter in the interval, for several values of the magnetic field and anisotropy parameter. For each value of we have two degeneracy fields, each bounded by a solid and a dotted line, shown in

We observe discontinuities in for fields in the shaded regions shown in

A novel feature of the four qubits case is the fact that both and present qualitative changes as we move in parameter space, thus implying a very reach structure of entanglement patterns. In

In the region denominated (I), monotonically decreases with. In region (II) we find that monotonically increases, while in region (III), is non-monotonic and exhibits a local maximum. In the

hibits a local maximum. Finally, in region (IV), we find a behavior that does not appear in: the emergence of a finite interval of values of, on which.

Global EntanglementAs in the three qubits case, we shall complete the analysis by describing the behavior of the global entanglement and the pairwise-stored entanglement as functions of the interpolation parameter in all regions associated with different entanglement patterns. In

with entanglement patterns characterized by the behaviors of and.

We find basically three types of behaviors: A monotonic increase with, which occurs in regions (I), a monotonic decrease in regions (II), and a non monotonic behavior with a local maximum in regions (III).

A notable behavior is observed in the regions (I), as the additional couplings always increases the global entanglement and the pairwise-stored entanglement decrease in this region, we can say that the additional couplins converts converts pairwise-stored entanglement in genuine multipartite entanglement. Remarkably, in region (III) this conversion occurs only for values of greater than the one corresponding to the maximum of. This behavior illustrates the changing on the nature of entanglement, which occurs also in three spins chains.

In this paper we showed how to identify entanglement patterns in qubit clusters that are induced by varying the couplings between next-nearest neighbors in the cluster. The procedure consists in separating points in parameter space according to the distinct types of behavior of the nearest neighbor and next-nearest neighbor concurrences as a function of the variable coupling parameter. We applied this procedure to clusters with three and four qubits. We found that points in parameter space associated with qualitatively similar behavior of the pairwise concurrences form a continuous multiply connected region, thus allowing its association to a pattern of entanglement. The results were presented in diagrams with continuous boundary lines separating the distinct entanglement patterns.

There are several ways in which the information contained in the diagrams of entanglement patterns could be used in practical applications. As an example, consider a cluster of three qubits in the trimer configuration. Suppose that the task is to produce a certain amount of entanglement between the extremal qubits, measured by, by switching on an interaction between them, but without reducing the amount of pairwise entanglement with the other qubit, measured by. From the diagram and graphs in

Clearly, as the cluster increases in size the diversity of entanglement patterns should also increase, thus making the full classification a very hard job. However, if a certain practical application requires that one concentrates on a particular type of pattern, finding its region in parameter space is a much simpler task. Studying concrete examples of practical applications of entanglement patterns in quantum networks is an interesting perspective for further research.

This work was supported by CNPq and Capes (Brazilian Agency), and by UEFS and UFPE (Brazilian Universities).